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

\[\eta_i(\omega)= \omega_i -\frac12\]


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


  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 ?