随机过程 草稿笔记

Stochastic process

大部分来自wiki
虽然想着英文好理解一些， 但是自己写还是会有好多用错词 的啊（
就当latex练习好了

a mathematical object defined as a family of random variables.

Definitions

Stochastic process

• a collection of random variables indexed by some set.
• numerical values of some system randomly changing over time.

Random Function

a stochastic process can also be interpreted as a random element in a function space.

Random Field

If the random variables are indexed by the Cartesian plane or some higher-dimensional Euclidean space, then the collection of random variables is usually called a random field instead.

Discrete-time & Continuous-time Stochastic Processes

(When interpreted as time, ) The index set has a finite or countable number of elements or not.

State Space

where each random variable takes values from.

Discrete/Integer-valued SP

• state space: integers or natural numbers

Real-valued SP

• state space: real line

N-dimensional Vector Process

• state space: n-d Euclidean space

Notation

probability space

$$(\Omega ,F,P)$$

where $\Omega$ is a sample space, $F$ is a $\sigma$-algebra, $P$ is a probability measure.

measurable space

$$(S,\Sigma )$$

while $S$ is the state space.

stochastic process

{X(t):t∈T} \{X(t):t\in{T}\} {X(t):t∈T}

while $X(t)$ refer to the random variable with the index $t$

$T$ is called the index set or parameter set.

distribution function $F_{t_1,t_2,\cdots ,t_i}(x_1,x_2,\cdots ,x_i)$

Ft1,t2,⋯ ,tn(x1,x2,⋯ ,xn)=P{X(t1)≤x1,X(t2)≤x2,⋯ ,X(tn)≤xn} F_{t_1,t_2,\cdots,t_n}(x_1,x_2,\cdots,x_n) = P\{X(t_1)\leq x_1,X(t_2)\leq x_2,\cdots,X(t_n)\leq x_n\} Ft1​,t2​,⋯,tn​​(x1​,x2​,⋯,xn​)=P{X(t1​)≤x1​,X(t2​)≤x2​,⋯,X(tn​)≤xn​}

If the distribution is independent,
P{X(t1)≤x1,X(t2)≤x2}=P{X(t1)≤x1}P{X(t2)≤x2} P\{X(t_1)\leq x_1,X(t_2)\leq x_2\}=P\{X(t_1)\leq x_1\}P\{X(t_2)\leq x_2\} P{X(t1​)≤x1​,X(t2​)≤x2​}=P{X(t1​)≤x1​}P{X(t2​)≤x2​}

mean function $m_X(t)$

$$m_X(t)=EX(t),t\in T$$

covariance function $B_X(s,t)$

$$B_X(s,t)=E[(X(s)-m_X(s))(X(t)-m_X(t))]$$

variance function $D_X(t)=B_X(t,t)$

$$D_X(t)={\sigma }_X^2(t)=E[(X(t)-m_X(t){)}^2]=E{X}^2(t)-m_X(t{)}^2=E{X}^2(t)-(EX(t){)}^2$$

Correlation coefficient $R_X(s,t)$

$$R_X(s,t)=E[X(s)X(t)]$$

while $m_X(t)$ is the mean value of $X(t)$, $D_X(t)$ is the offset of $X(t)$ to mean value at time $t$ ,

$B_X(s,t)$&$R_X(s,t)$ represents the relevance of SP {X(t),t∈T}\{X(t), t\in T\}{X(t),t∈T} from different time $s,t$ .

variance $D(x)$

$$DX=E{X}^2-(EX{)}^2$$

integral representation of $E[f(X)],X\sim U(0,T)$

if $X\sim U(0,T)$:
$$E[f(X)]=\frac{1}{T}{\int }_0^{T}f(x)dx$$
representing the mean value of every possible X in (0, T)

Process with Orthogonal Increments

stochastic process {X(t),t∈T}\{X(t), t\in T \}{X(t),t∈T} ,

if $EX(t)=0$, and $t_1<t_2\le t_3<t_4\in T:E[(X(t_2)-X(t_1))\overline{(X(t_4)-X(t_3)})]=0$,

$X(t),t\in T$ is a process with orthogonal increments.

Specially, if $T=[a,\infty )$ and $X(a)=0$,
$$B_X(s,t)=R_X(s,t)={\sigma }_X^2(min(s,t))$$

Normal distribution

$$\begin{gathered} N(\mu ,{\sigma }^2) \\ EX=\mu ,DX={\sigma }^2 \end{gathered}$$

Specially, in standard Normal distribution,
$$\mu =0,{\sigma }^2=1$$

Poisson process

It can be defined as a counting process, which represents the random number of events up to some time.
P{X(t+s)−X(s)=n}=e−λt(λt)nn! P\{X(t+s)-X(s)=n\}=e^{-\lambda t} \frac{(\lambda t)^n}{n!} P{X(t+s)−X(s)=n}=e−λtn!(λt)n​

• has the natural numbers as its state space and the non-negative numbers as its index set.

let $X(t),t\ge 0$ be a Poisson process, for $t,s \in [0,\infty) $ and $s\le t$ ,
$$E[X(t)-X(s)]=D[X(t)-X(s)]=\lambda (t-s)$$
since $X(0)=0$,
$$\begin{gathered} m_X(t)=\lambda t \\ {\sigma }_x^2(t)=\lambda t \\ {B}_X(s,t)=\lambda s \end{gathered}$$
normally,
$$B_X(s,t)=\lambda \min (s,t)$$

Poisson distribution

$$\begin{gathered} P(\lambda ) \\ P(X=k)=\frac{{\lambda }^k}{k!}{e}^{-\lambda } \\ EX=DX=\lambda \end{gathered}$$

Compound Poisson process

if {N(t),t≥0}\{N(t), t \geq 0 \}{N(t),t≥0} is a Poisson process of $\lambda$,

{Yk,k=1,2,⋯&ThinSpace;}\{Y_k, k=1,2,\cdots\}{Yk​,k=1,2,⋯} is a set of independent and identically distributed random variables,

and is independent to {N(t),t≥0}\{N(t), t \geq 0\}{N(t),t≥0},
$$X(t)=\sum _{k=1}^{N(t)}{Y}_k,t\ge 0,$$
{X(t),t≥0}\{X(t),t\geq0\}{X(t),t≥0} is a Compound Poisson process.
$$\begin{gathered} E[X(t)]=\lambda tE(Y_1) \\ D[X(t)]=\lambda tE(Y_1{)}^2 \end{gathered}$$

Markov Chain

probability transition matrix as
$$P=[p_{ij}]$$
Two-step transition probability matrix as
$$P^{(2)}=PP$$

State classification of Markov chain

assume state space I={1,2,⋯&ThinSpace;,9}I=\{1,2,\cdots ,9\}I={1,2,⋯,9} ,

for state 1, the step $T$ is the steps it takes to go back from state 1

for set {n:n≥1,pii(n)&gt;0}\{n:n\geq1,p_{ii}^{(n)}\gt0\}{n:n≥1,pii(n)​>0},
d=d(i)=G.C.D{n:pii(n)&gt;0} d = d(i)=G.C.D\{n:p_{ii}^{(n)}&gt;0\} d=d(i)=G.C.D{n:pii(n)​>0}
$d$ is the cycle of state $i$,

if $d>1$, state $i$ is periodic,

if $d=1$, state $i$ is aperiodic.
$$f_{ij}=\sum _{n=1}^{\infty }{f}_{ij}^n$$
$f_{ij}$ is the the probability that i can finally reach j,

when $f_{ii}=1$ , the state $i$ is recurrent. The necessary and sufficient condition is
$$\sum _{n=0}^{\infty }{p}_{ii}^{(n)}=\infty$$
Specially, state $i$ is ergodic state if it is aperiodic & recurrent.

stationary distribution

$$\{\begin{array}{l}{\pi }_j={\sum }_{i\in I}{\pi }_i{p}_{ij},\\ \\ {\sum }_{j\in I}{\pi }_j=1,{\pi }_j\ge 0,\end{array}$$

the expected time
$${\mu }_i=\frac{1}{{\pi }_i}$$

The Birth Death process

{X(t),t≥0}\{X(t),t\geq 0\}{X(t),t≥0} is a Birth Death process when
$$\{\begin{array}{ll}{p}_{i,i+1}(h)={\lambda }_{i}h+o(h),& {\lambda }_i>0,\\ p_{i,i-1}(h)={\mu }_{i}h+o(h),& {\mu }_i>0,{\mu }_0=0,\\ p_{ii}(h)=1-{\lambda }_{i}h-{\mu }_{i}h+o(h),& \\ p_{ij}(h)=o(h),& |i-j|\ge 2,\end{array}$$

Kolmogorov forward equation

$$p_{ij}^{\prime }(t)={\lambda }_{j-1}{p}_{i,j-1}(t)-({\lambda }_j+{\mu }_j)p_{ij}(t)+{\mu }_{j+1}{p}_{i,j+1}(t),i,j\in I$$

希望不会挂科吧。。