# Tag Archives: JCM_math545_HW3_S14

## 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 $$f$$.

## Ito to Stratonovich

Let’s think about different ways to make sense of $\int_0^t W(s)dW(s)$ were $$W(t)$$ is a standard Brownian motion. Fix any $$\alpha \in [0,1]$$define

\begin{equation*}
I_N^\alpha(t)=\sum_{j=0}^{N-1} W(t_j^\alpha)[W(t_{j+1})-W(t_j)]
\end{equation*}
were $$t_j=\frac{j t}N$$ and $$t_j^\alpha=\alpha t_j + (1-\alpha)t_{j+1}$$.
Calculate

1. $\lim_{N\rightarrow \infty}\mathbf E I_N^\alpha(t) \ .$
2. * $\lim_{N\rightarrow \infty}\mathbf E \big( I_N^\alpha(t)\big)^2$
3. * For which choice of $$\alpha$$ is $$I_N^\alpha(t)$$ a martingale ?

What choice of $$\alpha$$ is the standard It\^o integral ? What choice is the Stratonovich integral ?