Archive for May, 2007

Ito’s Lemma

May 31, 2007

Ito’s lemma can be thought as a generalization of Taylor’s expansion to stochastic processes. Taylor expansion connects the differential of a function F({x}) to d{{x}}.

dF({x}) = {\partial F \over \partial x } dx + {1\over 2 } {\partial^2 F \over \partial x^2} dx^2 + O(dx^3)

In the same spirit Ito’s lemma connects the differential of a function of a stochastic process F(X_t, t) to dX_t and dt.

dF(X_t, t)= \left( {\partial F \over \partial t} + {\partial^2 F \over \partial x^2} \right)dt + \left(\partial F\over \partial x \right) dX_t

This relation can be understood by applying the multiplication rule (dX_t)^2 =dt. The multiplication rule originates from the definition of the Brownian motion.

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