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 processU_t=u(X_t¹,X_t², …,X_tⁿ) satisfieswith the conventions (dt)²= 0 anddW^αdW^β= ∂_αβdt, where theXⁱare processes satisfying{W_t^α,t≥ 0}, α = 1, 2, … ,m,are independent Wiener processes; thedW_t^αare the corresponding random disturbances occurring in the infinitesimal time intervaldt; thea_tⁱandb_t^iαare independent of future disturbances, andu(x₁,x₂, …,x_n) is a function whose derivatives ∂_iuand ∂_i∂_juare continuous. Also known as Itô's formula.