Ito’s Lemma « Randomness

Ito’s Lemma

By Peyman Khorsand

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.

This entry was posted on May 31, 2007 at 4:22 am and is filed under Stochastic Processes. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.

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Ito’s Lemma

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