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American option under proportional transaction costs_pricing hedging and stopping algorithms for...
Acta Appl Math(2009)106:199–228DOI10.1007/s10440-008-9290-7American Options under Proportional TransactionCosts:Pricing,Hedging and Stopping Algorithmsfor Long and Short PositionsAlet Roux·Tomasz ZastawniakReceived:30July2008/Accepted:30July2008/Published online:21August2008©Springer Science+Business Media B.V.2008Abstract American options are studied in a general discrete market in the presence of pro-portional transaction costs,modelled as bid-ask spreads.Pricing algorithms and construc-tions of hedging strategies,stopping times and martingale representations are presented for short(seller’s)and long(buyer’s)positions in an American option with an arbitrary payoff. This general approach extends the special cases considered in the literature concerned pri-marily with computing the prices of American puts under transaction costs by relaxing any restrictions on the form of the payoff,the magnitude of the transaction costs or the discrete market model itself.The largely unexplored case of pricing,hedging and stopping for the American option buyer under transaction costs is also covered.The pricing algorithms are computationally efficient,growing only polynomially with the number of time steps in a re-combinant tree model.The stopping times realising the ask(seller’s)and bid(buyer’s)option prices can differ from one another.The former is generally a so-called mixed(randomised) stopping time,whereas the latter is always a pure(ordinary)stopping time.Keywords American options·Transaction costs·Randomised stopping·Superhedging1IntroductionIn this paper we study the seller’s and buyer’s positions in American options when trading in the underlying asset is subject to proportional transaction costs.The results apply to options with arbitrary payoffs in any discrete market model and proportional transaction costs of any magnitude.We are concerned with computing the seller’s price of an American option, also known as the upper hedging price or the ask price,as well as the buyer’s price,often referred to as the lower hedging price or the bid price.Apart from pricing,we construct op-timal strategies superhedging the positions of the option seller and buyer,together with the A.Roux( )·T.ZastawniakDepartment of Mathematics,University of York Heslington,York YO105DD,UKe-mail:ar521@T.Zastawniake-mail:tz506@200 A.Roux,T.Zastawniak respective stopping times realising the option prices,generally a mixed(randomised)stop-ping time for the seller and a pure(ordinary)stopping time for the buyer.We also consider martingale representations for the ask and bid option prices.Thefirst to examine American options under proportional transaction costs in a similar setting and level of generality as in the present paper were Chalasani and Jha[3].They estab-lished martingale representations for options with cash settlement,subject to the simplifying assumption that transaction costs apply at any time,except at any particular stopping time chosen by the buyer to exercise the option.An important feature that emerged in Chalasani and Jha’s representation for the option seller’s price was the role played by mixed stopping times in place of pure stopping times.Chalasani and Jha pointed out the non-trivial nature of computing the option prices in their representations and the need to develop algorithms to evaluate these prices.Our pricing algorithms solve this problem.Moreover,we put for-ward algorithms for constructing the corresponding hedging strategies,stopping times,and approximate martingales.Bouchard and Temam[2]established a dual representation for the set of initial endow-ments allowing to superhedge the seller’s position in an American option in a discrete time market model with proportional transaction costs in the setting of Kabanov,Rásonyi and Stricker[13],and Schachermayer[25].In particular,they reproduced Chalasani and Jha’s[3] martingale representation of the seller’s price.However,note that Bouchard and Temam[2] follow a different convention than Chalasani and Jha[3]in that they rebalance the portfolios in a hedging strategy before rather than after it becomes known whether or not the American option is to be exercised.Papers concerned with various special cases involving the hedging prices of American options under proportional transaction costs include Koci´n ski[14,15],who studied suffi-cient conditions for the existence of perfectly replicating strategies for American options, Perrakis and Lefoll[19,20],who investigated American calls and puts in the binomial model,and Tokarz and Zastawniak[27],who worked with general American payoffs in the binomial model under small proportional transaction costs.Another group of papers,using preference-based or risk minimisation approaches rather than superhedging for American options under proportional transaction costs,includes Davis and Zariphopoulou[8],Mercurio and V orst[17],Constantinides and Zariphopoulou [7],and Constantinides and Perrakis[6].The work by Levental and Skorohod[16],and Jakubenas,Levental and Ryznar[9]shows that superhedging in continuous time leads to unrealistic results for American options under proportional transaction costs,thus providing motivation for exploring discrete time approaches.The present paper complements and extends the results obtained by Chalasani and Jha [3]and Bouchard and Temam[2]by providing pricing,hedging,stopping and approximate martingale algorithms for arbitrary American options under proportional transaction costs in a general discrete setting.It also extends the work on hedging prices by several of the authors listed above,removing any restrictions imposed in the various special cases that have been considered in the literature.As a by-product,we establish the same martingale representations for American option prices under transaction costs as in[3]or[2]by a very different method based on an explicit construction of the stopping times and approximate martingales representing the ask and bid option prices.The construction provides a geomet-ric insight into the origin of mixed stopping times in the seller’s case.Some of the results presented here havefirst been established in[22];see also[23]for similar algorithms for European options.In the well-known case without transaction costs a stopping time that is best for the op-tion holder(the buyer)also happens to be the worst one for the option writer(the seller).American Options under Proportional Transaction Costs:Pricing201 Similarly,a strategy hedging a shorted option is essentially the opposite to a strategy hedg-ing a long position in the option.This kind of symmetry between the option seller and buyer breaks down in the presence of transaction costs.Hedging against a stopping time that is optimal for the buyer will generally no longer protect the seller against all other possible ex-ercise times.To hedge against all pure stopping times,the seller must in effect be protected against a certain mixed stopping time.Moreover,under transaction costs a simple relation-ship generally no longer exists between strategies hedging long and short positions in the option.These points are illustrated by the‘clinical’example in Sect.4.In the presence of transaction costs hedging against all stopping times can cost more than against the buyer’s optimal stopping time.If the seller knew with certainty that the option will be exercised at the buyer’s optimal stopping time,then it would only be necessary to hedge against this single stopping time,making the seller’s hedging strategy less expensive. However the option would then no longer be of American type.This situation is reminiscent of a Nash equilibrium.A deeper mathematical reason behind the apparent lack of symmetry between buyer and seller under transaction costs is that pricing for the seller as defined by(3.2)is a convex optimisation problem,whereas the buyer’s pricing problem(3.4)is not of this kind,in gen-eral.This is reflected in the pricing,hedging and stopping algorithms for the option seller presented in this paper,which operate within the space of convex functions(and thus have convex dual counterparts involving concave functions),whereas the corresponding buyer’s algorithms no longer act on convex functions alone.Computing the seller’s and buyer’s prices of an American option directly from the defini-tions(3.2)and(3.4)amounts to solving large optimisation problems over the corresponding set of superhedging strategies.Both these optimisation problems grow exponentially with the number of time steps,as observed(for European options)by Rutkowski[24]and Chen, Sheu and Palmer[4].In Algorithm3.1(and the equivalent convex dual Algorithm3.2)for the seller’s price and in Algorithm3.5for the buyer’s price we present computationally effi-cient dynamic programming type iterative procedures,which grow only polynomially with the number of time steps in a recombinant tree model.It is shown in Remark3.2that Algo-rithm3.2can be regarded as an extension of the familiar Snell envelope construction to the case with transaction costs.Numerical examples are provided to demonstrate theflexibility and efficiency of the pric-ing algorithms in a realistic market model approximation.The algorithms presented in this paper apply to options with arbitrary payoffs in general discrete market models,including incomplete ones,with arbitrary proportional transaction costs.The efficiency of the pricing algorithms(due to their polynomial growth)makes it possible to cover a considerably larger range of time steps and parameter values than in the latest numerical work by Perrakis and Lefoll[20],and to extend the numerical computations beyond the binomial tree model as well as beyond puts or calls to include long and short positions in option baskets(which are, of course,not equivalent to a combination of puts and calls in the presence of transaction costs).The contents of this paper are organised as follows.In Sect.2wefix the notation,spec-ify the market model with transaction costs,and present the necessary information on mixed stopping times,approximate martingales and the families of functions to be used throughout the paper.Section3is the main part of the paper.Following some definitions,pricing,hedg-ing,stopping and approximate martingale algorithms are presented here for both the seller and the buyer of an American option in the presence of proportional transaction costs,along with theorems proving the correctness of these algorithms.A simple illustrative example, which can be followed by hand,showing the algorithms in action can be found in Sect.4.202 A.Roux,T.Zastawniak In Sect.5we produce a number of more realistic numerical examples.Finally,Appendix contains some technical results.2Preliminaries2.1Market ModelConsider afinite probability space with thefield F=2 of all subsets of ,a probability measure Q on F such that Q{ω}>0for eachω∈ ,and afiltration{∅, }=F0⊂F1⊂···⊂F T=F,the time horizon T being a positive integer.For each t=0,1,...,T wedenote by t the set of atoms of F t,and identify any F t-measurable random variable X with a function defined on t.We shall write Xμto indicate the value of X atμ∈ t.Any probability measure P on F can be identified with the family of probability measures P t on F t such that P t(μ)=P(μ)for eachμ∈ t and t=0,1,...,T.Thefiltration can be represented as a tree,the atoms of F t corresponding to the nodes of the tree at time t.We shall say thatν∈ t+1is a successor node ofμ∈ t ifν⊂μ,this relationship corresponding to the branches of the tree.The set of successor nodes ofμ∈ t will be denoted bysuccμ={ν∈ t+1|ν⊂μ}.The market model consists of a risk-free bond and a risky stock.There are proportional transaction costs on stock trades expressed as bid-ask spreads,as in Jouini and Kallal[10].Shares can be bought at the ask price S at or sold at the bid price S bt,where S at≥S b t>0foreach t=0,1,...,T,the processes S a and S b being adapted to thefiltration.Without loss of generality,we can assume that all prices are discounted,the bond price being1for each t=0,1,...,T,so that a position in bonds can be identified with cash holdings.A portfolio(γ,δ)of cash(or bonds)and stock can be liquidated at time t by selling stockfor S bt per share to close a long positionδ≥0or buying stock for S atper share to close ashort positionδ<0.The liquidation value of the portfolio will beϑt(γ,δ)=γ+S btδ+−S a tδ−.The cost of setting up a portfolio(γ,δ)is−ϑt(−γ,−δ)=γ−S b tδ−+S a tδ+.A self-financing strategy is a predictable process(αt,βt)representing positions in cash (or bonds)and stock at t=0,...,T such thatϑt(αt−αt+1,βt−βt+1)≥0(2.1) for each t=0,...,T−1.The set of all self-financing strategies will be denoted by .An arbitrage opportunity is a self-financing strategy(α,β)∈ such that−ϑ0(−α0,−β0)≤0,ϑT(αT,βT)≥0,Q{ϑT(αT,βT)>0}>0.It was established by Jouini and Kallal[10]that the lack of arbitrage in the model with proportional transaction costs is equivalent to the existence of a probability measure P onequivalent to Q and a martingale S under P such that S bt ≤S t≤S a t for each t=0,1,...,T.This result also follows from Kabanov and Stricker[11],Ortu[18],Kabanov,Rásonyi and Stricker[12,13],Tokarz[26],and Schachermayer[25].American Options under Proportional Transaction Costs:Pricing2032.2Mixed Stopping TimesA stopping timeτis a random variable such that{τ=t}∈F t for each t=0,1,...,T.The set of stopping timesτwith values in{0,1,...,T}will be denoted by T.To distinguish them from mixed stopping times,defined below,we shall sometimes refer to suchτ’s as pure stopping times.A mixed stopping time(also called a randomised stopping time as in,for example,Chow, Robins and Siegmund[5],Baxter and Chacon[1],or Chalasani and Jha[3])is defined as a non-negative adapted processχsuch thatTt=0χt=1.The set of all mixed stopping times will be denoted by X.We have T⊂X in the sense that each pure stopping timeτcan be identified with a mixed stopping timeχτsuch that for any t=0,1,...,Tχτt=1{τ=t}.For any adapted process Z and any mixed stopping timeχthe time-χvalue of Z is defined asZχ=Tt=0χt Z t.Ifτis a pure stopping time,then Zχτis the familiar random variableZχτ=Tt=01{τ=t}Z t=Zτ.For any mixed stopping timeχ∈X and any adapted process Z we define processesχ∗and Zχ∗such that for each t=0,1,...,Tχ∗t =Ts=tχs,Zχ∗t=Ts=tχs Z s.(2.2)In addition,it will prove convenient to putχ∗T+1=0,Zχ∗T+1=0.(2.3) 2.3Approximate MartingalesAs observed in Sect.2.1,a market model with proportional transaction costs does not admit arbitrage if and only if there exists a pair(P,S)consisting of a probability measure P on equivalent to Q and a martingale S under P such that for each t=0,1,...,TS bt≤S t≤S a t.The family of such pairs(P,S)will be denoted by P.If the condition that P should be equivalent to Q is relaxed,then the corresponding family of pairs(P,S)is to be denoted204 A.Roux,T.Zastawniak by¯P.The families P and¯P can be used to represent the prices of European options under proportional transaction costs,see Jouini and Kallal[10].To represent the prices of Ameri-can options we need certain larger families than P or¯P.For any mixed stopping timeχ∈X we denote by P(χ)the family of pairs(P,S)con-sisting of a probability measure P on equivalent to Q and an adapted process S such that for each t=0,1,...,TS bt≤S t≤S a t,(2.4)χ∗t+1S bt≤E P(Sχ∗t+1|F t)≤χ∗t+1S a t,(2.5)where E P is the expectation under P.If the assumption that P should be equivalent to Q is relaxed,the corresponding family of pairs(P,S)will be denoted by¯P(χ).A pair(P,S)of this kind will be called an approximate martingale.For a pure stopping timeτ∈T we shall write P(τ)and¯P(τ)instead of P(χτ)and¯P(χτ).This notation and terminology resembles that in Chalasani and Jha[3].Form Proposition A.1we know that P⊂P(χ)and¯P⊂¯P(χ).It follows that the families P(χ)and¯P(χ)are non-empty for anyχ∈X in an arbitrage-free market model,since P is non-empty and P⊂¯P.2.4Families of Polyhedral FunctionsWe denote by the family of functions f:R→R∪{−∞}such that f≡−∞or f is an R-valued polyhedral function(i.e.continuous piecewise linear function with afinite number of pieces).For any f,g in the maximum and minimum of f and g also belong to .The epigraph of a function f∈ is given byepi f={(x,y)∈R2|x≥f(y)}.For any a≥b the functionh[b,a](y)=ay−−by+belongs to .Observe that the self-financing condition(2.1)can be written as(αt−αt+1,βt−βt+1)∈epi h[S bt,S a t ] .For each f∈ there is a unique function in ,denoted by gr[b,a](f),such thatepi[gr[b,a](f)]=epi h[b,a]+epi f.We shall call gr[b,a](f)the gradient restriction of f.This transformation is illustrated in Fig.1.If f∈ is a function withfinite values,then it hasfinite limits f (+∞)=lim x→+∞f (x) and f (−∞)=lim x→−∞f (x).If these limits satisfy the inequalitiesb≤−f (−∞)and−f (+∞)≤a,(2.6) then gr[b,a](f)is also a function withfinite values.Thefinancial meaning of gradient restric-tion is that portfolios in the epigraph of gr[S bt,S at](f)are precisely those that can be rebalancedin a self-financing manner at time t to yield a portfolio in the epigraph of f.American Options under Proportional Transaction Costs:Pricing205 Fig.1Gradient restriction of afunction f inComputer implementation of the three operations in mentioned above,namely the maximum,minimum,and gradient restriction,is straightforward.They will be used in pric-ing Algorithms3.1and3.5,and in the numerical examples in Sect.5.We denote by the family of all convex functions in .It is closed under the maximum and gradient restriction operations,but not the minimum.For any f∈ the convex dual is defined byf∗(x)=infy∈R(f(y)+xy)for each x∈R.The infimum is attained whenever it isfinite.Convex duality maps bijec-tively onto the family of concave functions v:R→R∪{−∞}such that v is polyhedral (continuous piecewise linear with afinite number of pieces)on its essential domaindom v={x∈R|v(x)>−∞}.The inverse transform from to is given byf(y)=supx∈R(f∗(x)−xy),(2.7)with the supremum attained wheneverfinite.For any v1,...,v n∈ we denote by cap{v1,...,v n}the concave cap of v1,...,v n∈ , defined as the smallest concave function v such that v≥v i for each i=1,...,n.It belongs to and for each x∈R can be represented ascap{v1,...,v n}(x)=maxni=1λi v i(x i),(2.8)where the maximum is taken over allλ1,...,λn≥0and x1,...,x n∈R that satisfyni=1λi=1,ni=1λi x i=x,and x i∈dom v i for each i=1,...,n such that dom v i=∅,see Rockafellar[21].Under convex duality the convex cap in corresponds to the maximum in ,max{f,g}∗=cap{f∗,g∗}206 A.Roux,T.Zastawniak for any f,g∈ .The operation in corresponding to gradient restriction in will be called domain restriction.For each v∈ and each x∈R it is defined bydr[b,a](v)(x)=v(x)if x∈[b,a],−∞if x/∈[b,a].For any f∈ we havegr[b,a](f)∗=dr[b,a](f∗).If f∈ hasfinite values,then[−f (+∞),−f (−∞)]=dom f∗,and(2.6)can be written as[b,a]∩dom f∗=∅.This condition guarantees that gr[b,a](f)hasfinite values,or,equivalently,that dr[b,a](f∗) has non-empty essential domain.3American Options under Proportional Transaction CostsLet us take an adapted process(ξt,ζt)with values in R2∪{(−∞,−∞)}defined for all t=0,1,...,T to be the payoff process of an American option.The seller of the option must deliver to the buyer a portfolio(ξτ,ζτ)of cash and stock at an exercise timeτ∈T chosen by the buyer.The pair(−∞,−∞)is included among the possible values of the payoff process to allow for the possibility that the option cannot be exercised at certain times or nodes of the tree. This ensures that the results of this paper are in fact valid not only for American options but also for European or Bermudan type derivatives.The seller can hedge a short position in the option by a self-financing strategy(α,β)∈ such that at each stopping timeτ∈T he or she will be left with a solvent portfolio (ατ−ξτ,βτ−ζτ)once the payoff(ξτ,ζτ)has been delivered to the buyer,that is,a portfolio such thatϑτ(ατ−ξτ,βτ−ζτ)≥0.(3.1) This is called a superhedging strategy for the seller.The cost of setting up such a strategy is−ϑ0(−α0,−β0),the lowest of which defines the seller’s price(ask price,upper hedging price)of the option:πa(ξ,ζ)=min{−ϑ0(−α0,−β0)|(α,β)∈ ,∀τ∈T:ϑτ(ατ−ξτ,βτ−ζτ)≥0}.(3.2) On the other hand,the buyer can hedge a long position in the option by a self-financing strategy(α,β)∈ such that there is a stopping timeτ∈T when he or she will be left with a solvent portfolio(ατ+ξτ,βτ+ζτ)after exercising the option and receiving the payoff (ξτ,ζτ),that is,a portfolio such thatϑτ(ατ+ξτ,βτ+ζτ)≥0.(3.3)American Options under Proportional Transaction Costs:Pricing207 This is called a superhedging strategy for the buyer.By setting up such a strategy the buyer can raise the amountϑ0(−α0,−β0).The highest amount that can be raised in this way is called the buyer’s price(bid price,lower hedging price)of the option:πb(ξ,ζ)=max{ϑ0(−α0,−β0)|(α,β)∈ ,∃τ∈T:ϑτ(ατ+ξτ,βτ+ζτ)≥0}.(3.4) In a discrete arbitrage-free market model the minimum in(3.2)and the maximum in(3.4) are attained.A strategy(α,β)∈ realising the minimum in(3.2)is referred to as the seller’s optimal strategy.A strategy(α,β)∈ and a stopping timeτ∈T realising the maximum in(3.4)are called the buyer’s optimal strategy and buyer’s optimal stopping time.The pricesπa(ξ,ζ)andπb(ξ,ζ)provide the upper and lower bounds of the no-arbitrage interval of option prices.Moreover,these are liquidity prices at which the option can be bought or,respectively,sold on demand.Liquidity is important because options are often traded as part of a strategy to hedge other derivatives.3.1Seller’s Case3.1.1Seller’s Pricing AlgorithmsLet(ξ,ζ)be the payoff process of an American option.For each t=0,1,...,T and each y∈R we putu t(y)=ξt+(y−ζt)−S a t−(y−ζt)+S b t.(3.5) This defines an adapted process u t∈ .Observe that a strategy(α,β)∈ satisfies seller’ssuperhedging condition(3.1)for a stopping timeτ∈T if and only if(ατ,βτ)∈epi uτ.Algorithm3.1For t=0,1,...,T take u t∈ given by(3.5)and construct adapted processes z t,v t,w t∈ by backward induction as follows:•For eachμ∈ T putzμT=vμT=wμT=uμT.•For each t=1,...,T andμ∈ t−1putzμt−1=max{vμt−1,uμt−1},(3.6) wherevμt−1=gr[S bμt−1,S aμt−1](wμt−1),(3.7)wμt−1=max{zνt|ν∈succμ}.(3.8)The resulting function z0will be related in Lemma3.1to hedging the seller’s position in the American option(ξ,ζ).In Theorem3.3it will be shown thatπa(ξ,ζ)=z0(0).This algorithm can also be stated in terms of the dual functionsZ t=z∗t,V t=v∗t,W t=w∗t,U t=u∗t,208 A.Roux,T.Zastawniak which belong to .Observe that for each t=0,1,...,T and x∈RU t(x)=ξt+xζt if x∈[S b t,S a t],−∞if x/∈[S b t,S a t].(3.9)By the duality between and ,Algorithm3.1is equivalent to the following procedure. Algorithm3.2For t=0,1,...,T take U t∈ given by(3.9)and construct adapted processes Z t,V t,W t∈ by backward induction as follows:•For eachμ∈ T putZμT=VμT=WμT=UμT.•For each t=1,...,T andμ∈ t−1putZμt−1=cap{Vμt−1,Uμt−1},(3.10) whereVμt−1=dr[S bμt−1,S aμt−1](Wμt−1),(3.11)Wμt−1=cap{Zνt|ν∈succμ}.(3.12)The function Z0will be related in Lemma3.1and Remark3.4to hedging the seller’s position in the American option(ξ,ζ).In Theorem3.3it will be shown thatπa(ξ,ζ)=maxx∈RZ0(x).Remark3.1If the payoff isfinite at some nodeμ∈ t,that is,(ξμt,ζμt)∈R2,then at eachancestor nodeν⊃μ,whereν∈ s for some s=0,1,...,t,the functions Wνs ,Vνs,Zνscon-structed in Algorithm3.2have non-empty effective domains,and wνs ,vνs,zνsin Algorithm3.1takefinite values.In particular,z0(0)and the maximum of Z0are thenfinite.Indeed,for any(P,S)∈P it can be shown by backward induction that the effective domains of Wνs ,Vνs,Zνsmust contain Sνs .In an arbitrage-free model P is non-empty,so that these effective domainsmust then also be non-empty.Remark3.2Algorithm3.2can be viewed as a natural extension of the familiar Snell en-velope construction.In the absence of transaction costs,when S at =S b t=S t for all t,for-mula(3.9)simply defines the cash equivalent U t=ξt+ζt S t of the payoff process(ξt,ζt) for an American option with physical delivery,(3.11)and(3.12)give the continuation value V t−1=E∗(Z t|F t−1),where E∗is the risk neutral expectation,and(3.10)becomes Z t−1=max{U t−1,V t−1}.The workings of Algorithms3.1and3.2will be illustrated in Example4.1and Figs.2 and3in a simple two-step binomial tree setting.The numerical results in Sect.5for an American put and a bull spread in the binomial and trinomial tree models are computed by implementing these algorithms.In a recombinant model these computations grow only poly-nomially with the number of time steps,resulting in efficient numerical work in a realistic setting.3.1.2Hedging Seller’s PositionThe following algorithm makes it possible to construct a strategy superhedging a short (seller’s)position in an American option with payoff process(ξ,ζ)by starting from any portfolio in epi z0.Algorithm3.3Construct a strategy(α,β)∈ by induction as follows:•Take any F0-measurable portfolio(α0,β0)∈epi z0.•Suppose that an F t-measurable portfolio(αt,βt)∈epi z t has already been constructed for some t=0,...,T−1.Since,by(3.6)and(3.7),epi z t⊂epi v t=epi h[S bt,S at]+epi w t,there is an F t-measurable portfolio(αt+1,βt+1)∈epi w t such that(αt−αt+1,βt−βt+1)∈epi h[S bt,S at].(3.13)Because of(3.8)we have(αt+1,βt+1)∈epi z t+1,and since(αt+1,βt+1)is F t-measurable, it is also F t+1-measurable,completing the induction step.Because(3.13)is equivalent to the self-financing condition(2.1),we know that (α,β)∈ .It will be shown in Lemma3.1that(α,β)is a superhedging strategy for the seller.Remark 3.3When implementing the iterative step in Algorithm 3.3,the portfolio (αt+1,βt+1)can be constructed from(αt,βt)as follows:•Ifαt≥w t(βt),then we put(αt+1,βt+1)=(αt,βt).No rebalancing of the portfolio occurs in this case.•Ifαt<w t(βt),then the equationαt+x−S b t−x+S a t=w t(βt+x)has a solution x,and we put(αt+1,βt+1)=(αt+x−S b t−x+S a t,βt+x),which amounts to buying x shares at the ask price S at if x>0or selling them at the bidprice S bt if x<0.The equation for x has a solution because(αt,βt)∈epi h[S bt,Sat]+epi w t.The following result shows that epi z0can be characterised as the set of endowments (γ,δ)consisting of cash and stock that are sufficient to initiate a superhedging strategy for the seller.Lemma3.1The following conditions are equivalent:1)(γ,δ)∈epi z0.2)There is a self-financing strategy(α,β)∈ such that(α0,β0)=(γ,δ)and(αt,βt)∈epi z t for each t=0,1,...,T.3)There is a superhedging strategy(α,β)∈ for the seller such that(α0,β0)=(γ,δ).Proof1)⇒2).This follows directly from the construction in Algorithm3.3.2)⇒3).This is so because epi zτ⊂epi uτby(3.6)and the seller’s superhedging condi-tion(3.1)can be written as(ατ,βτ)∈epi uτfor eachτ∈T.3)⇒1).If(α,β)∈ is a strategy as in condition3),we claim that(αt,βt)∈epi z t for all t=0,1,...,T.Condition1)then follows immediately.We prove this claim by backward induction on t.Since(αT,βT)∈epi u T=epi z T,the claim is valid for t=T.Suppose that the claim holds for some t=1,...,T,that is,(αt,βt)∈epi z t.Since(αt,βt)is F t−1-measurable,it follows by(3.8)that(αt,βt)∈epi w t−1.Because the strategy is self-financing,we have(αt−1−αt,βt−1−βt)∈epi h[S bt−1,S a t−1].As a result,(αt−1,βt−1)∈epi h[S bt−1,S a t−1]+epi w t−1=epi v t−1by(3.7).Moreover,since(α,β)is a superhedging strategy for the seller, (αt−1,βt−1)∈epi u t−1.We can conclude using(3.6)that(αt−1,βt−1)∈epi v t−1∩epi u t−1= epi z t−1.The claim has been verified.Remark 3.4By duality,since z t(y)=sup x∈R(Z t(x)−xy),conditions1)and2)in Lemma3.1can be written,equivalently,as follows:1∗)γ+xδ≥Z0(x)for each x∈R.2∗)There is a self-financing strategy(α,β)∈ such that(α0,β0)=(γ,δ)andαt+xβt≥Z t(x)for each x∈R and each t=0,1,...,T.3.1.3Seller’s Stopping Time and Approximate MartingaleOur aim in this section is to construct a mixed stopping timeˆχ∈X together with an ap-proximate martingale(ˆP,ˆS)∈¯P(ˆχ)so that the ask price of an American option with payoff process(ξ,ζ)can be expressed asπa(ξ,ζ)=EˆP((ξ+ˆSζ)ˆχ).(3.14)At the same time,we shall also construct certain auxiliary adapted processesˆλ,ˆp,ˆX,ˆY,ˆZ,ˆU,ˆV.Algorithm3.4Construct a mixed stopping timeˆχ∈X,a probability measureˆP and adapted processesˆλ,ˆp,ˆS,ˆX,ˆY,ˆZ,ˆU,ˆV by induction as follows:•For t=0there is aˆY0∈[S b0,S a0]such thatZ0(ˆY0)=maxx∈RZ0(x).By(2.8),since Z0=cap{V0,U0},there existˆX0,ˆS0∈[S b0,S a0]andˆλ0∈[0,1]such thatˆY0=(1−ˆλ)ˆX0+ˆλ0ˆS0,ˆZ0=(1−ˆλ)ˆV0+ˆλ0ˆU0,whereˆZ0=Z0(ˆY),ˆV0=V0(ˆX0),ˆU0=U0(ˆS0).Moreover,we can choose andˆλ0=0if U0≡−∞.We putˆχ0=ˆλ0,ˆP0=1.。
高中英语单词天天记：provide素材
高中英语单词天天记：·provide·v.[prə'vaɪd] ( provides; provided; providing )··双解释义·vt. & vi.提供,供给,供应cause or arrange from (sb) to have or use (sth needed or useful); supply·vt.规定state a special arrangement that must be fulfilled·基本要点•1.provide用作及物动词时,可表示“提供,供给,供应,装备”,后可接名词或代词作简单宾语; 还可接双宾语,常用介词for引出间接宾语,也可用介词with引出直接宾语,有时with可省略。
2.provide还可表示法律、规章、协议等的“规定”,往往作为条件提出使之能按照某人的意愿或某种规则履行,后常接that引起的从句,从句中可以用shall, should, must, may, ought to等,用should, must, ought to时,语气较重,用may时,语气较轻,用shall时,指当前,语气一般,也可用虚拟语气。
3.provide用作不及物动词时常与for, against 等介词搭配。
•·词汇搭配••provide a meal 提供一餐•provide blankets 提供毯子•provide education 提供教育•provide employment 提供就业•provide equipment 提供装备•provide money 提供金钱•provide support 提供支持•provide vehicle 提供交通工具•provide weapons 提供武器•••ill provide 供应差•well provide 供应好,准备得很好•••provide abundantly 充裕地供给•provide adequately 充分提供•provide copiously 大量地提供•provide duly 适当地准备•provide expressly 迅速准备•provide graciously 仁慈地提供•provide handsomely 优厚地提供•provide hospitably 殷勤地提供•provide liberally 充分地提供,充分地准备•provide properly 适当地提供•provide specifically 专门提供•provide suitably 适合提供•••provide against unemployment 防备失业•provide for 抚养…,为…提供,为…准备•provide for children 供养孩子•provide for family 养家•provide with 提供…给…•provide sb with money 为某人提供钱款·常用短语•provide against(v.+prep.)为…做好准备; 预防 take care for the future▲provide against sthThey provided against the attack.他们做好准备以防受攻击。
Statistical theory of the continuous double auction
A. Relationship of Price impact to cumulative depth 29 1. Moment expansion 30 2. Quantiles 31 B. Supporting calculations in density coordinates 1. Generating functional at general bin width a. Recovering the continuum limit for prices 2. Cataloging correlations a. Getting the intercept right b. Fokker-Planck expanding correlations Acknowledgments
Contents
I. Introduction 2 A. Motivation 2 B. Background: The continuous double auction 3 C. The model 3 D. Summary of prior work 5 II. Overview of predictions of the model A. Dimensional analysis B. Varying the granularity parameter ǫ 1. Depth profile 2. Liquidity for market orders: The price impact function 3. Spread 4. Volatility and price diffusion 5. Liquidity for limit orders: Probability and time to fill. C. Varying tick size dp/pc III. Theoretical analysis A. Summary of analytic methods B. Characterizing limit-order books: dual coordinates C. Frames and marginals D. Factorization tests E. Comments on renormalized diffusion F. Master equations and mean-field approximations 1. A number density master equation 2. Solution by generating functional 5 5 8 9 10 11 12 13 13 14 14 15 16 17 18 19 19 20
Sign-changing blowing-up solutions for the Brezis--Nirenberg problem in dimensions four and five
A. IACOPETTI, G. VAIRA Abstract. We consider the Brezis-Nirenberg problem:
1
SIGN-CHANGING BLOWING-UP SOL. FOR THE BREZIS–NIRENBERG PROB. IN DIM. N = 4, 5
2
sign-changing solutions of (1.1) when λ is smaller than λ∗ (B ) = λ1 (B )/4. A partial answer to this question posed by H. Brezis has been given in [10]. However, even in the case N = 4, 5, 6, some apparently strange phenomenon appears for what concerns radial sign-changing solutions in the ball. Indeed it was first proved by Atkinson, Brezis and Peletier in [5] that for N = 4, 5, 6 there exists λ∗ = λ∗ (N ) such that there are no sign-changing radial solutions of (1.1) for λ ∈ (0, λ∗ ). Later this result was proved in [1] in a different way. As it will be clear in the sequel, the nonexistence result of Atkinson, Brezis and Peletier is connected to the asymptotic analysis of low-energy sign-changing solutions of (1.1). Ben Ayed, El Mehdi and Pacella have investigated the latter question in [10],[11]. More precisely, denoting 1 1 by · the H0 (Ω)-norm and by S the best Sobolev constant for the embedding H0 (Ω) into 2∗ L (Ω), they studied the asymptotic behavior of sign-changing solutions uλ of (1.1) such that ¯ , if N = 3, where λ ¯ is the infimum of the values uλ 2 → 2S N/2 , as λ → 0 if N ≥ 4, or λ → λ of λ for which nodal low-energy solutions exist (see [10]). They proved that these solutions blow up at two different points a ¯1 , a ¯2 , which are the limit of the concentration points aλ,1 , aλ,2 of the positive and negative part of uλ . We point out that they need to assume the extra hypothesis that the concentration speeds of the two concentration points are comparable for N ≥ 4 (see [11]), while in dimension three this was derived without any extra assumption (see [10]). In view of the results of Ben Ayed, El Mehdi and Pacella we get that, for N ≥ 4, the question of proving the existence of sign-changing low-energy solutions (i.e. such that uλ 2 Ω converges to 2S N/2 as λ → 0) whose positive and negative part concentrate and blow up at the same point, was left open. In [30], by studying the asymptotic behavior, as λ → 0, of low-energy radial sign-changing solutions of (1.1) in the unit ball of RN , for N ≥ 7 (for these dimensions they do exist, as proved by Cerami, Solimini and Struwe in [20]), it has been proved that the positive and the negative part of such solutions concentrate and blow up at the center of the ball, and their concentration speeds are not comparable. Moreover, in the recent paper [33], it has been proved that for N ≥ 7 these so called “bubble-tower” solutions for (1.1), exist, as λ → 0, in general bounded domains with some symmetry. We point out that, in the previous result, the assumption N ≥ 7 on the dimension is not only technically crucial but it also is necessary. In fact, in the recent paper [31], the authors proved that for the low dimensions N = 4, 5, 6, and in general bounded domains, there cannot exist sign-changing “bubble-tower” solutions for (1.1), as λ → 0. This result is hence the counterpart, in general bounded domains, of the nonexistence theorem of Atkinson, Brezis and Peletier if we think of sign-changing “bubble-tower” solutions as the functions which play, in general bounded domains, the same role as the radial solutions in the case of the ball. In view of all these results it is natural to ask what kind of asymptotic profile we can expect for sign-changing solutions in the low dimensions N = 4, 5, 6, as λ goes to some strictly positive “limit” value. The case of radial sign-changing solutions in the ball, having two nodal regions, has been investigated in [32]. By studying the associated differential equation, and taking into account the results of [6], [7], the authors prove that if (uλ ) is a family of radial sign-changing solutions of (1.1) in the unit ball B1 of RN , having two nodal regions, such that uλ (0) > 0, and ¯ =λ ¯ (N ) the limit value of the parameter λ, which arises from the study of the denoting by λ related ordinary differential equation, then: ¯ = λ1 (B1 ), where λ1 (B1 ) is the first eigenvalue of −∆ in H 1 (B1 ), (i) if N = 4, 5, then λ 0 + and uλ concentrates and blows-up at the center of the ball having the limit profile of a
Complexity results for triangular sets
Complexity Results for Triangular Sets´Eric SchostLaboratoire GAGE,´Ecole polytechnique,91128Palaiseau Cedex,FranceAbstractWe study the representation of the solutions of a polynomial system by triangular sets,and concentrate on the positive-dimensional case.We reduce to dimension zero by placing the free variables in the basefield,so the solutions can be represented by triangular sets with coefficients in a rational functionfield.We give intrinsic-type bounds on the degree of the coefficients in such a triangular set,and on the degree of an associated degeneracy hypersurface.Then we show how to apply lifting techniques in this context,and point out the role played by the evaluation properties of the input system.Our algorithms are implemented in Magma;we present three applications,rele-vant to geometry and number theory.Key words:triangular sets,complexity,symbolic Newton operator1IntroductionThis article studies the triangular representation of the solutions of a polyno-mial system.Ourfirst focus is on complexity results and algorithms;we also present a series of applications that were treated with these techniques.To make things clear,let usfirst display a concrete example of a triangular set. An example in Q[X1,X2].Consider the polynomial system in Q[X1,X2]:F1=−X31X2+2X21−4X1X22+2X1X2−2,F2=X21X2−X1+4X22−2X2.Email address:Eric.Schost@polytechnique.fr(´Eric Schost).Preprint submitted to Elsevier Science27March2003It admits the following Gr¨o bner basis for the lexicographic order X1<X2:T1=X21−2,T2=X22−14X1.Since T1is in Q[X1]and T2in Q[X1,X2],we say that(T1,T2)form a triangular set.In particular,T1describes the projection of the zero-set of(F1,F2)on the X1-axis.From thefield-theoretic point of view,the system(F1,F2)generates a prime zero-dimensional ideal,so Q→B:=Q[X1,X2]/(F1,F2)defines afield ex-tension.We let x1,x2be the images of X1,X2in B;then T1is the minimal polynomial of x1in Q→B and T2,seen in Q(x1)[X2],is the minimal polyno-mial of x2in Q(x1)→B.Generalization andfirst complexity considerations.Consider now an arbitraryfield K,K its algebraic closure,and a zero-dimensional variety W⊂A n(K)defined over K.For simplicity,we take W irreducible over K;then just as above,the ideal defining W admits the following Gr¨o bner basis for the lexicographic order X1<···<X n:T1(X1),T2(X1,X2),...T n(X1,...,X n),with T k in K[X1,...,X k],and monic in X k,for k≤n.We will use this as an intuitive definition of a triangular set for the rest of this informal introduction. Note that if W is not irreducible,its defining ideal might not have such a triangular family of generators:several triangular sets may be necessary.For k≤n,the family T1,...,T k describes the projection of W on the affine subspace of coordinates X1,...,X k.In particular,as above,T1is the mini-mal polynomial of X1modulo the ideal defining W.This close link between projections and triangular representations is central in what follows.Let us turn to complexity considerations.The product of the degrees of thepolynomials T k in their“main variable”Πk≤n deg Xk T k equals the number ofpoints in W,and bounds the total degree of each polynomial T k.Thus,in terms of degrees in the variables X1,...,X n,there is not much more to say. New questions arise when the basefield K is endowed with a“size”function: if K is a rational functionfield,we may consider the degree of its elements;2if K is a numberfield,we can talk about the height of its elements.In this context,it becomes natural to ask how the size of the coefficients in T1,...,T n relates to some invariants measuring the“complexity”of the variety W.In view of the above remarks,a more accurate question is actually,for k≤n,the relation between the size of the coefficients in T1,...,T k and the complexity of the projection of W on the subspace of coordinates X1,...,X k.In this article,we focus on this question in the functionfield case.Here is the concrete situation from where the question originates.Polynomial systems with parameters.A variety of problems can be described by polynomial systems involving free variables,or parameters.In such situations,we also often know that there are onlyfinitely many solutions for a generic choice of the parameters.In other words,we are considering systems that are zero-dimensional over the field of rational functions on some parameter space;triangular sets with ra-tional functions coefficients can then be used to represent their solutions.The following applications motivated this approach;they are detailed in Section8.•Modular equations.In Gaudry and Schost[2002],we propose a definitionof modular equations for hyperelliptic curves,with a view towards point-counting applications.For a given curve,these equations come from the resolution of zero-dimensional polynomial systems,as the minimal polyno-mial of one of the unknowns.Thus,they can be obtained from a triangular set computation,as in the introductory example.An interesting question is that of modular equations for a curve with generic coefficients,which can be precomputed and stored in a database. This was already done in the elliptic case,and is now done for afirst hy-perelliptic modular equation in the Magma package CrvHyp.This naturally raises the question of triangular sets with coefficients in a rational function field.•Curves with split Jacobian.Curves of genus2with(2,2)-split Jacobian are of interest in number theory:over Q,torsion,rank and cardinality records are obtained for such curves,see Kulesz[1995,1999],Howe et al.[2000]. Roughly speaking,these curves are characterized by the presence of elliptic quotients of degree2of their Jacobian.We studied such curves in Gaudry and Schost[2001],and showed that the elliptic quotients can be read offtriangular sets coming from the resolution of a suitable polynomial system.Classification questions require treating this question for curves with generic coefficients,which leads again to the problem of computing triangular sets over a rational functionfield.•Implicitization.Finally,we will show that the implicit equation of a parame-trized surface in R3can be obtained using the triangular representation.3Contrary to the above,this question is not a priori formalized in terms of a parametric system.Nevertheless,this question actually reduces to the com-putation of a minimal polynomial over the rational functionfield Q(x1,x2), which can be done using triangular sets.These examples share the following property:only a partial information,such as a specific eliminating polynomial,is really wanted.We now see how trian-gular sets can answer this question with good complexity.Overview of our results.The above discussion is formalized as follows: we consider a polynomial system F defined over afield K,depending on m parameters P1,...,P m and n unknowns X1,...,X n.Geometrically speaking, F defines a variety W of dimension m in A m+n(K)and generates a zero-dimensional ideal,when extended over thefield of rational functions on A m(K). Then its”generic solutions”can be represented by a family of triangular sets with coefficients in this rational functionfield.For this short overview,we assume that the generic solutions are represented by a single triangular set T1,...,T ing additional regularity hypotheses,we will answer the following questions:How do the degrees in this triangular set relate to geometric degrees?How accurately does this triangular set describe the solutions of the parametric system F?How fast can it be computed?•Degree bounds.The coefficients of T1,...,T n are rational functions in thefree variables P1,...,P m.Wefirst show that their degrees are bounded by intrinsic geometric degrees,that is,independently of the B´e zout number of the system F.Precisely,for k≤n,the coefficients of T1,...,T k have degree bounded in terms only of the degree of the projection W k of W on the space of coordinates P1,...,P m,X1,...,X k.The precise bound is of order (deg W k)k.•Geometric degree of the degeneracy locus.A triangular set with coefficients in a rational functionfield describes generic solutions.Thus,there is an open subset in the parameter space where none of the denominators of these ra-tional functions vanishes,and where their specialization gives a description the solutions of the parametric system F.We show that the locus where the specialization fails is contained in an hypersurface whose degree is quadratic in the geometric degree of W.Note the difference with the above degree bounds,which are not polynomial in this degree.The analysis of the probabilistic aspects of our algorithms are based on this result.•Algorithms.Triangular sets are useful for structured problems.For instance, all the above examples can be reduced to the computation of thefirst k polynomials T1,...,T k,for some k≤n.We give probabilistic algorithms for computing these polynomials,whose complexity is polynomial in the4size of the ing the above upper bound,the complexity actually depends on the degree of the projection W k of W on the space of coordinates P1,...,P m,X1,...,X k,but not on the degree of W itself.Note nevertheless that our complexity results comprise an additional fac-tor which is exponential in n,inherent to computations with triangular sets.Following the series of articles Giusti et al.[1995,1997,1998],Heintz et al.[2000],Giusti et al.[2001],Heintz et al.[2001],our algorithms rely on symbolic Newton lifting techniques and the Straight-Line Program repre-sentation of polynomials.Their practical behavior matches their good com-plexity,as they enabled to solve problems that were otherwise out-of-reach. Comparison with primitive elements techniques.This work is in the continuation of Schost[2003],which focuses on a representation by primitive element techniques,the geometric resolution,in a similar context.Caution must be taken when comparing the two approaches.They answer different questions;as such,their complexities cannot be compared directly,since they are stated in terms of different quantities.We use again the above notation:the geometric object of interest is a variety W defined by polynomials in K[P1,...,P m,X1,...,X n],and for k≤n,W k is its projection on the space of coordinates P1,...,P m,X1,...,X k.The degree bound of the coefficients in a geometric resolution is linear in the degree of W.This is to be compared with the results for the triangular representation,which are not polynomial in this degree.On the other hand, triangular sets take into account the degrees of the successive projections W k, which cannot be reached using a primitive element.These degrees can be arbitrarily smaller than the degree of W,making the interest of the triangular representation.Consider now the algorithmic aspect.The algorithm in Schost[2003]computes a parametric geometric resolution with a complexity that depends on the de-gree of W.The algorithms proposed here compute k polynomials T1,...,T k, for any given k≤n;their complexity depends on the degree of the correspond-ing projection W k of W on the space of coordinates(P1,...,P m,X1,...,X k), but not on the degree of W.Again,this suggests that triangular sets are of interest for problems with a structure,where projections might induce degree drops.We refer to Section8for a practical confirmation for several applica-tions.Related work.In dimension zero,a landmark paper for the triangular rep-resentation is Lazard[1992].Our definition of triangular sets is inspired by5the one given there,as is the treatment of more technical questions such as splitting and combining triangular sets.In arbitrary dimension,several notions of triangular sets and algorithms ex-ist,see Lazard[1991],Kalkbrener[1991],Maza[1997],Aubry[1999],Delli`e re [1999],Szanto[1999].For a comparison of some of these approaches,see Aubry et al.[1999];we also refer to the detailed survey of Hubert.Our choice to re-duce the question to dimension zero over afield of rational functions yields algorithms with good complexity,and easy to implement.Yet,our output is not as strong as for instance that of Lazard[1991],Maza[1997],Delli`e re[1999]: ours is only generically valid.Upper bounds on the degrees of the polynomials in a triangular set were given in Gallo and Mishra[1990]and Szanto[1999];we recall these results in the next section.In particular,the approach of Gallo and Mishra[1990] inspired Theorem1below.We also use results from Schost[2003],which follow notably Sabia and Solern´o[1996].Lifting techniques for polynomial systems were introduced in Trinks[1985], Winkler[1988].They were used again in the series of articles by Giusti,Heintz, Pardo and collaborators,Giusti et al.[1995,1997,1998],Heintz et al.[2000], Giusti et al.[2001],Heintz et al.[2001].The conjoint use of the Straight-Line Program representation led there to algorithms with the best known complexity for primitive element representations.The present work is in the continuation of the above;see also the survey of Pardo[1995]for a histori-cal presentation of the use of Straight-Line Programs in elimination theory. Finally,let us mention the results of Lecerf[2002],which extend lifting tech-niques to situations with multiplicities.We note that the article Heintz et al.[2000]precedes Schost[2003]and the present work,and considers similar questions of parametric systems.Never-theless,we noted in Schost[2003]that the geometric hypotheses made in that article are not satisfied in many“real life”applications,and this is again the case for the applications treated here.It should be noted that our complexity statements are of an arithmetic nature, that is,we only estimate the number of basefield operations.When the base field is the rationalfield,the notion of binary complexity will give a better description of the expected computation time.We have not developed this aspect,which requires arithmetic-geometric considerations.We refer to Krick and Pardo[1996],Giusti et al.[1997],Krick et al.[2001]where such ideas are presented.This work is based on a shorter version published in Schost[2002].The degree bounds given here are sharper.The whole analysis of the degeneracy locus and the subsequent error probability analyses for the algorithms are new.The6complexity results are now precisely stated in terms of basic polynomial and power series arithmetic.Acknowledgements.I wish to thank L.M.Pardo for his useful remarks on the first version of this paper.2Notation,Main ResultsTriangular sets in dimension zero.We first define triangular sets over a ring R .Our definition is directly inspired by that of reduced triangular sets given in Lazard [1992]:a triangular set is a family of polynomials T =(T 1,...,T n )in R [X 1,...,X n ]such that,for k ≤n :•T k depends only on X 1,...,X k ,•T k is monic in X k ,•T k has degree in X j less than the degree in X j of T j ,for all j <k .Let now K be a field,K its algebraic closure and W ⊂A n (K )a zero-dimensional variety.Recall that W is defined over K if its defining ideal in K [X 1,...,X n ]is generated by polynomials in K [X 1,...,X n ].In this case,a family {T 1,...,T J }of triangular sets with coefficients in K represents the points of W if the radical ideal defining W in K [X 1,...,X n ]is the intersection of the ideals generated by T 1,...,T J ,and if for j =j ,T jand T j have no common zero.In this situation,all ideals (T j )are radical by the Chinese Remainder Theo-rem.We then relate the degrees of the polynomials in the family {T 1,...,T J }and the cardinality of W :•If W is irreducible,the family {T 1,...,T J }is actually reduced to a sin-gle triangular set T =(T 1,...,T n )and the product Πk ≤n deg X k T k is the cardinality of W .Here,deg X k T k denotes the degree of T k in the variable X k .•If W is not irreducible,a family {T 1,...,T J }satisfying our conditions exists but is not unique [Lazard,1992,Proposition 2and Remark 1];now the sum j ≤J Πk ≤n deg X k T j k is the cardinality of W .Hereafter,note thatthe superscript in the notation T j k does not denote a j -th power.Note that it necessary to work over the algebraically closed field K ,or more generally to impose separability conditions,to obtain equalities as above,re-lating the degrees in the triangular sets T or {T 1,...,T J }and the number of7points in the variety W.The basic geometric setting.We now turn to more geometric considera-tions.All along this article,wefix afield K,K its algebraic closure,and work in the affine space A m+n(K).We denote by P=P1,...,P m thefirst m coor-dinates in A m+n(K)and by X=X1,...,X n the last n coordinates.We use the notion of geometric degree of an arbitrary affine variety(not necessarily irreducible,nor even equidimensional),introduced in Heintz[1983].In what follows,the affine space A m+n(K)is endowed with two families of projections.For k≤n,we defineµk andπk as follows;hereafter,p denotes a point in A m(K).µk:A m+n(K)→A m+k(K)πk:A m+k(K)→A m(K) (p,x1,...,x n)→(p,x1,...,x k)(p,x1,...,x k)→p.Note in particular thatπn maps the whole space A m+n(K)to A m(K).The main geometric object is a m-dimensional variety W⊂A m+n(K).Our first results are of an intrinsic nature,so we do not need an explicit reference to a defining polynomial system.The assumptions on W follow the description made in the introduction:Assumption1Let{W j}j≤J denote the irreducible components of W.We assume that for j≤J:(1)the imageπn(W j)is dense in A m(K).(2)the extension K(P1,...,P m)→K(W j)is separable.Assumption1.1implies that thefibers of the restriction ofπn to each compo-nent of W are genericallyfinite;this justifies treating thefirst m coordinates as distinguished variables and calling them parameters.Assumption1.2is of a more technical nature,and will help to avoid many difficulties;it is always satisfied in characteristic zero.Under Assumption1,we can define the generic solutions of the variety W. Let J⊂K[P,X]be the radical ideal defining W and J P its extension in K(P)[X].We call generic solutions of W the roots of J P,which are infinite number.We now refer to the previous paragraph,taking K=K(P),and for W the finite set of generic ing Assumption1.2,the ideal J P remains radical in K[X],so the generic solutions are indeed defined over K=K(P). Thus,they can be represented by a family of triangular sets in K(P)[X];our8purpose in this article is to study their complexity properties,and provide algorithms to compute with them.Let us immediately note some particular cases:•If W is irreducible,a single triangular set is enough to represent its generic solutions.•If W is defined over K,it can be written W=∪j≤J W j,where for all j, W j is defined over K,and the defining ideal of W j is prime in K[P,X]. Then the generic solutions of each W j are represented by a triangular set in K(P)[X];the generic solutions of W are represented by their reunion.Projections of W.Before presenting the main results,we introduce some notation related to W and its successive projections.Let k be in1,...,n.First of all,we denote by X≤k thefirst k variables X1,...,X k;if T is a triangular set,T≤k is the sub-family T1,...,T k.We denote by W k⊂A m+k(K)the closure ofµk(W),so in particular W n coincides with W.It is a routine check that for all k,W k satisfies Assumption1 as well.Let J k⊂K[P,X≤k]be the ideal defining W k,and J P,k its extension in K(P)[X≤k].Under Assumption1.1,J P,k coincides with J P∩K(P)[X≤k].Thus if the generic solutions of W are defined by a triangular set T,J P,k is generated by T≤k.For p in A m(K),we denote by W k(p)thefiberπ−1k (p)∩W k and by D k thegeneric cardinality of thefibers W k(p).Finally,let B k be the quotient K(P)[X≤k]/J P,k;by Assumption1.2,the ex-tension K(P)→B k is a product of separablefield ing the separability,B k has dimension D k,by Proposition1in Heintz[1983]. Degree bounds.With this notation,we now present our main results.We assume that the generic solutions of W are represented by a triangular set T=(T1,...,T n)in K(P)[X].In view of the above remarks,this is not a strong limitation:if this assumption is not satisfied,as soon as W is defined over K,the following upper bounds apply to all the K-defined irreducible components of W.As mentioned in the preamble,the degree bounds of T in the X variables areeasily dealt with:for all k≤n,the productΠi≤k deg Xi T i is the dimension ofB k over K(P),that is,the generic cardinality D k of thefibers W k(p).9We will thus concentrate on the dependence with respect to the P variables. For k≤n,the polynomial T k depends only on the variables X1,...,X k,and has coefficients in K(P)=K(P1,...,P m).It is then natural to relate the degrees of these coefficients to the degree of the projection of W on the space of coordinates P1,...,P m,X1,...,X k,that is,W k.This is the object of ourfirst theorem.In all that follows,we call degree of a rational function the maximum of the degrees of its numerator and denomi-nator.Theorem1Let W be a variety satisfying Assumption1,and suppose that the generic solutions of W are represented by a triangular set T in K(P)[X].For k≤n,all coefficients in T k have degree bounded by(2k2+2)k(deg W k)2k+1.This result improves those of Gallo and Mishra[1990]and Szanto[1999]for re-spectively Ritt-Wu’s and Kalkbrener’s unmixed representations.If W is given as the zero-set of a system of n equations of degree d,then Gallo-Mishra’s bound is2n(8n)2n d(d+1)4n2and Szanto’s is d O(n2).With this notation,the B´e zout inequality(Theorem1in Heintz[1983])implies that the degree of W k is at most d n for all k.Thus according to Theorem1, for k≤n,in a worst-case scenario the coefficients in the polynomial T k have degree bounded by(2k2+2)k d2kn+n.Hence the estimate is better for low indices k than for higher indices;this contrasts with the previous results,which gave the same bounds for all T k.For the worst case k=n,our estimates are within the class d2n2+o(n2),to be compared with Gallo and Mishra’s bound of d4n2+o(n2).Any of these bounds are polynomial in d n2;we do not know if this is sharp.More importantly,Theorem1reveals that the degrees of the coefficients of T are controlled by the intrinsic geometric quantities deg W k,rather than by the degrees of a defining polynomial system.For instance,this indicates a good behavior with respect to decomposition,e.g.into irreducible.Also,these degrees may be bounded a priori:in the example presented in Subsection8.3, the B´e zout bound is1024,but an estimate based on the semantics of the problem gives deg W k≤80.Degree of the degeneracy locus.We still assume that the generic solu-tions of W are represented by a triangular set T=(T1,...,T n)in K(P)[X]. Since the coefficients of T are rational functions,there exists an open subset of the parameter space where they can be specialized,and give a description of thefibers ofπn.Theorem2below gives an upper bound on the degree of an hypersurface where this specialization fails.10Theorem2Let W be a variety satisfying Assumption1,and suppose that the generic solutions of W are represented by a triangular set T in K(P)[X]. There exists a polynomial∆W∈K[P]of degree at most(3n deg W+n2)deg W such that,if p∈A m(K)does not cancel∆W:(1)p cancels no denominator in the coefficients of(T1,...,T n).We denoteby(t1,...,t n)⊂K[X]these polynomials with coefficients specialized at p.(2)(t1,...,t n)is a radical ideal.Let Z n⊂A n(K)be the zero-set of the poly-nomials(t1,...,t n);then thefiber W n(p)is{p}×Z n⊂A m+n(K).Just as Theorem1,this result is of an intrinsic nature,since it depends only on geometric quantities.Nevertheless,in strong contrast with the previous result,these bounds are polynomial in the geometric degree of W.In particular,Theorem2shows that the reunion of the zero-sets of all de-nominators of the coefficients of T is contained in an hypersurface of degree bounded polynomially in terms of the degree of W.Thus,the zero-set of any such denominator has degree bounded by the same quantity.Theorem1does not give such a polynomial bound for the degrees of the denominators.Were the upper bounds of Theorem1to be sharp,this would indicate that these denominators are(high)powers of polynomials of moderate degree.Algorithms.The above results are purely geometric,and independent of any system of generators.For algorithmic considerations,we now assume that W is given as the zero-set of a polynomial system F=F1,...,F n in K[P,X]. We make the additional assumption that the Jacobian determinant with re-spect to X is invertible on a dense subset of W.Then Assumption1is satisfied, and we consider the problem of computing triangular sets that represent the generic solutions of W.The underlying paradigm is that solving a zero-dimensional system over K by means of triangular sets is a well-solved task.Thus,the basic idea isfirst to specialize the indeterminates P in the system F,and solve the corresponding system in the remaining variables X,by means of triangular sets in K[X].A lifting process then produces triangular sets with coefficients in a formal power series ring,from which we can recover the required information.Ourfirst contribution treats the case when W is irreducible:its generic so-lutions are then represented by a single triangular set T=(T1,...,T n),and we propose a probabilistic algorithm that computes T1,...,T k for any k.If W is not irreducible,we compute the minimal polynomial of X1modulo the extended ideal(F1,...,F n)in K(P)[X],using similar techniques.We do not treat the general question of computing a whole family of triangular11sets when W is not irreducible.From the practical point of view,this might not be a strong restriction:our results cover all the applications that we had to treat.We use the following complexity notations:•We suppose that F is given by a Straight-Line Program of size L,and that F1,...,F n have degree bounded by d.•We say that f is in O log(g)if there exists a constant a such that f is in O(g log(g)a)—this is sometimes also expressed by the notation f∈O˜(g).•M(D)denotes the cost of the multiplication of univariate polynomials of degree D,in terms of operations in the base ring.M(D)can be taken in O(D log D log log D),using the algorithm of Sch¨o nhage and Strassen[1971].We denote by C0a universal constant such that for any ring R,any integer D and any monic polynomial T in R[X]of degree D,all operations(+,×) in R[X]/(T)can be done in at most C0M(D)operations,see Chapter9 in[von zur Gathen and Gerhard,1999].We assume that there exists constants C1andαsuch that M(D)M(D )≤C1M(DD )log(DD )αholds for all D,D .This assumption is satisfied for all commonly used multiplication algorithms.•M s(D,M)denotes the cost of M-variate series multiplication at precision D.This can be taken less than M((2D+1)M)using Kronecker’s substitu-tion.If the basefield has characteristic zero,this complexity becomes linear in the size of the series,up to logarithmic factors;see[Lecerf and Schost, 2003,Theorem1].We assume that there exists a constant C2<1such that M s(D,M)≤C2M s(2D,M)holds for all D and M.This is the case for all commonly used estimates,for instance for the ones mentioned above.Apart from the above constants,the complexities below are stated in terms of the degrees D k of the rational functions that appear in the output,and the number D n.This number was defined earlier as the generic cardinality of thefibers W n(p);it is thus the generic number of solutions of the parametric system F.Theorem3Assume that W is irreducible.Let p,p be in K m;assume that a description of the zeros of the systems F(p,X),F(p ,X)by triangular sets is known.For k≤n,let D k be the maximum of the degrees of the coefficients of T1,...,T k.Then T1,...,T k can be computed withinO log (nL+n3)(C0C1)n M(D n)M s(4D k,m)+km2D n M(D k)M s(4D k,m−1) operations in K.The algorithm chooses3m−1values in K,including the coordinates of p and p .IfΓis a subset of K,and these values are chosen inΓ3m−1,then the algorithm fails for at most50n(k2+2)3k d6kn+4n|Γ|3m−2 choices.12。
Generalized parton distributions of the pion in a Bethe-Salpeter approach
Our paper is organized as follows. In section II we give some general definitions for scalar partons and we introduce our kinematical variables. In section III and IV we will define our approach for the scalar and the NJL model respectively. Section V presents our results and Section VI our conclusions.
Generalized parton distributions of the pion in a Bethe-Salpeter approach
S. Noguera,∗ L. Theußl,† and V. Vento‡
Departamento de F´ ısica Te` orica and Instituto de F´ ısica Corpuscular, Universidad de Valencia - CSIC, E-46100 Burjassot (Valencia), Spain. (Dated: February 8, 2008) We calculate generalized parton distribution functions in a field theoretic formalism using a covariant Bethe-Salpeter approach for the determination of the bound-state wave function. We describe the procedure in an exact calculation in scalar Electrodynamics proving that the relevant corrections outside our scheme vanish. We extend the formalism to the Nambu–Jona-Lasinio model, a realistic theory of the pion. We go in both cases beyond all previous calculations and discover that all important features required by general physical considerations, like symmetry properties, sum rules and the polynomiality condition, are explicitly verified. We perform a numerical study of their behavior in the weak and strong coupling limits.
Splitting an operator Algebraic modularity results for logics with fixpoint semantics
1. INTRODUCTION An important aspect of human reasoning is that it is often incremental in nature. When dealing with a complex domain, we tend to initially restrict ourselves to a small subset of all relevant concepts. Once these “basic” concepts have been figured out, we then build another, more “advanced”, layer of concepts on this knowledge. A quite illustrative example of this can be found in most textbooks on computer networking. These typically present a seven-layered model of the way in which computers communicate. First, in the so-called physical layer, there is only talk of hardware and concepts such as wires, cables and electronic pulses. Once these low-level issues have been dealt with, the resulting knowledge becomes a fixed base,
Splitting an operator: Algebraic modularity results for logics with fixpoint semantics
Universal data access through OLE DB
TRANSACTIONS OF THEAMERICAN MATHEMATICAL SOCIETYVolume358,Number10,October2006,Pages4225–4250S0002-9947(06)03960-2Article electronically published on May9,2006PALEY–WIENER THEOREMS FOR THE DUNKL TRANSFORMMARCEL DE JEUAbstract.We conjecture a geometrical form of the Paley–Wiener theoremfor the Dunkl transform and prove three instances thereof,by using a reductionto the one-dimensional even case,shift operators,and a limit transition fromOpdam’s results for the graded Hecke algebra,respectively.These Paley–Wiener theorems are used to extend Dunkl’s intertwining operator to arbitrarysmooth functions.Furthermore,the connection between Dunkl operators and the Cartan mo-tion group is established.It is shown how the algebra of radial parts of in-variant differential operators can be described explicitly in terms of Dunkloperators.This description implies that the generalized Bessel functions coin-cide with the spherical functions.In this context of the Cartan motion group,the restriction of Dunkl’s intertwining operator to the invariants can be inter-preted in terms of the Abel transform.We also show that,for certain valuesof the multiplicities of the restricted roots,the Abel transform is essentiallyinverted by a differential operator.1.Introduction and overviewIn recent times the study of special functions associated with root systems has developed to a considerable degree.Starting with a number of conjectures by Mac-donald,and the work of Heckman and Opdam on multivariable hypergeometric functions in the late1980’s,the development of the theory was greatly enhanced by the introduction of rational Dunkl operators by Dunkl[7].Through various intermediate steps of generalization,these operators can even be said to have ulti-mately provided crucial building blocks for Cherednik’s work on double affine Hecke algebras and the q-Macdonald conjectures.Originally,before the introduction of Dunkl operators,the idea when studying special functions related to root systems was to consider root multiplicities in the theory of spherical functions on Lie groups as parameters,and then to develop a theory for Weyl group invariant objects in this more general situation,without the aid of the presence of the group.It was this point of view which underlay the Macdonald conjectures and which led Heckman and Opdam to the development of their theory of hypergeometric functions in higher dimension.One of the main technical problems in this context is the description of the generalized radial parts of invariant differential operators.Apart from an explicit formula for the generalized radial part of the Laplacian—an expression which was in fact the starting point for Heckman and Opdam—the other operators remain somewhat intangible.Received by the editors April23,2004.2000Mathematics Subject Classification.Primary33C52;Secondary43A32,33C80,22E30.Key words and phrases.Dunkl operator,Paley–Wiener theorem,graded Hecke algebra,Cartan motion group,spherical function,multivariable Bessel function.c 2006American Mathematical SocietyReverts to public domain28years from publication42254226MARCEL DE JEUThis problem disappeared when Dunkl found the operators which have come to bear his name.They are parameterized deformations of the ordinary derivatives,in-volving afinite reflection group,for which it is still relatively easy to study the spec-tral problem and develop the theory of the corresponding Fourier transformation—the Dunkl transform.The invariant part of the theory then answers questions in a generalized theory of spherical functions—for the Cartan motion group,to be precise—as described above.The operators under consideration being explicitly given,it is actually easier to study the general context of Dunkl operators,and then specialize to invariant objects later on,than it is to restrict oneself to the invariants from the outset.The same holds true for the modification of Dunkl op-erators into Cherednik operators,which gives rise to a representation of the graded Hecke algebra[28]and the non-invariant“envelope”of the work of Heckman and Opdam on hypergeometric functions.Quite remarkably,the theory of spherical functions in analysis on Lie groups can in a number of situations thus be regarded as only the invariant part of a general non-invariant theory for Dunkl-type operators.At the time of this writing,it is unknown to the author whether there is an underlying reason for this phenomenon.In view of all this,these Dunkl operators and their modifications have attracted considerable attention in various areas of mathematics and mathematical physics during the last decade.To get an impression of their influence on the development of special functions associated with root systems—also in the general non-invariant context—we refer to,e.g.,[18],[29],[16],[31].For their use in the study of integrable quantum many body systems of Calogero–Moser–Sutherland type we refer to[5] and the bibliography therein.In this paper,we are mainly concerned with the further development of the general theory of the Dunkl transform,notably with the Paley–Wiener theorem.In addition,we describe the relation between Dunkl operators and the Cartan motion group.We will now turn to an overview of the contents of this paper with regard to these two subjects.Thefirst substantial results for the Dunkl transform,i.e.,the Plancherel the-orem and inversion theorem,were obtained by Dunkl[10]and the author[22]. Two Paley–Wiener theorems were established in[23,Chapter3](unpublished),of which this paper is an extension.Whereas the proof of the inversion theorem in [22]is partly a formal argument based on various symmetry properties of the eigen-functions of Dunkl’s operators,it will become apparent below that quite some more work is required in Paley–Wiener theory,due to the lack of adequate asymptotic re-sults in the spectral domain.We will establish three Paley–Wiener theorems,each of these being a special case of a conjectured geometrically more precise general Paley–Wiener theorem,which can be found below as Conjecture4.1.Each version requires a different technique.Thefirst and most general version relies on a reduc-tion to the one-dimensional even case—where asymptotic results are available—and some non-trivial results from general representation theory,and from the represen-tation theory of the orthogonal group.The second version,which is proved for a discrete set of parameters only,is established through shift operators.Finally,the third version,which we prove for Weyl groups,follows from a limit transition from results of Opdam[28].This limit transition has some interest in itself,and for a discrete set of parameters its validity has also been established by Ben Sa¨ıd and Ørsted;cf.[2]and[3].PALEY–WIENER THEOREMS FOR THE DUNKL TRANSFORM4227 Turning to the Cartan motion group(where the material is taken from[23]), we will show how the algebra of radial parts of invariant differential operators can be described explicitly in terms of Dunkl operators with suitable values of the parameters.From this description it is easy to see that the invariant components of the eigenfunctions of the Dunkl operators are precisely the spherical functions. We can then also describe Dunkl’s intertwiner operator,or rather its restriction to the invariants,in terms of the Abel transform.For certain multiplicities,the results on shift operators imply that this Abel transform is essentially inverted by a differential operator.The organization of this paper is as follows.In Section2we establish the neces-sary general definitions and notations,and we recall some previous results.Section3 contains a number of useful formulas.We do not claim any originality in particular for these formulas,but we believe that the proofs are considerably simpler than the ones in the existing literature.The conjectured Paley–Wiener theorem can be found in Section4,as can the proven three instances of it that were described above.These Paley–Wiener theorems are then used in Section5to extend Dunkl’s intertwining operator to the smooth functions.To conclude,Section6contains the details of the connection between Dunkl operators and the Cartan motion group, including the results on the Abel transform.2.Notations and previous resultsLet a be a real vector space offinite dimension N which is equipped with an inner product(.,.),inducing a Lebesgue measure dx on a.We let a C=a⊗R C denote the complexification of a,and we extend the form(.,.)to a bilinear form on a C,again denoted by(.,.).Both a and a C can be identified with their duals via (.,.);forξin a or a C the corresponding linear functional is then denoted byξ∗. Define the orthogonal group O(N)=O(a,(.,.)).The norm|.|which is induced on a by(.,.)extends to an O(N)-invariant norm on a C,also denoted by|.|.There is a natural action of O(N)on functions:(g·f)(x)=f(g−1x).Let G⊂O(N)be afinite(real)reflection group with corresponding root system R.We may and will assume that(α,α)=2for allα∈R.Ifα∈R,then rαis the orthogonal reflection in the hyperplane perpendicular toα.We choose andfix a positive system R+in R.A function k:R→C is called a multiplicity function if k is G-invariant.We write k≥0if all values of k are non-negative,with analogous notations Re k≥0 and k>0.We let K denote the vector space of multiplicity functions on R.Let k∈K andξ∈a.Then the corresponding Dunkl operator Tξ(k)is defined bykα(α,ξ)M(α∗)−1(1−rα).Tξ(k)=∂ξ+α∈R+Here∂ξ(ξ∈a)is the(unnormalized)directional derivative operator;we have used the notation M f for pointwise multiplication by a function f.The definition of the operators is independent of the choice of the positive sys-tem.The Tξ(k)leave the polynomials P invariant,mapping the homogeneous polynomials P n of degree n into P n−1.Furthermore,the spaces C∞(a)of smooth4228MARCEL DE JEUfunctions,D (a )of compactly supported smooth functions and S (a )of rapidly de-creasing smooth functions are also invariant [22,Lemma 2.1].Quite remarkably,the T ξ(k )form a commutative family:T ξ(k )T η(k )=T η(k )T ξ(k )for all ξ,η∈a and all k ∈K ,as was proved by Dunkl [7];see [14]for a different proof.As a consequence of the commutativity the map ξ→T ξ(k )extends to P ;for p ∈P the corresponding operator is then denoted by T p (k ).The important k -Laplacian T |.|2(k )is denoted by ∆k .For k ≥0,Dunkl [8]has constructed a linear isomorphism V k :P →P ,homoge-neous of degree 0,such that V k 1=1and T ξ(k )V k =V k ∂ξ(ξ∈a ).An alternative approach for more general k can be found in [14].A concrete description of this intertwiner operator V k is presently still unknown,with the exception of the one-dimensional case [8]and the case A 2[11].Significant abstract results were obtained by R¨o sler [30],who showed,amongst others,that V k is for k ≥0a positive operator which can be described in terms of measures.Let λ∈a C and consider the simultaneous eigenfunction problem(2.1)T ξ(k )f =(λ,ξ)f (ξ∈a ).This problem was studied first by Dunkl [9]for k ≥0;later Opdam [27]treated the general case.One more definition is needed to state the general result:a multiplicity function k ∈K is said to be singular if the simultaneous kernel of {T ξ(k )}ξ∈a in P is non-trivial,i.e.,if it properly contains the constants.We will use the self-evident notations K sing and K reg .The set K sing has been determined in all cases [14],and some partial information about the simultaneous kernel for singular multiplicities can also be found in [loc.cit.].The general nature of the simultaneous kernel for singular multiplicities is,however,still unknown,with the exception of the case A n of the symmetric groups which has been solved by Dunkl [12,13].In this paper we will mainly be concerned with multiplicities satisfying Re k ≥0.Such multiplicities are regular,as is most easily seen by considering the operator N i =1e ∗i T e i (k )for an orthonormal basis {e 1,...,e N }of a [14].For regular multiplicities,the result from [27]for the eigenfunction problem is as follows.Theorem 2.1.For all k ∈K reg the eigenfunction problem (2.1)has a 1-dimension-al solution space for all λ∈a C .This space contains a (unique )function Exp G (λ,k,.)such that Exp G (λ,k,0)=1.Furthermore,Exp G (λ,k,.)extends to a holomorphic function on a C ,andExp G (.,.,.):a C →Cis a meromorphic function with poles precisely in K sing .In order to be able to define the corresponding Fourier transform—the Dunkl transform—for sufficiently general functions,one needs non-trivial bounds for the eigenfunctions.It is shown in [22]that for Re k ≥0one has (2.2)|Exp G (λ,k,z )|≤ |G |exp(max g ∈GRe (gλ,z ))(λ,z ∈a C ),in particular(2.3)|Exp G (iλ,k,x )|≤ |G |(λ,x ∈a ).If k ≥0,then the constant |G |in (2.2)and (2.3)can in fact be improved to 1,as a consequence of R¨o sler’s results on the intertwiner operator [30].PALEY–WIENER THEOREMS FOR THE DUNKL TRANSFORM 4229Following Dunkl,we define for Re k ≥0the G -invariant complex-valued weight function w k = α∈R +|α∗|2k α.The T ξ(k )are anti-symmetric with respect to this weight function:if f ∈S (a )and g is smooth such that both g and T ξ(k )g are of at most polynomial growth,then (2.4) a (T ξ(k )f )gw k dx =−af (T ξ(k )g )w k dx.In view of (2.3),the Dunkl transform for Re k ≥0is meaningfully defined on L 1(a ,|w k (x )|dx )as D k f (λ)=1c k af (x )Exp G (−iλ,k,x )w k (x )dx (λ∈a ,f ∈L 1(a ,|w k (x )|dx )).Its alleged inverse is E k f (x )=1c k af (λ)Exp G (iλ,k,x )w k (λ)dλ(x ∈a ,f ∈L 1(a ,|w k (λ)|dλ)).Here the normalizing constant c k is the Mehta integral,which is defined as c k = ae −|x |22w k (x )dx.There is a closed expression for this integral;this former Macdonald conjecture has been proved by Opdam,first for Weyl groups [26]and later for finite reflection groups in general [27].For our needs it suffices to know that c k =0if Re k ≥0,which can be proved by more elementary means [22,Corollary 4.17].The first results for the transform,and notably a version of the Plancherel the-orem,were obtained by Dunkl [10].Later on,a more systematic study was under-taken in [22].The main properties and results are as follows.Theorem 2.2.Let Re k ≥0and ξ∈a .Then:(1)D k T ξ(k )f =M iξ∗D k f (f ∈S (a )).(2)E k T ξ(k )f =−M iξ∗E k f (f ∈S (a )).(3)D k M iξ∗f =−T ξ(k )D k f (f ∈S (a )).(4)E k M iξ∗f =T ξ(k )E k f (f ∈S (a )).(5)D k is a linear homeomorphism of S (a ),with inverseE k .(6)If f ∈L 1(a ,|w k (x )|dx )and D k f ∈L 1(a ,|w k (x )|dx ),then D k E k f =E k D k f=f a.e.(7)If k ≥0,then D k maps L 1(a ,w k (x )dx )∩L 2(a ,w k (x )dx )into L 2(a ,w k (x )dx ),isometrically with respect to the two-norm corresponding to w k (x )dx ,and extends uniquely from L 1(a ,w k (x )dx )∩L 2(a ,w k (x )dx )to a unitary oper-ator on L 2(a ,w k (x )dx ).Remark 2.3.The definition of Dunkl operators can be generalized to complex re-flection groups [15].3.FormulariumIn this section we establish some useful formulas.Some of them are known from the work of Dunkl,who based his proofs to a large extent on the existence of a k -harmonic decomposition for polynomials [6,Theorem 1.7].However,with the4230MARCEL DE JEUbenefit of hindsight we can simplify some of the original proofs considerably,by systematically exploiting the following commutator relation [7,Proposition 2.2]:(3.1) M ξ∗,∆k 2 =−T ξ(k )(ξ∈a ).To start,we first note that ∆k :P →P is homogeneous of degree −2,hencelocally nilpotent.This enables us to define the linear automorphisms e ±∆k 2of P .These automorphisms will play an important part in what follows,together with the Gaussian ψ(x )=exp(−|x |2/2).It is known [10],[22]that the Gaussian is an eigenfunction of both D k and E k with eigenvalue 1.Lemma 3.1.Let ξ∈a .Then for arbitrary k we have (1)in End C (P ): M ξ∗,e −∆k 2 =T ξ(k )◦e −∆k 2;(2)in Hom C (P ,S (a )):T ξ(k )◦M ψ◦e −∆k 2=−M ψ◦e −∆k 2◦M ξ∗.Proof.The first part follows immediately from (3.1)and the obvious fact that[∆k ,T ξ(k )]=0.As to the second part,note that T ξ(k )◦M ψ=M ψ◦T ξ(k )−M ψ◦M ξ∗,as a consequence of the G -invariance of ψ.HenceT ξ(k )◦M ψ◦e −∆k 2=M ψ◦T ξ(k )◦e −∆k 2−M ψ◦M ξ∗◦e −∆k 2=M ψ◦T ξ(k )◦e −∆k 2−M ψ◦ M ξ∗,e −∆k 2 +e −∆k 2◦M ξ∗ =M ψ◦T ξ(k )◦e −∆k 2−M ψ◦ T ξ(k )◦e −∆k 2+e −∆k 2◦M ξ∗ =−M ψ◦e −∆k 2◦M ξ∗.Repeated application of the second part of Lemma 3.1yields the formula(3.2)T p (k ) e−∆k 2q ψ =(−1)deg p e −∆k 2(pq ) ψ,for homogeneous p ∈P and arbitrary q ∈P .Taking q =1this implies,together with Theorem 2.2,the following result,which is equivalent to [10,Proposition 2.1].Corollary 3.2.If p ∈P is homogeneous and Re k ≥0,then (1)D k (pψ)=(−i )deg p e −∆k 2p ψ.(2)E k (pψ)=i deg p e −∆k 2p ψ.The second part of Lemma 3.1also enables us to reprove the symmetry of a bilinear form which was introduced by Dunkl [9],as follows.For p,q ∈P ,put (p,q )k =(T p (k )q )(0).Although it is not obvious from the definition,this bilin-ear form on P is actually symmetric.This symmetry follows from the following generalization by Dunkl [loc.cit.]of a result of Macdonald:(3.3)(p,q )k =1c k ae −∆k 2p e −∆k 2q ψw k dx (Re k ≥0,p,q ∈P ),PALEY–WIENER THEOREMS FOR THE DUNKL TRANSFORM 4231in which the right-hand side is obviously symmetric.In order to re-establish (3.3),denote the right-hand side by [p,q ]k .Now (2.4)and Lemma 3.1imply that [p,T ξ(k )q ]k =1c k a e −∆k 2p T ξ(k )e −∆k 2q ψw k dx =1c k a −T ξ(k ) e −∆k 2p ψ e −∆k 2q w k dx =1c k ae −∆k 2(M ξ∗p ) e −∆k 2q ψw k dx =[M ξ∗p,q ]k .But the other form (.,.)k also has this property:(M ξ∗p,q )k =(p,T ξ(k )q )k ,as a direct consequence of its definition.Since it is easy to see that (1,q )k =[1,q ]k ,an induction with respect to deg p then proves that (p,q )k =[p ,q ]k ,which is (3.3).The following proposition will be used in the reduction of the proof of the Paley–Wiener theorem,Theorem 4.10,to the one-dimensional even case.Proposition 3.3.If Re k,Re k ≥0,then E k D k (pψ)= e −∆k 2e ∆k 2p ψ(p ∈P ).Proof.We may assume that p is homogeneous.Then Corollary 3.2implies that E k D k (pψ)=(−i )deg p E k ∞ n =01n ! −∆k 2 np ψ =(−i )deg p ∞ n =0i deg p −2n n !e −∆k 2 −∆k 2 n pψ= e −∆k 2e ∆k 2p ψ.To conclude this section,we re-establish a formula of Heckman [17]which ex-presses the pivotal role of the k -Laplacian ∆k :(3.4)T p (k )=1n ! ad ∆k 2n M p ,for p ∈P n .The proof in [loc.cit.]is based on representation theory for sl(2),but it can also be seen directly,as follows.We may assume that M p = n i =1M ξ∗i .Then ad ∆k 2 n M p = j i =nn !j 1!···j n ! ad ∆k 2 j 1M ξ∗1 ◦···◦ ad ∆k 2 j n M ξ∗n .But (ad ∆k )2M ξ∗i =0as a consequence of (3.1)and the commutativity of the T ξ(k ),so the only surviving term in the summation is the one with j 1=...=j n =1.Using (3.1)once more,this proves (3.4).4.Paley–Wiener theoremsIn this section,we conjecture a geometrical form of the Paley–Wiener theorem for the Dunkl transform,and present several theorems to support it.Establishing notation,for S ⊂a we let D (S )denote the smooth compactly supported functions with support contained in S .If S is compact and non-empty,then we define the indicator I S :a →R as I S (x )=max y ∈S (x,y )for x ∈a .For4232MARCEL DE JEUsuch S,let H S be the functions on a C of Paley–Wiener type corresponding to S, i.e.,those entire functions with the property that for each integer M≥0there exists a constantγM such that|f(λ)|≤γM(1+|λ|)−M exp I S(Imλ)for allλ∈a C. We then conjecture the following.Conjecture4.1(Paley–Wiener conjecture).Let G be afinite reflection group.If Re k≥0and S is a non-empty G-invariant compact convex subset of a,then D k is a linear isomorphism between D(S)and H S.There is some evidence supporting this conjecture:•The inversion theorem and[22,Corollary4.10]show that D k is an injective map from D(S)into H S.•For k=0the statement holds[21,Theorem7.3.1].•If S is a ball centered at the origin,then a reduction to the one-dimensional even case,where asymptotics can be used,enables us to establish the state-ment as Theorem4.10below.•If the kαare all strictly positive integers,then the statement can be estab-lished,using shift operators,as Theorem4.11below.•If G is a Weyl group and S is the intersection of convex hulls of orbits, then a limit transition from results of Opdam establishes the statement as Theorem4.15below.The main obstacle for a possible proof of the conjecture along the usual lines, using a contour shift,is the absence of adequate asymptotic results for the Dunkl kernel.There are some asymptotic results available[32],but these fall far short of what is needed.It is to be expected that better results could be obtained if more was known about R¨o sler’s representing measures[30],but as yet these remain elusive.But even if much stronger asymptotic results became available,the proof of Theorem4.4below seems to suggest that additional monodromy arguments maythen still be necessary.Remark 4.2.In[40],a proof of Conjecture4.1if k≥0andα∈R+kα>0ispresented.That proof,however,is not correct,and to our knowledge Conjecture4.1 is at the time of writing still open.4.1.The case of arbitrary G and Re k≥0.Throughout this section,B R will denote the closed ball in a with radius R and the origin as center.The space D(B R) carries the usual Fr´e chet topology of uniform convergence of all derivatives.Our approach of the Paley–Wiener theorem for such sets—under the assumption that Re k≥0—consists of three steps;cf.the proof of Theorem4.10.First,we prove the result for even functions in one dimension.Second,it is shown that this implies the theorem for radial functions in arbitrary dimension.In the third step wefinally prove that the result for radial functions implies the theorem for general smooth functions with support in a closed ball B R.Remark4.3.The special role of radial functions in the theory has been noted by several authors[23],[41],[33],[39],e.g.,the Dunkl transform of a radial function is again radial[33].In Paley–Wiener theory this phenomenon is encountered once more,when reducing the radial case in arbitrary dimension to the even case in one dimension.The one-dimensional even case can be handled by either Weyl fractional integral operators[39]or by our approach below,which uses classical results for the asymptotics of Bessel functions combined with a contour shift.Thefinal stepPALEY–WIENER THEOREMS FOR THE DUNKL TRANSFORM4233 in our approach,from the radial to the general case,and for which a reference to[23]was given in[39],is in a sense the deepest,since it ultimately rests on the general theory of representations of compact groups in Fr´e chet spaces.Recently,an alternative reduction to the one-dimensional case was introduced which circumvents the inference of the general case from the radial one[36].This reduction is based on results for k-harmonic polynomials,and for functions supported in B R it gives a proof of the Paley–Wiener theorem which is independent of the results in the present paper.Starting with the proof,the one-dimensional even case is settled in the following theorem,where,as usual,invariance is denoted by superscripts.Theorem4.4.Let a be one dimensional and Re k≥0.If f∈H Z2B R ,then E k f∈D(B R)Z2.Proof.Let f be as in the statement.Then obviously E k f is an even rapidly de-creasing smooth function.We have to show that∞−∞f(λ)Exp Z2(iλ,k,x)|λ|2k dλ=0if x>R.Fix such x.From[10]we haveExp Z2(iλ,k,x)=Γ(k+12)λx212−kJ k−12(λx)+iJ k+12(λx).Using the invariance of f we see that we are left to show that(4.1)∞−∞f(λ)(λx)12−k J k−12(λx)|λ|2k dλ=0.This expression makes it obvious that there are two obstructions for a direct appli-cation of the classical argument of shifting the contour to infinity.First,the Bessel function has exponential growth in both the positive and negative imaginary di-rections,and second,the weight function|λ|2k is in general not the restriction of a holomorphic function on the upper or lower half-plane.So we proceed indirectly.Recall the definition of the Bessel functions of the third kind:H(1)ν=J−ν−e−νπi Jνi sinνπ,H(2)ν=J−ν−eνπi Jν−i sinνπ,so thatJν=H(1)ν+H(2)ν2.Ifνis an integer,then a limit has to be taken.In our case,this occurs if k is a half-integer,but by continuity in k we may and will assume that this is not the case.For our purposes,the important property of these functions is the asymptotic behaviour[1,9.2.7]:(4.2)H(1)ν(z)=2πze i(z−νπ2−π4)1+O1z,4234MARCEL DE JEUvalid if−π<arg z<2π(which is to be interpreted in the sense of analytic continuation).Note that the range of validity of this asymptotic development contains the entire upper half-plane and that(in contrast to the ordinary Bessel function)this Hankel function has exponential decrease in the positive imaginary direction.Defineφ(1),φ(2):(0,∞)→C byφ(1)(λ)=(λx)12−k H(1)k−12(λx)λ2kandφ(2)(λ)=(λx)12−k H(2)k−12(λx)λ2k.Letφ(1)c denote the analytic continuation ofφ(1)from(0,∞)to C\{it|t≤0}.If one recalls that Jν(z)=zν˜Jν(z)with˜Jνentire and Z2-invariant,one notes that φ(1)c(λ)remains bounded asλ→0in C\{it|t≤0},since Re k≥0.A small computation will also make it clear thatφ(1)c(λ)=φ(2)(−λ)forλ<0.Using the invariance of f and the weight function,we then compute as follows: ∞−∞f(λ)(λx)12−k J k−12(λx)|λ|2k dλ=2∞f(λ)(λx)12−k J k−12(λx)λ2k dλ=∞f(λ)(λx)12−kH(1)k−12(λx)+H(2)k−12(λx)λ2k dλ=∞f(λ)φ(1)(λ)+φ(2)(λ)dλ=lim↓0∞f(λ)φ(1)(λ)dλ+∞f(λ)φ(2)(λ)dλ=lim↓0∞f(λ)φ(1)(λ)dλ+−−∞f(−λ)φ(2)(−λ)dλ=lim↓0∞f(λ)φ(1)c(λ)dλ+−−∞f(λ)φ(1)c(λ)dλ.Now fφ(1)c is holomorphic on C\{it|t≤0},and,since x>R,it has exponential decrease in the positive imaginary direction,as a consequence of(4.2).The classical argument therefore establishes(4.1),with a minor modification involving a semi-circle of radius around0in the upper half-plane and using the fact that fφ1c is bounded around0.Next,we proceed with the reduction of the general radial case to the one-dimensional even case.To this end,fix x0=0in a,and define the map Res x0: S(a)O(N)→S(R)Z2by restricting to the line passing through x0:(Res x0f)(s)=fsx0|x0|s∈R,f∈S(a)O(N).The following proposition,which implies the Paley-Wiener theorem for radial functions(as will become apparent in the proof of Theorem4.10),involves the Dunkl transform for general a and R both at the same time.We therefore add aPALEY–WIENER THEOREMS FOR THE DUNKL TRANSFORM4235 subscript Z2in the latter case for clarity.Let S(a)O(N)denote the O(N)-invariantsin S(a).Proposition4.5.Suppose Re k≥0.Let f∈S(a)O(N)and putγ=α∈R+kα.ThenRes x0E k D0f=Eγ,Z2D0,Z2Res xffor all non-zero x0∈a.Proof.Let S0(a)={pψ|p∈P},withψdenoting the Gaussian as in Section3.It is known that S0(a)is dense in S(a);see[35,p.263]for this particular result,or[24] for a general framework for this type of problem.Note that the canonical projection from S(a)onto S(a)O(N)is continuous as a consequence of the closed graph theorem, implying that S0(a)O(N)is dense in S(a)O(N).It is therefore,by linearity and continuity,sufficient to prove the proposition for a function f of the form|x|2qψ, where q is a non-negative integer.This can be done using Proposition3.3and an identity for Laguerre polynomials.Recall the definition:L(α) n (x)=nm=0(−1)mm!n+αn−mx m.Then the following identity holds[1,22.12.6]:(4.3)nm=0L(α)m(x)L(β)n−m(y)=L(α+β+1)n(x+y).It is known[9,Proposition 3.9]that e∆02|x|2q=2q q!L(N/2−1)q−|x|2/2ande−∆k2|x|2q=(−2)q q!L(N/2+γ−1)q|x|2/2.Using Proposition3.3we thereforefindE k D0|x|2qψ=e−∆k2e∆02|x|2qψ=2q q!qm=0(−1)mN/2+q−1q−mL(N/2+γ−1)m|x|2/2ψ=(−2)q q!qm=0−N/2−mq−mL(N/2+γ−1)m|x|2/2ψ.The special case n=q,y=0,α=N/2+γ−1andβ=−N/2−q of(4.3)thenshows thatE k D0|x|2qψ=(−2)q q!L(γ−q)q|x|2/2ψ.Curiously enough,the dimension N has dropped out,and the proposition follows immediately from this observation.It follows easily from this result that the Dunkl transform of a radial function is again radial,retrieving this result from[33].Let D0(B R)=Span{T p(0)f|p∈P,f∈D(B R)O(N)}.The following density result is crucial in the step from the radial to the general case.Proposition4.6.D0(B R)is dense in D(B R).This proposition implies a special case of[20,Cor.7.8,p.310],but the latter result does not seem to imply the proposition.The proof of Proposition4.6is based on representation theory for compact groups in general and O(N)in particular;we recall a few facts to start with.。
Informed and Strategic Order Flow in the Bond Markets
RFS Advance Access published August 27, 2007 Informed and Strategic Order Flow in theBond MarketsPaolo PasquarielloRoss School of Business,University of MichiganClara VegaSimon School of Business,University of Rochester and FRBG We study the role played by private and public information in the process of price formation in the U.S.Treasury bond market.To guide our analysis,we develop a parsimonious model of speculative trading in the presence of two realistic market frictions—information heterogeneity and imperfect competition among informed traders—and a public signal.We test its equilibrium implications by analyzing the response of two-year,five-year,and ten-year U.S.bond yields to orderflow and real-time U.S.macroeconomic news.Wefind strong evidence of informational effects in the U.S.Treasury bond market:unanticipated orderflow has a significant and permanent impact on daily bond yield changes during both announcement and nonannouncement days.Our analysis further shows that,consistent with our stylized model,the contemporaneous correlation between orderflow and yield changes is higher when the dispersion of beliefs among market participants is high and public announcements are noisy.(JEL E44;G14)Identifying the causes of daily asset price movements remains a puzzling issue infinance.In a frictionless market,asset prices should immediately adjust to public news surprises.Hence,we should observe price jumps only during announcement times.However,asset pricesfluctuate significantly during nonannouncement days as well.This fact has motivated the introduction of various market frictions to better explain the behavior The authors are affiliated with the Department of Finance at the Ross School of Business,University of Michigan(Pasquariello)and the University of Rochester,Simon School of Business and the Federal Reserve Board of Governors(Vega).We are grateful to the Q Group forfinancial support.We benefited from the comments of an anonymous referee,Sreedhar Bharath,Michael Brandt,Michael Fleming, Michael Goldstein,Clifton Green,Joel Hasbrouck(the editor),Nejat Seyhun,Guojun Wu,Kathy Yuan, and other participants in seminars at the2005European Finance Association meetings in Moscow, the2006Bank of Canada Fixed Income Markets conference in Ottawa,the2007American Finance Association meetings in Chicago,Federal Reserve Board of Governors,George Washington University, the University of Maryland,the University of Michigan,the University of Rochester,and the University of Utah.Any remaining errors are our own.Address correspondence to Paolo Pasquariello,Ross School of Business,University of Michigan,701Tappan Street,Room E7602,Ann Arbor,MI48109-1234,or e-mail:ppasquar@ and Clara.Vega@.©The Author2007.Published by Oxford University Press on behalf of The Society for Financial Studies. All rights reserved.For Permissions,please email:journals.permissions@.doi:10.1093/rfs/hhm034The Review of Financial Studies/v20n52007of asset prices.One possible friction is asymmetric information.1When sophisticated agents trade,their private information is(partially)revealed to the market,via orderflow,causing revisions in asset prices even in the absence of public announcements.The goal of this article is to theoretically identify and empirically measure the effect of these two complementary mechanisms responsible for daily price changes:aggregation of public news and aggregation of orderflow. In particular,we assess the relevance of each mechanism conditional on the dispersion of beliefs among traders and the public signals’noise.To guide our analysis,we develop a parsimonious model of speculative trading in the spirit of Kyle(1985).The model builds upon two realistic market frictions:information heterogeneity and imperfect competition among informed traders(henceforth,speculators).In this setting,more diverse information among speculators leads to lower equilibrium market liquidity,since their trading activity is more cautious than if they were homogeneously informed,thus making the market makers(MMs)more vulnerable to adverse selection.We then introduce a public signal and derive equilibrium prices and trading strategies on announcement and nonannouncement days.The contribution of the model is two-fold.To our knowledge,it provides a novel theoretical analysis of the relationship between the trading activity of heterogeneously informed,imperfectly competitive speculators,the availability and quality of public information, and market liquidity.Furthermore,its analytically tractable closed-form solution,in terms of elementary functions,generates several explicit and empirically testable implications on the nature of that relationship.2In particular,we show that the availability of a public signal improves market liquidity(the more so the lower that signal’s volatility)since its presence reduces the adverse selection risk for the MMs and mitigates the quasimonopolistic behavior of the speculators.1According to Goodhart and O’Hara(1997,p.102),‘‘one puzzle in the study of asset markets,either nationally or internationally,is that so little of the movements in such markets can be ascribed to identified public‘news.’In domestic(equity)markets thisfinding is often attributed to private information being revealed.’’This friction has been recently studied by Fleming and Remolona(1997,1999),Brandt and Kavajecz(2004)and Green(2004)in the US Treasury bond market,by Andersen and Bollerslev(1998) and Evans and Lyons(2002,2003,2004)in the foreign exchange market,by Berry and Howe(1994)in the US stock market,and by Brenner et al.(2005)in the US corporate bond market,among others.2Foster and Viswanathan(1996)and Back et al.(2000)extend Kyle(1985)to analyze the impact of competition among heterogeneously informed traders on market liquidity and price volatility in discrete-time and continuous-time models of intraday trading,respectively.Foster and Viswanathan(1993)show that,when the beliefs of perfectly informed traders are represented by elliptically contoured distributions, price volatility and trading volume depend on the surprise component of public information.Yet,neither model’s equilibrium is in closed-form,except the(analytically intractable)inverse incomplete gamma function in Back et al.(2000).Hence,their implications are sensitive to the chosen calibration parameters. Further,neither model,by its dynamic nature,generates unambiguous comparative statics for the impact of information heterogeneity or the availability of public information on market liquidity.Finally,neither model can be easily generalized to allow for both a public signal of the traded asset’s payoff and less than perfectly correlated private information.2Informed and Strategic Order Flow in the Bond MarketsThis model is not asset-specific,that is,it applies to stock,bond,and foreign exchange markets.In this study,we test its implications for the US government bond market for three reasons.First,Treasury market data contains signed trades;thus,we do not need to rely on algorithms[e.g.Lee and Ready(1991)]that add measurement error to our estimates of order flow.Second,government bond markets represent the simplest trading environment to analyze price changes while avoiding omitted variable biases.For example,most theories predict an unambiguous link between macroeconomic fundamentals and bond yield changes,with unexpected increases in real activity and inflation raising bond yields[e.g.Fleming and Remolona(1997)and Balduzzi et al.(2001),among others].In contrast, the link between macroeconomic fundamentals and the stock market is less clear[e.g.Andersen et al.(2004)and Boyd et al.(2005)].Third,the market for Treasury securities is interesting in itself since it is among the largest,most liquid USfinancial markets.Our empirical results strongly support the main implications of our model.During nonannouncement days,adverse selection costs of unanticipated orderflow are higher when the dispersion of beliefs—measured by the standard deviation of professional forecasts of macroeconomic news releases—is high.For instance,we estimate that a one standard deviation shock to abnormal orderflow decreases two-year,five-year,and ten-year bond yields by7.19,10.04,and6.84basis points, respectively,on high dispersion days compared to4.08,4.07,and2.86 basis points on low dispersion days.These differences are economically and statistically significant.Consistently,these higher adverse selection costs translate into higher contemporaneous correlation between order flow changes and bond yield changes.For example,the adjusted R2of regressing dailyfive-year Treasury bond yield changes on unanticipated orderflow is41.38%on high dispersion days compared to9.65%on low dispersion days.Intuitively,when information heterogeneity is high,the speculators’quasimonopolistic trading behavior leads to a‘‘cautious’’equilibrium where changes in unanticipated orderflow have a greater impact on bond yields.The release of a public signal,a trade-free source of information about fundamentals,induces the speculators to trade more aggressively on their private information.Accordingly,wefind that the correlation between unanticipated orderflow and day-to-day bond yield changes is lower during announcement days.For example,comparing nonannouncement days with Nonfarm Payroll Employment release dates,the explanatory power of orderflow decreases from15.31%to6.47%,21.03%to19.61%,and6.74% to3.59%for the two-year,five-year,and ten-year bonds,respectively.Yet, when both the dispersion of beliefs and the noise of the public signal —measured as the absolute difference between the actual announcement and its last revision—are high,the importance of orderflow in setting3The Review of Financial Studies/v20n52007bond prices increases.All of the above results are robust to alternative measures of the dispersion of beliefs among market participants,as well as to different regression specifications and the inclusion of different control stly,our evidence cannot be attributed to transient inventory or portfolio rebalancing considerations,since the unanticipated government bond orderflow has a permanent impact on yield changes during both announcement and nonannouncement days in the sample.Our article is most closely related to two recent studies of orderflow in the U.S.Treasury market.Brandt and Kavajecz(2004)find that orderflow accounts for up to26%of the variation in yields on days without major macroeconomic announcements.Green(2004)examines the effect of order flow on intraday bond price changes surrounding U.S.macroeconomic news announcements.We extend both studies by identifying a theoretical and empirical link between the price discovery role of orderflow and the degree of information heterogeneity among investors and the quality of macroeconomic data releases.In particular,we document important effects of both dispersion of beliefs and public signal noise on the correlation between daily bond yield changes and orderflow during announcement and nonannouncement days.This evidence complements the weak effects reported by Green(2004)over30-minute intervals around news releases. Since the econometrician does not observe the precise arrival time of private information signals,narrowing the estimation window may lead to underestimating the effect of dispersion of beliefs on market liquidity.3 Our work also belongs to the literature bridging the gap between asset pricing and market microstructure.Evans and Lyons(2003)find that signed orderflow is a good predictor of subsequent exchange rate movements;Brandt and Kavajecz(2004)show that this is true for bond market movements;and Easley et al.(2002)argue that the probability of informed trading(PIN),a function of orderflow,is a pricedfirm characteristic in stock returns.These studies enhance our understanding of the determinants of asset price movements,but do not provide any evidence on the determinants of orderflow.Evans and Lyons(2004)address this issue by showing that foreign exchange order flow predicts future macroeconomic surprises(i.e.it conveys information about fundamentals).We go a step further in linking the impact of order flow on bond prices to macroeconomic uncertainty(public signal noise) and the heterogeneity of beliefs about real shocks.We proceed as follows.In Section1,we construct a stylized model of trading to guide our empirical analysis.In Section2,we describe the data. In Section3,we present the empirical results.We conclude in Section4. 3For instance,heterogeneously informed investors may not trade immediately after public news releases but instead wait to preserve(and exploit)their informational advantage as long(and as much)as possible, as in Foster and Viswanathan(1996)4Informed and Strategic Order Flow in the Bond Markets1.Theoretical ModelIn this section we motivate our investigation of the impact of the disper-sion of beliefs among sophisticated market participants and the release of macroeconomic news on the informational role of trading.We first describe a one-shot version of the multiperiod model of trading of Foster and Viswanathan (1996)and derive closed-form solutions for the equi-librium market depth and trading volume.Then,we enrich the model by introducing a public signal and consider its implications for the equilib-rium price and trading strategies.All proofs are in the Appendix unless otherwise noted.1.1Benchmark:no public signalThe basic model is a two-date,one-period economy in which a single risky asset is exchanged.Trading occurs only at the end of the period (t =1),after which the asset payoff,a normally distributed random variable v with mean zero and variance σ2v ,is realized.The economy is populated by three types of risk-neutral traders:a discrete number (M )of informed traders (that we label speculators),liquidity traders,and perfectly competitive MMs.All traders know the structure of the economy and the decision process leading to order flow and prices.At time t =0there is neither information asymmetry about v nor trading.Sometime between t =0and t =1,each speculator k receives a private and noisy signal of v ,S vk .We assume that the resulting signal vector S v is drawn from a multivariate normal distribution (MND)with mean zero and covariance matrix s such that var (S vk )=σ2s and cov S vk ,S vj =σss .We also impose that the speculators together know the liquidation value of the risky asset: M k =1S vk =v ;therefore,cov (v,S vk )=σ2v M .This specification makes the total amount of information available to the speculators independent from the correlation of theirprivate signals,albeit still implying the most general information structure up to rescaling by a constant [see Foster and Viswanathan (1996)].These assumptions imply that δk ≡E (v |S vk )=σ2v Mσ2s S vk and E δj |δk =γδk ,where γ=σss σ2s is the correlation between any two private information endowments δk and δj .As in Foster and Viswanathan (1996),we parametrize the degree of diversity among speculators’signals by requiring that σ2s −σss =χ≥0.This restriction ensures that s is positive definite.If χ=0,then speculators’private information is homogeneous :Allspeculators receive the same signal S vk =v M such that σ2s =σss =σ2v M 2and γ=1.If χ=σ2v M ,then speculators’information is heterogeneous :σ2s =χ,σss =0,and γ=0.Otherwise,speculators’signals are only 5The Review of Financial Studies /v 20n 52007partially correlated:Indeed,γ∈(0,1)if χ∈ 0,σ2v M and γ∈ −1M −1,0 if χ>σ2v M .4At time t =1,both speculators and liquidity traders submit their orders to the MMs,before the equilibrium price p 1has been set.We define the market order of the k -th speculator to be x k .Thus,that speculator’s profit is given by πk (x k ,p 1)=(v −p 1)x k .Liquidity traders generate a random,normally distributed demand u ,with mean zero and variance σ2u .For simplicity,we assume that u is independent from all other random variables.MMs do not receive any information,but observe the aggregate order flow ω1= M k =1x k +u from all market participants and set the market-clearing price p 1=p 1(ω1).1.1.1Equilibrium.Consistently with Kyle (1985),we define a Bayesian Nash equilibrium as a set of M +1functions x 1(·),...,x M (·),and p 1(·)such that the following two conditions hold:1.Profit maximization :x k (S vk )=arg max E (πk |S vk );and2.Semistrong market efficiency :p 1(ω1)=E (v |ω1).We restrict our attention to linear equilibria.We first conjecture general linear functions for the pricing rule and speculators’demands.We then solve for their parameters satisfying conditions 1and 2.Finally,we show that these parameters and those functions represent a rational expectations equilibrium.The following proposition accomplishes this task.Proposition 1.There exists a unique linear equilibrium given by the price functionp 1=λω1(1)and by the k -th speculator’s demand strategyx k =λ−12+(M −1)γδk ,(2)where λ=σ2vσu σs √M [2+(M −1)γ]>0.The optimal trading strategy of each speculator depends on the information received about the asset payoff (v )and on the depth of the 4The assumption that the total amount of information available to speculators is fixed ( Mk =1S vk =v )implies that σ2s =σ2v +M(M −1)χM 2and σss =σ2v −MχM 2,hence γ=σ2v −Mχσ2v +M(M −1)χ.Further,the absolute bound to the largest negative private signal correlation γcompatible with a positive definite s , −1M −1 ,is a decreasing function of M .6Informed and Strategic Order Flow in the Bond Marketsmarket (λ−1).If M =1,Equations (1)and (2)reduce to the well-known equilibrium of Kyle (1985).The speculators,albeit risk-neutral,exploit their private information cautiously (|x k |<∞),to avoid dissipating their informational advantage with their trades.Thus,the equilibrium market liquidity in p 1reflects MMs’attempt to be compensated for the losses they anticipate from trading with speculators,as it affects their profits from liquidity trading.1.1.2Testable implications.The intuition behind the parsimonious equilibrium of Equations (1)and (2)is similar to that in the multiperiod models of Foster and Viswanathan (1996)and Back et al.(2000).Yet,its closed-form solution (in Proposition 1)translates that intuition into unambiguous predictions on the impact of information heterogeneity on market depth.5The optimal market orders x k depend on the number of speculators (M )and the correlation among their information endowments (γ).The intensity of competition among speculators affects their ability to maintain the informativeness of the order flow as low as possible.A greater number of speculators trade more aggressively —that is,their aggregate amount of trading is higher—since (imperfect)competition among them precludes any collusive trading strategy.For instance,when M >1speculators are homogeneously informed (γ=1),then x k =σu σv √M v ,which implies that the finite difference Mx k =(M +1)x k −Mx k =σu σv √M +1−√M v >0.This behavior reduces the adverse selection problem for the MMs,thus leading to greater market liquidity (lower λ).The heterogeneity of speculators’signals attenuates their trading aggressiveness.When information is less correlated (γcloser to zero),each speculator has some monopolistic power on the private signal,because at least part of the information is exclusively known.Hence,as a group,they trade more cautiously—that is,their aggregate amount of trading is lower—to reveal less of their own information endowments δk .For example,when M >1speculators are heterogeneously informed (γ=0),then x k =σu σv S vk ,which implies that M k =1x k =σu σv v <σu M σv √M v ,that is,lower than the aggregate amount of trading by M >1homogeneously informed speculators (γ=1)but identical to the trade of a monopolistic speculator (M =1).This ‘‘quasimonopolistic’’behavior makes the MMs more vulnerable to adverse selection,thus the market less liquid (higher λ).The following corollary summarizes the first set of empirical implications of our model.5This contrasts with the numerical examples of the dynamics of market depth reported in Foster and Viswanathan (1996,Figure 1C)and Back et al.(2000,Figure 3A).7The Review of Financial Studies /v 20n 52007Corollary 1.Equilibrium market liquidity is increasing in the number of speculators and decreasing in the heterogeneity of their information endowments.To gain further insight on this result,we construct a simple numerical example by setting σv =σu =1.We then vary the parameter χto study the liquidity of this market with respect to a broad range of signal correlations γ(from very highly negative to very highly positive)when M =1,2,and4.By construction,both the private signals’variance (σ2s )and covariance (σss )change with χand M ,yet the total amount of information available to the speculators is unchanged.We plot the resulting λin Figure 1.Multiple,perfectly heterogeneously informed speculators (γ=0)collectively trade as cautiously as a monopolist speculator.Under these circumstances,adverse selection is at its highest,and market liquidity atits lowest (λ=σv 2σu ).A greater number of competing speculators improvesmarket depth,but significantly so only if accompanied by more correlated private signals.However,ceteris paribus ,the improvement in market liquidity is more pronounced (and informed trading less cautious)when speculators’private signals are negatively correlated.When γ<0,each speculator expects her competitors’trades to be negatively correlated toFigure 1Equilibrium without a public signalIn this figure we plot the market liquidity parameter defined in Proposition 1,λ=σ2v σu σs √M [2+(M −1)γ],as a function of the degree of correlation of the speculators’signals,γ,in the presence of M =1,2,or 4speculators,when σ2v =σ2u =1.Since σ2s =σ2v +M(M −1)χM 2,σss =σ2v −MχM 2,and γ=σ2v −Mχσ2v +M(M −1)χ,the range of correlations compatible with a positive definite s is obtained by varying the parameterχ=σ2s −σss within the interval [0,10]when M =2,and the interval [0,5]when M =4.8Informed and Strategic Order Flow in the Bond Marketsher own (pushing p 1against her signal),hence trading on it to be more profitable.1.2Extension:a public signalWe now extend the basic model of Section 1.1by providing each player with an additional,common source of information about the risky asset before trading takes place.According to Kim and Verrecchia (1994,p.43),‘‘public disclosure has received little explicit attention in theoretical models whose major focus is understanding market liquidity.’’6More specifically,we assume that,sometime between t =0and t =1,both the speculators and the MMs also observe a public and noisy signal,S p ,of the asset payoffv .This signal is normally distributed with mean zero and variance σ2p >σ2v .We can think of S p as any surprise public announcement (e.g.macroeco-nomic news)released simultaneouslyto all market participants.We further impose that cov S p ,v =σ2v ,so that the parameter σ2p controls for the quality of the public signal and cov S p ,S vk =σ2v M .The private informa-tion endowment of each speculator is then given by δ∗k≡E v |S vk ,S p −E v |S p =αS vk +βS p ,where α=Mσ2v σ2p −σ2v σ2p [σ2v +M(M −1)χ]−σ4v >0and β=−σ4v σ2p −σ2v σ2p {[σ2v +M(M −1)χ]−σ4v}<0.Thus,E δ∗j |δ∗k =E δ∗j |S vk ,S p =γp δ∗k ,where γp =Mα2σss +2αβσ2v +Mβ2σ2pMα2σ2s +2αβσ2v +Mβ2σ2p ≤γ.1.2.1Equilibrium.Again we search for linear equilibria.The following proposition summarizes our results.Proposition 2.There exists a unique linear equilibrium given by the price functionp 1=λp ω1+λs S p(3)and by the k -th speculator’s demand strategyx k =λ−1p 2+(M −1)γp δ∗k ,(4)6Admati and Pfleiderer (1988)and Foster and Viswanathan (1990)consider dynamic models of intraday trading in which the private information of either perfectly competitive insiders or a monopolistic insider is either fully or partially revealed by the end of the trading period.McKelvey and Page (1990)provide experimental evidence that individuals make inferences from publicly available information using Bayesian updating.Diamond and Verrecchia (1991)argue that the disclosure of public information may reduce the volatility of the order flow,leading some MMs to exit.Kim and Verrecchia (1994)show that,in the absence of better informed agents but in the presence of better information processors with homogeneous priors,the arrival of a public signal leads to greater information asymmetry and lower market liquidity.9The Review of Financial Studies /v 20n 52007where λp =α12σv σ2p −σ2v 12σu σp [2+(M −1)γp ]>0and λs =σ2v σ2p.The optimal trading strategy of each speculator in Equation (4)mirrors that of Proposition 1[Equation (2)],yet it now depends only on δ∗k ,the truly private—hence less correlated (γp ≤γ)—component of speculator k ’s original private signal (S vk )in the presence of a public signal of v .Hence,the MMs’belief update about v stemming from S p makes speculators’private information less valuable.The resulting equilibrium price p 1in Equation (3)can be rewritten as:p 1=α2+(M −1)γp v +λp u + λs +Mβ2+(M −1)γp S p .(5)According to Equation (5),the public signal impacts p 1through two channels that [in the spirit of Evans and Lyons (2003)]we call direct ,related to MMs’belief updating process (λs >0),and indirect ,via the speculators’trading activity (β<0).Since λs 2+(M −1)γp >−Mβ>0,the former always dominates the latter.Therefore,public news always enter the equilibrium price with the ‘‘right’’sign.1.2.2Additional testable implications.Foster and Viswanathan (1993)generalize the trading model of Kyle (1985)to distributions of the elliptically contoured class (ECC)and show that,in the presence of a discrete number of identically informed traders,the unexpected realization of a public signal has no impact on market liquidity regardless of the ECC used.This is the case for the equilibrium of Proposition 2as well.7Nonetheless,Proposition 2allows us to study the impact of the availability of noisy public information on equilibrium market depth in the presence of imperfectly competitive and heterogeneously informed speculators.To our knowledge,this analysis is novel to the financial literature.We start with the following result.Corollary 2.The availability of a public signal of v increases equilibrium market liquidity.The availability of the public signal S p reduces the adverse selection risk for the MMs,thus increasing the depth of this stylized market,for two reasons.First,the public signal represents an additional,trade-free source of information about v .Second,speculators have to trade more aggressively to extract rents from their private information.In Figure 27Specifically,it can be shown that the one-shot equilibrium in Foster and Viswanathan (1993,Proposition1)is a special case of our Proposition 2when private signal correlations γ=1for any ECC.10Informed and Strategic Order Flow in the Bond MarketsFigure 2Equilibrium with a public signalIn this figure we plot the difference between the sensitivity of the equilibrium price to the order flow in the absence and in the presence of a public signal S p ,λ−λp ,as a function of the degree of correlation ofthe speculators’signals,γ,in the presence of M =1,2,or 4speculators,when σ2v =σ2u =1.Accordingto Proposition 1,λ=σ2v σu σs √M [2+(M −1)γ],while λp =α12σv σ2p −σ2v 12σu σp 2+(M −1)γpin Proposition 2,where γp =σ2p σ2v −Mχ −σ4v σ2p σ2v +M(M −1)χ −σ4v and α=Mσ2v σ2p −σ2v σ2p σ2v +M(M −1)χ −σ4v .Since σ2s =σ2v +M(M −1)χM 2,σss =σ2v −MχM 2,and γ=σ2v−Mχσ2v +M(M −1)χ,the range of correlations compatible with a positive definite s is obtained by varying the parameter χ=σ2s −σss within the interval [0,10]when M =2,and the interval [0,5]whenM =4.we plot the ensuing gain in liquidity,λ−λp ,as a function of private signalcorrelations γwhen the public signal’s noise σp =1.25,that is,by varying χand M (so σ2s and σss as well,but not the total amount of information available to the speculators)as in Figure 1.The increase in market depth is greater when γis negative and the number of speculators (M )is high.In those circumstances,the availability of a public signal reinforces speculators’existing incentives to place market orders on their private signals,S vk ,more aggressively.However,greater σ2p ,ceteris paribus ,increases λp ,since the poorer quality of S p (lower information-to-noise ratioσ2vσ2p)induces the MMs to rely more heavilyon ω1to set market-clearing prices,hence the speculators to trade less aggressively.Remark 1.(The increase in)market liquidity is decreasing in the volatility of the public signal.11。
最新关于“普罗帕酮”的认识
1. 普罗帕酮的传统合成方法
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1. 普罗帕酮的传统合成方法 主要通过多步化学反应，包 括硝化、还原、酯化和酰胺 化等步骤。
2. 在普罗帕酮的传统合成过 程中，需要使用到大量的有 机溶剂和催化剂，对环境有 一定的污染。
3. 传统合成方法中，每一步 反应的产率和纯度都对最终 产品的质量有直接影响，因 此需要进行严格的质量控制 。
最新关于“普罗帕酮”的认 识
目录 CONTENTS
01 一、普罗帕酮的基本信息 02 二、普罗帕酮的制备方法
03 三、普罗帕酮的药理作用 04 四、普罗帕酮的安全性评价
一、普罗帕酮的基本信息
1. 普罗帕酮的定义和性质
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1. 普罗帕酮是一种常用 的抗心律失常药物，主要 用于治疗室上性和室性心 动过速。
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2. 对于普罗帕酮的副 作用，医生通常会根据 患者的具体情况调整药 物剂量或更换其他抗心 律失常药物。
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3. 若出现普罗帕酮副 作用，患者应立即停止 服用并及时就医，避免 自行处理或忽视症状导 致病情恶化。
3. 普罗帕酮的安全用药指导
1. 普罗帕酮是一种常用的抗心律失常药物，其安全用药需要在 医生的指导下进行。
1. 普罗帕酮的毒性研究
1. 普罗帕酮的毒性研究显示，该药物在高剂量下可能对心脏产 生不良影响。
2. 长期使用普罗帕酮可能导致肝脏和肾脏功能受损，需要定期 进行相关检查。
3. 普罗帕酮的毒性研究还发现，该药物可能引起视觉障碍，如 模糊视觉或视力下降。
2. 普罗帕酮的副作用及处理
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1. 普罗帕酮的常见副 作用包括恶心、呕吐、 头痛、疲劳等，严重时 可能出现心律失常、呼 吸困难等症状。
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Ｐ ｒ ｏ ｖ ｉ ｄ ｅ ｎ ｔ Ａ ｇ ｒ ｏ公 司 今 年 首 季 度 记 载 收 入 １ ５ ２ ２ ． ３亿 盾 ， 比去 年 同期 的 １ ３ ２ ９ ． １亿 盾 增 加
１ ２ ％。 Ｐ ｒ ｖ ｉ ｎ Ａ ｎ ｔ ｏ ｎ ｉ ｏ
投 入 运 行 后 ，该 工 厂 将 需要 聘 用 逾 ５ ０ ０名 直 接 劳 动 者 和 数 千 名 间 接 劳 动 者 ，年 均 营 收 额 预 测 达
２ ３ ０ ０万美元 。
ｆ 摘自 《 越南共产党 电子报》，２ ０ １ ３ — ０ ５ — ２ ８ ）
高 ，油棕仁产量也提高。 ”
面 积达 到 ４ ． ４ ８万 ｈ ｍ 。
棕糖——越南安江省 高棉族人的 “ 招牌菜”
棕糖 是越南安江 省静边县 高棉族人 的特产 ， 是使用糖棕花序 中的糖 汁熬制 而成 的。深黄色 的 棕糖清甜可 口，是 当地高棉族人 辛苦创造 出来 的 劳动成果 ，与 由其他原料熬成的糖类味道完全不
间 内就 有 收获 ” 因此 ，他 呼 吁印 尼 国营农 园公 司
司 的收 入达 ５ ９ ９ ２ ． ３ 亿盾 ，比前年 同期 的 ３ ９ ９ ５ ． ７ 亿盾提高 ５ ０ ％。今年首季度收入达 １ ５ ２ ５ ． ６ 亿盾 ， 比去年同期的 １ ３ ２ ９ ． １ 亿盾提高 １ ２ ％。 ”
５月 ２ ６日，柬 埔 寨 最 大 的糖 、 乙醇 及热 电综 合 生 产 厂在 桔 井 省 三 坡 县 奥 良乡 正 式竣 工 。该 工
印尼 Ｐ ｒ ｏ ｖ ｉ ｄ ｅ ｎ ｔ Ａ ｇ ｒ ｏ 公司 今年首季度棕 油产量提高 ２ ９ ％
印度 尼西 亚 Ｐ ｒ ｏ ｖ ｉ ｄ ｅ ｎ ｔ Ａ ０ 公 司 记 载 ，今 年 首
Ｒ ｉ ｄ ｗ ａ ｎ 称 ，今年首季度公 司的油棕新鲜果 串、粗 棕油与内核 的产量都 比去年 同期高 。去年 ，果 串 产量有 ２ １ ． ６ ４万 ｔ ，比前年 同期 的 １ ９ ． ８ ５万 ｔ 增 加
厂 由越 南 合 资 企 业 Ｋ ａ ｍ ｄ ｈ ｅ ｎ ｕ Ｖ ｅ ｎ ｔ ｕ ｒ ｅ ｓ Ｃ ａ ｍ ｂ ｏ ｄ ｉ ａ
Ｌｍ ｉ ｔ ｅ ｄ责 任有 限公 司（ Ｋ Ｖ Ｃ Ｕ投资 。
Ｋ Ｖ Ｃ Ｌ的报告显示 ，这家糖 、乙醇及热电综合
生 产 厂 是外 国在 柬 埔 寨 投 资 的 最 大 项 目之一 。该
世界热带农业信息
２ ０ １ ３ 年第 ７ 期
９ ． １ ％，粗 棕 油 的 产 量 有 ７ ． ０ ９万 ｔ ，比前 年 同 期 的
４ ． ４ ６万 ｔ 增加 ５ ８ ． ８ ％ ，内核 的产 量 为 １ ． ４ ９万 ｔ ， 比
在雅加达表示 ： “ 菲律宾邀请印尼 国营企业 种植
园发 展 油 棕业 。在 初 级 阶段 ，该 国 已经 准 备 好 总
还未达到极 限，因为一部分油棕树的树龄仍幼小 。 为提高往后 的油棕产量 ，该公司称 ，将持续增 加
油棕 种 植 土 地 面积 。２ ０ １ １年该 公 司 的油 棕 种植 土 地 面积 为 ４ ． ２ ４万 ｈ ｍ ，去 年 提 高 至 ４ ． ４ ５万 ｈ ｍ２ ，
今年将再持续提 高 ，至今年首季 ，油棕种植 土地
项 目投资总额约为 ９ ０ ７ ０ 万美元 ， 其中Ｋ Ｖ Ｃ Ｌ出资
２ ５ ３ ０万 美 元 （ 占该项 目投资总额的 ２ ７ ． ８ ％ ） ，从 越 南 投
季度的粗棕油产量提高 ２ ９ ％，达到 １ ． ８ ５ 万ｔ ，去
年该 公 司的粗 棕油 产量 为 １ ． ４ ３ 万ｔ 。 日前 ，Ｐ ｒ ｏ ｖ ｉ ｄ ｅ ｎ ｔ Ａ ｇ ｒ ｏ总经 理 特 利 ・ 布沃诺称 ： “ 粗 棕 油 产 量 提 高 ，是 因为 油棕 新 鲜 果 串 产 量 提
（ 摘 自印度尼西亚 《 商报》 ， ２ ０ １ ３ — ０ ５ — ３ １ ）
（ Ｐ Ｔ Ｐ Ｎ ） 进一步扩展计划。 “ 国营农 园公司能与菲律 宾 国营企业成立合资企业 。 ”
（ 摘 自印度尼西亚 《 商报》 ，２ ０ １ ３ — ０ ６ — ０ ４ ）
越南在柬埔寨投资 兴建 的最大糖厂竣工
今 年 首 季 度 ，Ｐ ｒ ｏ ｖ ｉ ｄ ｅ ｎ ｔ Ａ ｇ ｒ ｏ的粗 棕 油 产 量 几
乎与同期的油棕新鲜果串产量提高 ２ ９ ． ３ ％相同，该 公司已生产 ５ ． ４ ４万 ｔ 油棕新鲜果串 ，比去年首季 度落实的 ４ ． ２ １ 万ｔ 高 ，而油棕仁产量 比去年落 实
的３ １ ９ ９ ｔ 提高 １ ０ ． ８ ％ ，达 到 ３ ５ ４ ５ ｔ 。该公 司产 量
面 积达 ２ ． ５万 ｈ ｍ 的农 地 。 ”
前 年 同期 的 ８ ８ ０ ０ ｔ 增加 ６ ８ ． ８ ％。 “ 去 年 年 底 公
他认 为 ，菲律宾希望 印尼能够发展该 国的油 棕业 ，特别是在菲律 宾南 区 ，从种 植油棕 开始 ， 直至使用高科技兴 建棕油厂 。菲律宾政府评价印 尼是成功发展油棕业 的国家 ，其工作业绩甚至 已 超过马来西亚。余 世甘说 ： “ 他们赞扬印尼油棕 业 的经验和技术 ，能发展油棕树而且在 ２ 年半时
资暨发展银行和柬埔寨投资与发展银行￣ Ｉ Ｄ Ｃ ） 贷款
６ ５ ４ ０万 美 元 （ 占该项 目投资总额 的 ７ ０ ％） 。该 项 目于
２ ０ １ ０ 年年底动工兴建 ，设厂面积为 ７ ６ ３ ５ ｈ ｍ 。根 据设计 ，糖厂产量为 ３ ５ ０ ０ ｔ 蔗糖 ／ ｄ ；乙醇厂产量 为３ 万 ／ ｄ ；热电厂功率为 ２ ０ Ｍ Ｗ。 除 了提高经济效益 以外 ，该 项 目还有助于确 保地 方社会 民生 、提供 就业机会 和稳定 的收入 。