Itô Differentials

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Itô Differentials

Infinitesimal increments dW_t in a Wiener process W_t are sampled from

$\begin{eqnarray} \frac{1}{\sqrt{2\pi dt}} e^{-dW_t^2/2dt}\nonumber \end{eqnarray}$

and obey $\overline{dW_t^2}=dt$ . The variance of dW_t² is proportional to dt² and hence vanishes for infinitesimal dt. This means that dW_t² = dt in Itô calculus. Another peculiarity is that if dW_t^a and dW_t^b are statistically independent infinitesimal increments for two different Wiener processes W_t^a and W_t^b then dW_t^adW_t^b = 0. These properties force us to alter the normal rules of calculus.

For example, consider the Itô differential:

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

Taylor expanding f (t + dt, W_t + dW_t) about t, W_t gives

$\begin{eqnarray} f(t+dt,W_t+dW_t)&=&f(t,W_t)+\frac{\partial f(t,W_t)}{\partial t... ...}\frac{\partial^2 f(t,W_t)}{\partial W_t^2}dW_t^2+\dots \nonumber \end{eqnarray}$

and substituting into the formula for the differential and using dW_t² = dt gives

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

which differs from the rule in normal calculus.

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