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# 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.

- Follow given steps to show that for fixed \(n\), \(W^{(n)}_t\) is random walk on \(\mathbf R\) with Gaussian steps.
- Show \(\mathbf E \eta_{k}^{(n)} = 0\) and \(\mathbf E \big[ (\eta_{k}^{(n)})^2\big] = 2^{-n}\)
- 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 \(j\neq k\).)

- 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:

- \[ \mathbf{E} \Big(\sum_{k=1}^n W_{t_k} \big[ W_{t_{k}} – W_{t_{k-1}} \big] \Big)\]

Hint: you might want to do the second part of the problem first and then return to this question and write

\[W_{t_k} \big[ W_{t_{k}} – W_{t_{k-1}} \big]= W_{t_{k-1}} \big[ W_{t_{k}} – W_{t_{k-1}} \big]+ \big[W_{t_{k}} -W_{t_{k-1}}\big]\big[ W_{t_{k}} – W_{t_{k-1}}\big]\] - \[ \mathbf{E} \Big(\sum_{k=1}^n W_{t_{k-1}} \big[ W_{t_{k}} – W_{t_{k-1}} \big] \Big)\]

## 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?