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E. A~ ;:::: 2klln hi. 34) has the meansquare order 1/2. 4. 37). Then the following inequality holds: o :::; E(e - d) = 1 - Ed : :; (1 + 2J2klln hl)h k . 38) 40 1 Mean-square approximation for stochastic differential equations Proof. 38) follows. 39) Since IChl :::; yl21lnhl, this method is realizable for all h satisfying the inequality 1 2hllnhl < 2". 5. 39) is of the mean-square order 1/2. Proof. 32). We get 00 00 m=O m=O It is obvious from here that the principal term in the expansion of E(X - X) is equal to xa 2 h(Ed - 1).
Therefore, 18 1 Mean-square approximation for stochastic differential equations we need both derivation of various methods and additional investigation of properties of derived methods. When we derive a method, it is natural to use some convenient and at the same time sufficiently broad conditions which allow us to prove convergence of the method. The proof of convergence under less restrictive conditions and the investigation of its properties are further problems which have to be considered both theoretically and experimentally.
3) is N(O, h 3 /4)-distributed, and the second term is N(O, h 3 /12)-distributed. 1) with a single noise is rather simple. 4 Modeling of Ito integrals 47 w2(O)dO. In , the characteristic function of these random variables is found. However, it is very complicated and cannot be useful in practice. Thus, the exact modeling has bad perspectives, and therefore we need to be able to model these variables approximately. 1. 3)) Xt,x(t + h) = x + A(t, x, hj Wi(O) - Wi(t), i = 1, ... 4) generates a method with order of accuracy m.