# Tag Archives: JCM_math545_HW2_S14

## A PDE example

Observe that for $$k=0,1,\dots$$

$\phi_k(x) = \sin(\pi k x/2)$

form an orthonormal basis of  function for $$L^2([0,2]$$ with $$\phi(0)=\phi(2)=0$$. Here the inner-product of  two functions in  $$f,g \in L^2([0,2]$$  is

$\langle f,g\rangle =\int_0^2 f(x)g(x) dx$

Define the operator $$L$$ acting on a function $$\phi(x)$$ by

$L\phi(x)=\frac12 \frac{\partial^2 \phi}{\partial^2x}(x) – 5 \phi(x)$

To solve the equation

$\frac{\partial u}{\partial t}(x,t) = (L u)(x,t)$

with

$u(0,t)=u(2,t)=0 \qquad\text{and}\qquad u(x,0)=F(x)$

assume that $$u(x,t)$$ takes the from

$u(x,t)=\sum_{k=0}^\infty a_k(t) \phi_k(x)$

Find the equations for the $$a_k$$ and solve then find an expression for $$u(x,t)$$.

## Levy’s construction of Brownian Motion

Let $$\{ \xi_k^{(n)} : n =0,1,\dots ; k =1,\dots,2^n\}$$ be a collection of independent Gaussian random variables with  $$\xi_k^{(n)}$$ having mean zero and variance $$2^{-n}$$. Define the random variable $$\eta_k^{(n)}$$ recursively by

$\eta_1^{(0)} = Z \qquad\text{with}\quad Z\sim N(0,1) \quad\text{and independent of the $$\xi$$’s}$

$\eta_{2k}^{(n+1)} = \frac12\eta_{k}^{(n)} -\frac12 \xi_{k}^{(n)}$

$\eta_{2k-1}^{(n+1)} = \frac12\eta_{k}^{(n)} +\frac12 \xi_{k}^{(n)}$

For any time $$t \in [0,1]$$ of the form $$t=k 2^{-n}$$ define

$W^{(n)}_t = \sum_{j=1}^k \eta_{j}^{(n)}$

For $$t \in [0,1]$$ not of this form we connect the two nearest defined points with a line.

1. Follow given steps to show that for fixed $$n$$, $$W^{(n)}_t$$ is random walk on $$\mathbf R$$ with Gaussian steps.
1. Show $$\mathbf E \eta_{k}^{(n)} = 0$$ and  $$\mathbf E \big[ (\eta_{k}^{(n)})^2\big] = 2^{-n}$$
2. Argue that $$\eta_{k}^{(n)}$$ is Gaussian and that for any fixed $$n$$,
$\{ \eta_{k}^{(n)} : k=1,\dots, 2^n\}$
are a collection of mutually independent random variables. (To show independence show that they are mean zero Gaussians  with correlation $$\mathbf E [\eta_{k}^{(n)}\eta_{j}^{(n)}]=0$$ when $$j\neq k$$.)
2. To understand the relationship between $$W^{(n)}$$ and $$W^{(n+1)}$$, simulate a collection of random $$\xi_k^{(n)}$$ and plot $W^{(0)}, W^{(1)}, W^{(2)}, W^{(3)}, W^{(4)}$
over the time interval $$[0,1]$$. Notice that at $$n$$ increases the functions seem to converge. Try a few different realizations to get a feeling for how the limiting function might look.

## Calculating with Brownian Motion

Let $$W_t$$ be a standard brownian motion. Fixing an integer $$n$$ and a terminal time $$T >0$$, let $$\{t_i\}_{i=1}^n$$ be a partition of the interval $$[0,T]$$ with

$0=t_0 < t_1< \cdots< t_{n-1} < t_n=T$

Calculate the following two expressions:

1. $\mathbf{E} \Big(\sum_{k=1}^n W_{t_k} \big[ W_{t_{k}} – W_{t_{k-1}} \big] \Big)$
Hint: you might want to do the second part of the problem first and then return to this question and write
$W_{t_k} \big[ W_{t_{k}} – W_{t_{k-1}} \big]= W_{t_{k-1}} \big[ W_{t_{k}} – W_{t_{k-1}} \big]+ \big[W_{t_{k}} -W_{t_{k-1}}\big]\big[ W_{t_{k}} – W_{t_{k-1}}\big]$
2. $\mathbf{E} \Big(\sum_{k=1}^n W_{t_{k-1}} \big[ W_{t_{k}} – W_{t_{k-1}} \big] \Big)$

## Kolmogorov Continuity Theorem : Ilustrative Example

The standard Poisson process  is just the process $$N(t)$$ where $$N(t)$$ takes integer values, the increments are independent ( $$N(t_2)-N(t_1)$$ is independent of $$N(t_4)-N(t_3)$$ for $$t_1 < t_2 \leq t_3 < t_4$$) and for $$t> s \geq 0$$ and $$n \in \mathbf{N}$$

$\mathbf{P}\big(N(t)-N(s)=n\big)=e^{-(t-s)}\frac{(t-s)^n}{n!}$

Here “standard” just means  rate one Poisson process.

Define the process $X(t)=\xi \cdot (-1)^{N(t)}\, ,$ where $$\xi$$ is a random variable independent of the standard Poisson process $$N(t)$$ that take values $$\pm 1$$ with probability $$\frac12$$. Clearly $$X(t)$$ takes only two values, $$\pm 1$$. Show that $$X(t)$$ is stationary and that its covariance is $$e^{-2|t-s|}$$.

The stationary Ornstein-Uhlenbeck process is a Gaussian process with mean zero and covariance $$R(t,s)=\frac{1}{2}e^{-|t-s|}$$. Thus the OU process and $Y(t)=\frac{1}{\sqrt 2}X(\frac{t}{2})$ are both stationary and have the same covariance but are very different processes. Does $$X(t)$$ satisfy the Kolmogorov condition for path continuity? Does the OU process?