Itô Stochastic Taylor Expansion

* A Brief Introduction to Stochastic Differential Equations *

Itô Stochastic Taylor Expansion

Consider a set of sdes with strong solutions. The solutions can therefore be expanded in Taylor series. Keeping terms of order dt or less then gives

$\begin{eqnarray} X^j_{t+dt}&=&X^j_t+\frac{\partial X^j_t}{\partial t}~dt+\sum_{k... ...ial^2 X^j_t}{\partial W^k_t\partial W^l_t}~dW^k_tdW^l_t.\nonumber \end{eqnarray}$

The product of differentials dW^k_tdW^l_t is equivalent to $\delta_{k,l}dt$ in the Itô formulation of stochastic calculus, so that

$\begin{eqnarray} dX^j_{t+dt}=X^j_{t+dt}-X^j_t&=&[\frac{\partial X^j_t}{\partial ... ...sum_{k=1}^m\frac{\partial X^j_t}{\partial W^k_t}~dW^k_t.\nonumber \end{eqnarray}$

Comparison to the original sdes allows us to identify the first derivatives

$\begin{eqnarray} \frac{\partial X^j_t}{\partial W^k_t}&=&b^j_k({\bf X}_t,t) \non... ..._t,t)\frac{\partial b^j_k({\bf X}_t,t)}{\partial X_t^i}.\nonumber \end{eqnarray}$

From these first order derivatives, expressed in terms of a^j and b^j_k, higher order derivatives can be computed. Thus a Taylor expansion of the solutions

$\begin{eqnarray} X^j_{t+\Delta t}&=&X^j_t+\frac{\partial X^j_t}{\partial t}\Delt... ...rtial W^k_t\partial W^l_t}\Delta W^k_t\Delta W^l_t+\dots\nonumber \end{eqnarray}$

can be obtained for finite displacements $\Delta$ t and $\Delta$ W^k_t.

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