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\title{Derivatives Lab}
\author{Session 15}
\date{November 8, 2011}
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\def\e{\mathrm e}
\def\d{\mathrm d}

Let $X=X(t)$ be an It\=o process, i.e., a solution of the
stochastic differential equation
\[
  \d X = u(X,t) \, \d t + v(X,t) \, \d W \,,
\]
interpreted in the sense of the It\=o stochastic integral.  Let
$f(X,t)$ be twice continuously differentiable.
Then the stochastic chain rule, also known as the It\=o formula, reads
\[
  \d f(X,t)
  = \biggl(
      \frac{\partial f(X,t)}{\partial t}
      + u(X,t) \, \frac{\partial f(X,t)}{\partial X}
      + \frac12 \, v(X,t)^2 \, \frac{\partial^2 f(X,t)}{\partial X^2}
    \biggr) \, \d t
  + v(X,t) \, \frac{\partial f(X,t)}{\partial X} \, \d W \,.
\]

Verify the It\=o formula numerically for the example when $X(t)$ is
geometric Brownian motion and 
\[
  f(X,t) = t \, \sqrt X \,.
\]

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