# Tag Archives: JCM_math545_HW2_S17

## Diffusion and Brownian motion

Let $$B_t$$ be a standard Brownian Motion  starting from zero and define

$p(t,x) = \frac1{\sqrt{2\pi t}}e^{-\frac{x^2}{2t} }$

Given any $$x \in \mathbf R$$, define $$X_t=x + B_t$$ . Of course $$X_t$$ is just a Brownian Motion stating from $$x$$ at time 0. Fixing a smooth, bounded, compactly supported function $$f:\mathbf R \rightarrow \mathbf R$$, we define the function $$u(x,t)$$ by

$u(x,t) = \mathbf E_x f(X_t)$

where we have decorated the expectation with the subscript $$x$$ to remind us that we are starting from the point $$x$$.

1. Explain why $u(x,t) = \int_{\infty}^\infty f(y)p(t,x-y)dy$
2. Show by direct calculation using the formula from the previous question that for $$t>0$$, $$u(x,t)$$ satisfies the diffusion equation
$\frac{\partial u}{\partial t}= c\frac{\partial^2 u}{\partial x^2}$
for some constant $$c$$. (Find the correct $$c$$ !)
3. Again using the formula from part 1), show that
$\lim_{t \rightarrow 0} u(t,x) = f(x)$
and hence the initial condition for the diffusion equation is $$f$$.

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

## Simple Numerical Exercise

Let $$\omega_i$$ , $$i=1,\cdots$$ be a collection of mutually independent, uniform on $$[0,1]$$ random variables. Define

$\eta_i(\omega)= \omega_i -\frac12$

and

$X_n(\omega) = \sum_{i=1}^n \eta_i(\omega)\,.$

1. What is $$\mathbf{E}\,X_n$$ ?
2. What is $$\mathrm{Var}(X_n)$$ ?
3. What is $$\mathbf{E}\,X_{n+k} | X_n$$ for $$n, k >0$$ ?
4. What is $$\mathbf{E}(\,X_5^2 \,|\, X_3)$$ ?
5. [optional] Write a computer program to simulate some realizations of this process viewing $$n$$ as time. Plot some plots of $$n$$ vs $$X_n$$.
6. [optional] How do you simulations agree with the first two parts ?