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* A Brief Introduction to Stochastic Differential Equations *

Itô Chain Rule

In ordinary calculus the differential of a function f (t) is defined via

\begin{eqnarray} df(t)&=&f(t+dt)-f(t).\nonumber \end{eqnarray}

The differential of a product of two functions f (t) and g(t) can therefore be calculated via

\begin{eqnarray} d(f(t)g(t))&=&f(t+dt)g(t+dt)-f(t)g(t) \nonumber \\ &=&(f(t+dt)... ...t)-g(t))\nonumber \\ &=&df(t)dg(t)+df(t)g(t)+f(t)dg(t).\nonumber \end{eqnarray}

Since df (t) = $ {\frac{{df(t)}}{{dt}}}$dt and dg(t) = $ {\frac{{dg(t)}}{{dt}}}$dt it follows that

\begin{eqnarray} d(f(t)g(t))=\frac{df(t)}{dt}\frac{dg(t)}{dt} dt^2+\frac{df(t)}{dt}g(t)dt+f(t)\frac{dg(t)}{dt} dt.\nonumber \end{eqnarray}

Clearly the first term is order dt2 while the second and third are order dt. Hence, the first term is zero when dt is infinitesimal and hence we get the usual rule

\begin{eqnarray} d(f(t)g(t))=(\frac{df(t)}{dt}g(t)+f(t)\frac{dg(t)}{dt}) dt.\nonumber \end{eqnarray}

Now, consider functions of time and a Wiener process. Similar reasoning leads to the equality

\begin{eqnarray} d(f(t,W_t)g(t,W_t))&=&df(t,W_t)dg(t,W_t)+df(t,W_t)g(t,W_t)+f(t,W_t)dg(t,W_t)\nonumber \end{eqnarray}

but now

\begin{eqnarray} df(t,W_t)&=&(\frac{\partial f(t,W_t)}{\partial t}+\frac{1}{2}\f... ...ial W_t^2})dt+\frac{\partial g(t,W_t)}{\partial W_t}dW_t\nonumber \end{eqnarray}

and so we obtain

\begin{eqnarray} d(f(t,W_t)g(t,W_t))&=&[(\frac{\partial f(t,W_t)}{\partial t}+\f... ... g(t,W_t)}{\partial W_t} dW_t^2+{\rm higher~order~terms}\nonumber \end{eqnarray}

Higher order terms proportional to dWtdt and dt2 can be neglected but dWt2 = dt and so

\begin{eqnarray} d(f(t,W_t)g(t,W_t))&=&[(\frac{\partial f(t,W_t)}{\partial t}+\f... ...t)+f(t,W_t)\frac{\partial g(t,W_t)}{\partial W_t}) dW_t \nonumber \end{eqnarray}

is the rule for the stochastic differential.

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