Chapter 2(stochastic process)
金融随机分析 II Stochastic process
n
i
bn 0
P 1 (ii) bn E X | F ni i1 0, and 2 n
(iii ) b
EX
n i 1
2 ni
E E X ni | Fi 1
2
0
12
Ref: Hall . P, Heyde C., Martingale limit theory and its applicatiom.1980, Academic Press. Inc
M k M a.s and
M
k
M dP 0 as k
10
Corollary 2.2.9 Let X L P ,N k k 1 be an
1
increasing family of σ-algebras, N k F and define N to be the σ-algebra generated by
h X k 1 k h X k 1 X k (b) for every bounded Borel-measurable function h :
(c) (d)
euX k1 k euX k1 X k , u
(Agreement of Laplace transforms)
2.3 Markov Processes
Definition 2.3.1 Let (, , ) be a filtration under . Let {Xk,k=0,1,…}be a stochastic process on (, , ) . This process is said to be Markov if:
StochasticProcess
2
Discrete Time and Continuous Time Processes
•X(t) is a discrete time process if X(t) is defined only for a set of time instants tn=nT where T is a constant and n is an integer
According to Theorem 1 p( j)= [ P(0, 1)P(1,2) … P( j-1, j)]T p(0) In the Stationary case, P(0, 1)P(1,2) … P( j-1, j) = P j
Example: Random Walk With Barriers
Markov Diagram
•In the stationary form
–the Chapman-Kolmogorov equations has a graphical interpretation in terms of one-step probabilities p1,1 p1 p2 p2,2 p2,1 2 1 p1,2 p2,3 p3,2 p1,3 p3,1 3
t ,t ' pm P[ X (t ' ) n | X (t ) m] ,n
•For discrete-time process, define
i, j pm P[ X ( j ) n | X (i ) m] ,n
Markov Chain
•A stochastic process is a Markov chain if
•X(t) is a discrete value process if the set of all possible values of X(t) at all times t is a countable set SX
随机过程讲义（第二章）（PDF）
第二章 随机过程的一般概念2.1 随机过程的基本概念和例子定义2.1.1：设(P ,,F )Ω为概率空间，T 是某参数集，若对每一个，是该概率空间上的随机变量，则称为随机过程(Stochastic Process)。
T t ∈),(w t X ),w t (X 随机过程就是定义在同一概率空间上的一族随机变量。
随机过程可以看成定义在),(w t X Ω×T 上的二元函数，固定Ω∈0w ，即对于一个特定的随机试验，称为样本路径(Sample Path)，或实现(realization)，这是通常所观测到的过程；另一方面，固定，是一个随机变量，按某个概率分布随机取值。
),(0w t X T t ∈0),(0w t X抽象一点：令，即∏∈=Tt T R R T R 中的元素为),(T t x X t t ∈=,为其Borel域(插乘)(T R B σ域)，随机过程实质上是()F ,Ω到())(,T T R R B 上的一个可测映射，在())(,T TR RB 上诱导出一个概率测度：T P ()B X P B P R B T T T ∈=∈∀)(),(B 。
一般代表的是时间。
根据参数集T 的性质，随机过程可以分为两大类： t 1)为可数集，如T {}L ,2,1,0=T 或{}L L ,1,0,1,−=T ，称为离散参数随机过程，也称为随机序列；2)为不可数集，如T {}0≥=t t T 或{}∞<<∞−=t t T ，称为连续参数随机过程。
随机过程的取值称为过程所处的状态(State)，所有状态的全体称为状态空间(State Space)。
通常以表示随机过程的状态空间。
根据状态空间的特征，一般把随机过程分为两大类：T t t X ∈),(S 1) 离散状态，即取一些离散的值； )(t X 2)连续状态，即的取值范围是连续的。
)(t X离散参数离散状态随机过程： Markov 链 连续参数离散状态随机过程： Poisson 过程 离散参数连续状态随机过程： *Markov 序列连续参数连续状态随机过程： Gauss 过程，Brown 运动例2.1.1：一醉汉在路上行走，以的概率向前迈一步，以q 的概率向后迈一步，以p r 的概率在原地不动，1=++r q p ，选定某个初始时刻，若以记它在时刻的位置，则就是直线上的随机游动(Random Walk)。
随机过程--Chapter 2
customer in (0,t] is t-Si. Adding the revenues generated by all
arrivals in (0,t]
N (t)
N (t )
(t Si ) ,
i 1
E (t Si )
i1
14
2.2 Properties of Poisson processes
Solution:
(a) E[S10]=10/= 10 days
(b) P{X11>2} = e -2 = e-2 0.1333
9
2.2 Properties of Poisson processes
Arrival time distribution
Proposition 2.2 :
The arrival time of the nth event Sn follows a Γ distribution
f (t) e t
each interarrival time {Xn, n1} follows an exponential
distribution with parameter .
8
2.2 Properties of Poisson processes
Example 1 Suppose that people immigrate into a territory at a Poisson
and X1 and X2 are independent
7
2.2 Properties of Poisson processes
Similarly, we obtain: P{ Xn>tXn-1=s} = e-t P{ Xn tXn-1= s} = 1- e-t
流程密码第二章如何把制度转化为流程读后感
流程密码第二章如何把制度转化为流程读后感（中英文实用版）英文文档：After reading Chapter 2 of "Process Password - How to Transform Systems into Processes", I gained a deeper understanding of the importance of converting organizational systems into efficient processes.This chapter highlighted the significance of aligning daily operations with the overall strategic goals of a company.The chapter emphasized the need for a clear and well-defined process framework that outlines the steps and guidelines for completing tasks.It discussed the benefits of having a standardized process, such as increased productivity, reduced errors, and improved customer satisfaction.One key concept that stood out to me was the importance of process documentation.The chapter highlighted that documenting processes not only helps in ensuring consistency but also serves as a valuable resource for training new employees and maintaining organizational knowledge.Furthermore, the chapter introduced the concept of process mapping, which I found to be a useful tool for visualizing and understanding the flow of activities within a process.By mapping out the steps and decision points, organizations can identify bottlenecks, inefficiencies, and areas for improvement.In conclusion, Chapter 2 of "Process Password" provided valuable insights on how to transform organizational systems into effective processes.It highlighted the importance of process documentation, standardization, and process mapping in achieving operational excellence.This chapter has motivated me to further explore and implement these concepts in my own work to improve efficiency and effectiveness.中文文档：阅读了《流程密码- 如何把制度转化为流程》的第二章后，我深刻体会到了将组织制度转化为高效流程的重要性。
Stochastic Process - Introduction
γ (s ) = cov( X t , X t + s ) = E (( X t − µ )( X s − µ ))
• The correlation between Xt and Xt+s is
ρ (s ) =
cov( X t , X s ) =
var( X t ) var( X s )
γ (s ) γ (0 )
FX t
1
,..., X t k
(x1 ,..., xk ) = P(X t
1
≤ x1 ,..., X tk ≤ x k
)
for any t1 ,..., t k ∈ I and any real numbers x1, …, xk . • The distribution function tells us everything we need to know about the process {Xt }.
week 2 4
Stationary Processes
• A process is said to be strictly stationary if (X t ,..., X t joint distribution as (X t + ∆ ,..., X t + ∆ ) . That is, if
week 2 5
Weak Stationarity
• Strict stationarity is too strong of a condition in practice. It is often difficult assumption to assess based on an observed time series x1,…,xk. • In time series analysis we often use a weaker sense of stationarity in terms of the moments of the process. • A process is said to be nth-order weakly stationary if all its joint moments up to order n exists and are time invariant, i.e., independent of time origin. • For example, a second-order weakly stationary process will have constant mean and variance, with the covariance and the correlation being functions of the time difference along. • A strictly stationary process with the first two moments finite is also a second-ordered weakly stationary. But a strictly stationary process may not have finite moments and therefore may not be weakly stationary.
机械工程学专业词汇英语翻译（S）2
specimen 试样 speckle 散斑 speckle holography 散斑全息照相术 speckle interferometry 散斑⼲涉法 spectral condition 谱条件 spectral curve 谱曲线 spectral density 谱密度 spectral displacement 谱位移 spectral distribution 谱分布 spectral frequency 谱频率 spectral function 谱函数 spectral line 谱线 spectral method 谱⽅法 spectral position 谱位置 spectral shift 谱位移 spectral theory 谱理论 spectrum 谱 spectrum matrix 谱矩阵 spectrum tensor 谱张量 speed 速率 speed change 变速 speed control 速率第 speed error 速率误差 speed governor 蒂机 speed indicator 速度计 speed limiting device 限速器 speed of autorotation ⾃转速度 speed of heat propagation 热传播速率 speed of light 光速 speed of perception 感觉速率 speed of response 响应速度 speed of sound 声速 speed per hour 时速 speed range 变速范围 speed reduction 减速 speed regulation 速度第 speed regulator 蒂机 speedometer 转速表 sphere 球 sphere of reflection 反射球 sphere shaped 球形的 spherical angle 球⾯⾓ spherical bearing 球⾯⽀承 spherical coordinates 球⾯坐标 spherical function 球⾯函数 spherical joint 球形接头 spherical motion 球⾯运动 spherical pendulum 球摆 spherical shell 球壳 spherical shock wave 球⾯激波 spherical space 对映空间 spherical strain tensor 球⾯应变张量 spherical stress tensor 球⾯应⼒张量 spherical surface 球⾯ spherical tensor 球⾯张量 spherical top 球形陀螺 spherical wave 球⾯波 spheroidal wave function 球波函数 spillway 溢淋 spillway dam 溢劣 spin angle ⾃旋⾓ spin angular momentum ⾃旋⾓动量 spin axis ⾃旋轴 spin coordinate ⾃旋坐标 spin eigenfunction ⾃旋本寨数 spin eigenstate ⾃旋本宅 spin interaction ⾃旋相互酌 spin orbit potential ⾃旋轨道势 spin relaxation ⾃旋弛豫 spin resonance ⾃旋共振 spin resonance frequency ⾃旋共振频率 spin slip spectrum ⾃旋反转谱 spin tensor ⾃旋张量 spin tensor operator ⾃旋张量算符 spin wave ⾃旋波 spin wave theory ⾃旋波理论 spindle 轴 spinning 尾旋 spinning detonation 旋焰爆炸 spinning obstacle 旋转障碍 spinning top 旋转陀螺 spinning wind tunnel 螺旋风洞 spinor field 旋量场 spinor gravitation 旋量引⼒ spinor wave 旋量波 spinor wave equation 旋量波⽅程 spinor wave function 旋量波函数 spiral 螺线 spiral dislocation 螺旋形位错 spiral fiber structure 螺旋形纤维结构 spiral flow 螺旋流 spiral of archimedes 阿基⽶德螺线 spiral orbit 螺线轨道 spiral propeller 螺旋推进器 spiral spring 螺旋形弹簧 spiral trajectory 螺线轨道 spiral vortex 螺旋形涡流 spiral vortex model 螺旋涡模型 spire 尖塔 spirit level ⽔准器 split hopkinson bar 分离式霍普⾦森杆 split pattern 分离模式 splitting 分裂 spoiler 绕菱 sponge 海绵 spongy 海绵状的 spontaneous combustion ⾃燃 spontaneous crack propagation ⾃发裂缝传播 spontaneous emission ⾃发发射 spout 瘤⼝ spouted bed 喷出床 spouting spring 喷泉 spray 喷雾 spray pump 喷淋泵 spray resistance 喷雾阻⼒ spraying effect 雾化效应 spraying nozzle 喷雾嘴 spring 弹簧 spring back 弹性回复 spring balance 弹簧秤 spring buffer 弹簧绶冲器 spring constant 弹簧常数 spring dynamometer 弹簧秤 spring force 弹簧弹⼒ spring hammer 弹簧锤 spring manometer 弹簧式压⼒计 spring model 弹簧模型 spring oscillator 弹簧振⼦ spring pressure 弹簧压⼒ spring pressure gage 弹簧式压⼒计 spring ring 弹簧圈 spring scales 弹簧秤 spring tension 弹簧张⼒ spurious frequency 寄⽣频率 spurious oscillation 寄⽣振荡 spurious scattering 虚散射 squagging ⾃锁 square measure ⾯积单位 square wave ⽅形波 square wave oscillation 矩形波振荡 square wave pulse 矩形脉冲 squashing 压碎 squeezing 压榨 stability 稳定性 stability condition 稳定性条件 stability constant 稳定性常数 stability curve 稳定性曲线 stability layer 稳定层 stability limit 稳定性极限 stability line 稳定性曲线 stability of equilibrium 平衡稳定性 stability of motions 运动稳定性 stability of parallel flow 平⾏寥定性 stability of stratified flow 分层寥定性 stability of structures 结构稳定性 stability of vibration 振动稳定性 stability tensor 稳定性张量 stability theorem 稳定性定理 stabilization 稳定化 stabilized equilibrium 稳定平衡 stabilizer 稳定器 stabilizing force 稳定⼒ stabilizing gyroscope 稳定陀螺 stable axis 稳定轴 stable equilibrium 稳定平衡 stable equilibrium position 稳定平衡位置 stable motion 稳定运动 stable orbit 稳定轨道 stable state 稳定状态 stable system 稳定系 stable wave 稳定波 stage 阶段 stage discharge curve ⽔位量曲线 stage efficiency 分级效率 staged rocket 多级⽕箭 staggered arrangement 交错配置 staggered riveting 交错铆接 staggering 摇摆 stagnant water 静⽔ stagnation 停滞 stagnation curve 静⽔曲线 stagnation density 滞⽌密度 stagnation point 驻点 stagnation point flow 驻点流 stagnation pressure 驻点压⼒ stagnation temperature 滞⽌温度 stake 杆 stalled airfoil 失速机翼 stalled condition 失速条件 stalled flow 失速流 stalled wing 失速机翼 stalling 失速 stalling angle 失速迎⾓ stalling characteristics 失速特性 stalling flight 失速飞⾏ stalling point 失速点 stalling speed 失速速率 stand by power 备⽤功率 standard atmosphere 标准⼤⽓ standard condition 标准条件 standard density 标准密度 standard deviation 标准偏差 standard equation 正规⽅程 standard frequency 标准频率 standard frequency spectrum 标准频谱 standard isobaric surfaces 标准等压⾯ standard linear solid 标准线形固体 standard load 正常负载 standard measure 标准衡器 standard orifice plate 标准孔板 standard pressure 标准压⼒ standard resistance 标准阻⼒ standard solid 标准固体 standard state 标准状态 standard test specimen 标准试样 standard water column 标准⽔柱 standard wave 标准波 standard wind 标准风 standing wave 驻波 star connection y 形接法 starboard 右舷 starling hypothesis 斯塔林假说 start up time 起动时间 starting air 起动空⽓ starting lever 起动杆 starting of oscillations 振荡起动 starting oscillation 起动振动 starting period 起动时间 starting power 起动功率 starting pulse 起动脉冲 starting resistance 起动阻⼒ starting torque 起动转矩 state continuity 状态连续性 state curve 状态曲线 state equation 物态⽅程 state feedback control 状态反馈控制 state function 态函数 state of aggregation 聚集状态 state of energy 能量状态 state of plane stress 平⾯应⼒状态 state of rest 静态 state of strain 应变状态 state of stress 应⼒状态 state of suspension 悬浮状态 state parameter 状态参数 state space 状态空间 state vector 态⽮量 static 静的 static accuracy 静态准确度 static balance 静态平衡 static balancing 静⼒平衡 static balancing machine 静平衡试验机 static constraint reaction 静反⼒ static deflection 静载挠度 static elasticity 静弹性 static equilibrium 静态平衡 static fatigue 静疲劳 static field 静场 static force 静⼒ static fracture 静态破裂 static friction 静摩擦 static head 静压头 static hysteresis curve 静态滞后曲线 static imbalance 静⼒不平衡 static indeterminateness 超静定性 static instability 静⼒不稳定性 static lift 静升⼒ static load 静负载 static magnetic field 静磁场 static modulus of elasticity 静弹性模量 static moment 静⼒矩 static moment of the surface 静⼒表⾯矩 static pressure 静压 static pressure tube 静压管 static reaction 静反酌 static rolling friction 静滚动摩擦 static slip 静滑移 static stability 静⼒稳定性 static strength 静⼒强度 static stress 静应⼒ static surface tension 静表⾯张⼒ static temperature 静温 static test 静⼒试验 static unbalance 静⼒不平衡 statical condition 静⼒条件 statically defined 静定的 statically determinate 静定的 statically determinate beam 静定梁 statically determinate reaction 静定反⼒ statically determinate structure 静定结构 statically determinate system 静定系 statically indeterminate beam 静不定梁 statically indeterminate structure 静不定结构 statically indeterminate system 静不定系 statics 静⼒学 stationary axle 固定轴 stationary blade 固定叶⽚ stationary creep 定常蠕变 stationary distribution 定常分布 stationary equilibrium 定常平衡 stationary field 恒定场 stationary flow 定常流 stationary motion 定常运动 stationary orbit 静⽌轨道 stationary point 平稳点 stationary potential 稳定势 stationary random process 平稳随机过程 stationary satellite 静⽌卫星 stationary state 平稳态 stationary vibration 平稳振动 stationary vortex 定常涡旋 stationary wave 驻波 statistic test 统计检验 statistical accuracy 统计精度 statistical analysis 统计分析 statistical fluctuation 统计涨落 statistical mechanics 统计⼒学 statistical noise 统计噪声 statistical theory of turbulence 湍脸计理论 statistical thermodynamics 统计热⼒学 statistical weight 统计重量 statistics 统计学 staude cone 斯陶德锥⾯ steady detonation 定常爆轰 steady flow 定常流 steady free surface flow 定常⽆压流 steady gas flow 定常⽓流 steady load 不变负荷 steady motion 定常运动 steady potential 稳定势 steady precession 稳定旋进 steady rest 稳定架 steady state 平稳态 steady state creep 定常蠕变 steady state magnetic field 稳定磁场 steady state oscillation 稳态振荡 steady vorticity 定常涡流 steam 蒸汽 steam bleeding 抽汽 steam drive 蒸汽传动 steam ejector 蒸汽喷射器 steam extraction 抽汽 steam hammer 蒸汽锤 steam injector 蒸汽喷射器 steam jet 蒸汽喷流 steam meter 蒸汽量计 steam nozzle 蒸汽喷嘴 steam power 汽⼒ steam pressure 蒸汽压 steep throw 陡抛 steepness of wave edge 波前陡度 steering angle 转向⾓ steering column 转向柱 steiner theorem 平⾏轴定理 stellar dynamics 恒星动⼒学 stellar energy 恒星能量 stellar guidance 天⽂导航 stellar structure 恒星结构 step disturbance 阶跃扰动 step rocket 多级⽕箭 step velocity 阶跃速度 stepped shaft 梯级式轴 stepwise loading 逐步加载 stereographic net 伍尔夫经纬圈 stereographic projection 球⾯投影 stereophotogrammetry ⽴体照相测量术 steric acceleration 空间加速度 stiff chain 刚链 stiff joint 刚节点 stiffener 加劲材 stiffening 加劲 stiffening plate 加强板 stiffening rib 加强肋 stiffening ring 加强环 stiffness 刚度 stiffness coefficient 刚度系数 stiffness matrix 刚度矩阵 stiffness method 刚度法 stiffness modulus 劲度模量 stiffness reactance 劲度⼒抗 stiffness rotor 刚性转⼦ stiffness term 刚度项 stochastic acceleration 随机加速度 stochastic force 随机⼒ stochastic hydraulics 随机⽔⼒学 stochastic process 随机过程 stochastic similarity 随机相似 stochastic simulation 随机模拟 stockmayer potential 史托克梅耶势 stokes approximation 斯托克斯近似 stokes flow 斯托克斯怜 stokes fluid 斯托克斯铃 stokes law 斯托克斯定律 stokes stream function 斯托克斯怜数 stokes theorem 斯托克斯定理 stokes wave 斯托克斯波 stoma ⼩孔 stone stream 岩⽯流 stop 停⽌ stopping distance 停⽌距离 storage 存储 storage modulus 存储模量 stored energy 储能 stored energy function 储能函数 storm surge 风暴潮 storm wave 风暴波 straight angle 平⾓ straight line motion 直线运动 straight line plot of the bending stress 弯曲应⼒直线图 straight line wedge 直线楔 straightening 矫正 straightness 平直度 strain 应变 strain ageing 应变时效 strain amplitude 应变幅度 strain anisotropy 应变蛤异性 strain anneal method 应变退⽕法 strain component 应变分量 strain crack 应变裂缝 strain deviator 形变偏量 strain ellipsoid 应变椭球 strain energy 应变能 strain energy density 应变能密度 strain energy function 应变能函数 strain energy method 应变能法 strain energy theory 应变能理论 strain fatigue 应变疲劳 strain field 应变场 strain free lattice ⽆应变点阵 strain gage 应变计 strain gradient 应变梯度 strain hardening 应变硬化 strain hardening capacity 应变硬化能⼒ strain hardening coefficient 应变硬化系数 strain hardening curve 应变硬化曲线 strain hardening index 应变硬化指数 strain history 应变历程 strain intensity 应变强度 strain invariant 应变不变量 strain matrix 应变矩阵 strain measure 应变量度 strain measurement 应变测定 strain potential 应变势 strain rate 应变率 strain rate effect 应变率效应 strain rate history 应变率历程 strain relaxation 应变弛豫 strain space 应变空间 strain tensor 应变张量 strain theory 应变理论 strain work 形变功 strainer 式过滤器 stranded wire 绞线 strap 带 stratification 分层 stratification coefficient 分层系数 stratification of atmosphere ⼤⽓分层 stratification of water mass ⽔团分层 stratification of wind 风分层 stratiform structure 层状结构 stratosphere 平零 stream 怜 stream bed 河床 stream cross section center 怜截⾯中⼼ stream flow 河流 stream function 怜数 stream sheet 怜层 stream surface 伶 stream tube 淋 streamer 闪流等离⼦柳 streaming 怜 streaming around 绕流 streaming potential 怜位势 streamline 吝 streamline analogy 吝相似 streamline field 吝场 streamline flow ⽚流层流 streamli n e f o r m T b_ / p >。
AdventuresInStochasticProcessesSolutionManual
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Chapter 2
Conditional Expectation
2.1 A Binomial Model for Stock
Price Dynamics
n Figure:A three coin period binomial model.
n Note that the following notation:
1. Sample space
2. is stock price at time k
2.2 Information
n Definition 2.1 (Sets determined by the first k tosses) We say that a set A is determined by the first k coin tosses if, knowing only the outcome of the first k tosses, we can decide whether the outcome of all tosses is in A.
n Note that
n1. is the collection of set determined by the first k tosses
n2.
n3. the random variable is -measurable, for each k=1,2,…n
n Example 2.1
n
n Definition 2.2 (Information carried by a random variable.) Let X be a random variable We say that a set is determined by the random variable X if, knowing only the value of the random variable, we can decide whether or not . Another way of saying this is that for
every , either
n.
n Note that
n 1. The collection of subsets of determined by X is a algebra, denote by
n 2. If the random variable X takes finitely many different values, then is generated by the
collection of sets
n these sets are called the atoms of the algebra .
n 3.if X is a random variable then is given by
n Example 2.2
2.3 Conditional Expectation
n Definition 2.3 (Expectation.)
n And
n We can think of as a partial average of X over the set A.
n2.3.1 An example
n Let us estimate , given . Denote the estimate by .
n is a random variable Y whose value at is defined by
n where .
n Properties of n.
n.
n.
n.
n We then take a weighted average:
n Furthermore,
n In conclusion, we can write n Where
n2.3.2 Definition of Conditional Expectation
n Existence. There is always a random variable Y satisfying the above properties (provided that
i.e., conditional expectations always exist.
n Uniqueness. There can be more than one random variable Y satisfying the above properties, but if is another one, then almost surely, i.e.,
n Notation 2.1 For random variables X, Y , it is standard notation to write
n.
n.
n2.3.3 Further discussion of Partial Averaging
n.
n2.3.4 Properties of Conditional Expectation
n We compute
n We can also write
n A similar argument shows that
n We can verify the Tower Property,
2.4 Martingales
n The ingredients are:
n A super martingale
n A sub martingale
n A Martingale
n Example 2.3 (Example from the binomial model.)
n For k = 1;2 we already showed that
n The right hand side is , and so we have。