Introduction to Stochastic Processes

Abstract

In this chapter we provide mathematical tools to study the stochastic process from the physical point of view.

Problems

2.1

Statistical moment-generating function

(a) Let $\mathbf{x} = (x_1,\ldots , x_n)^{T}$ be a random vector, and $\mathbf{u} = (u_1, \ldots , u_n)^T \in \mathbb {R}^n$, where $(\cdots )^T$ be the transpose of vector $(\cdots )$. The statistical moment-generating function is defined by

$Z_\mathbf{x} (\mathbf {u}) = \langle e^\mathbf{{u}^T \mathbf {x}} \rangle$

for all $\mathbf {u}$ for which the average exists (is finite). Show that the statistical moments of order k can be determined using the following relation:

$$ \frac{\partial ^k}{\partial u_1^{k_1} \cdots \partial u_n^{k_n}} Z_\mathbf{x} (\mathbf{u})|_\mathbf{{u}=0} = \langle x_1 ^{k_1} \cdots x_n ^{k_n} \rangle $$

where $k=k_1+\cdots +k_n$.

(b) The density function of the univariate normal distribution is given by

$$ f(x)=\frac{1}{\sqrt{2\pi }\sigma }\exp \left\{ -\frac{1}{2}\left( \frac{x-\mu }{\sigma }\right) ^2\right\} $$

for $-\infty<x<\infty$, where $\mu$ is the mean and $\sigma ^2>0$ is the variance. Show that

$$ Z_x(u) = \exp \{ \mu u + \frac{\sigma ^2 u^2}{2} \} $$

and prove that, $\langle (x-\mu )^{2n} \rangle = \frac{2n!}{2^n n!} \langle (x-\mu )^{2} \rangle ^n$ and $\langle (x-\mu )^{2n+1} \rangle = 0$, where $n=1,2,\ldots$.

2.2

Bivariate normal distribution

The density function of the bivariate normal distribution is given by

$$ p(x,y)=\frac{\exp \left\{ -\frac{1}{2(1-\rho ^2)}\left[ \left( \frac{x-\mu _x}{\sigma _x}\right) ^2-2\rho \left( \frac{x-\mu _x}{\sigma _x}\right) \left( \frac{y-\mu _y}{\sigma _y}\right) +\left( \frac{y-\mu _y}{\sigma _y}\right) ^2\right] \right\} }{2\pi \sigma _x\sigma _y\sqrt{1-\rho ^2}} $$

where $(\mu _x,\mu _y)$ is the mean vector and the variance-covariance matrix is

$$ \left( \begin{array}{cc} Var(x) &{} Cov(x,y) \\ Cov(x,y) &{} Var(y) \end{array} \right) =\left( \begin{array}{cc} \sigma _x^2 &{} \rho \sigma _x\sigma _y \\ \rho \sigma _x\sigma _y &{} \sigma _y^2 \end{array} \right) . $$

The constraints are $\sigma _x^2>0,\sigma _y^2>0$ and $-1<\rho <1$, where $\rho$ is the correlation coefficient, $\rho =Cov(x,y)/ \sigma _x \sigma _y$ and $Cov(x,y) = \langle (x - \mu _x) (y- \mu _y) \rangle$.

Derive the following conditional averaging and show that,

(a) $\langle y| x \rangle =\mu _y + \rho ~ \frac{\sigma _y}{\sigma _x} ~ (x- \mu _x)$

(b) $\sigma ^2_{y| x} = \sigma _y^2 ~ (1-\rho ^2)$.

2.3

p-variate normal distribution

The density function of the p-variate normal distribution is given by

$$ f(\mathbf {x})=\frac{1}{\left( 2\pi \right) ^{p/2}\left| \mathbf {g}\right| ^{1/2}}\exp \left\{ -\frac{1}{2}\left( {\mathbf x}-{\varvec{\mu }}\right) ^T\mathbf {g}^{-1}\left( {\mathbf x}-{\varvec{\mu }}\right) \right\} $$

where $\mathbf {x}^T=(x_1,\ldots ,x_p)$, ${\varvec{\mu }}^T=( \mu _1,\ldots ,\mu _p)$ and $\mathbf {g}$ is a full rank variance-covariance matrix, i.e.,

$$ \mathbf {g}_{ij}=Cov(x_i,x_j) = \langle (x_i - \mu _i) (x_j - \mu _j) \rangle $$

where $\mathbf {g}^{-1}$ and $\left| \mathbf {g}\right|$ are the inverse and the determinant of $\mathbf {g}$.

(a) Derive the statistical moment-generating function for p-variate normal distribution.

(b) Prove the following relation for the fourth-order correlation function (Wick’s Theorem):

$$ \langle (x_i - \mu _i) (x_j - \mu _j) (x_k - \mu _k) (x_l -\mu _l) \rangle = \mathbf {g}_{ij} \mathbf {g}_{kl} + \mathbf {g}_{ik} \mathbf {g}_{jl} + \mathbf {g}_{il} \mathbf {g}_{jk} . $$

2.4

Chapman–Kolmogorov equation

Show that the following conditional density functions, (a) Brownian motion and (b) Cauchy process satisfy the Chapman–Kolmogorov equation

$$\begin{aligned} (a) \quad p(x_2,t_2|x_1,t_1)= & {} \frac{1}{\sqrt{ 2 \pi (t_2-t_1)}} \exp \left\{ - \frac{(x_2-x_1)^2}{2(t_2-t_1)} \right\} , \\\nonumber \\ \\\nonumber \\ (b) \quad p(x_2,t_2|x_1,t_1)= & {} \frac{1}{\pi } \frac{t_2-t_1}{(t_2-t_1)^2 + (x_2-x_1)^2}\end{aligned}$$

where $t_2>t_1$.