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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 \(x\).
- Explain why \[ u(x,t) = \int_{\infty}^\infty f(y)p(t,x-y)dy\]
- 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 \(c\) !) - 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
- \[\lim_{N\rightarrow \infty}\mathbf E I_N^\alpha(t) \ .\]
- * \[\lim_{N\rightarrow \infty}\mathbf E \big( I_N^\alpha(t)\big)^2\]
- * 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 ?
