Mean-square approximation for stochastic differential equations

Abstract

The simplest approximate method for solving the Ito system

$$dX = a(t,X)dt + \sum\limits_{r = 1}^q {\sigma _r } (t,X)dw_r (t)$$

(0.1)

is Euler’s method:

$$X_{k + 1} = X_k + \sum\limits_{r = 1}^q {\sigma rk\Delta _k } w_r (h) + |a_k h,$$

(0.2)

where Δ _k w _r (h) = w _r (t _{k +1}) − w _r (t _k ), and the index k at σ _r and a indicates that these functions are evaluated at the point (t _k , X _k ).