stochastic chain rule

stochastic chain rule

[stō′kas·tik ′chān ‚rül]
(mathematics)
A generalization of the ordinary chain rule to stochastic processes; it states that the process Ut= u (Xt 1, Xt 2, …, Xt n ) satisfies with the conventions (dt)2= 0 and dW α dW β= ∂αβ dt, where the X i are processes satisfying {Wt α, t ≥ 0}, α = 1, 2, … , m, are independent Wiener processes; the dWt αare the corresponding random disturbances occurring in the infinitesimal time interval dt ; the at i and bt i αare independent of future disturbances, and u (x1, x2, …, xn) is a function whose derivatives ∂i u and ∂i j u are continuous. Also known as Itô's formula.
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