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

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

  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 ?