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An asymptotically efficient difference formula for solving stochastic differential equations. (English) Zbl 0618.60053
Time discretizations of the vector stochastic differential equation $dx_ t=a(x_ t)dt+b(x_ t)dy_ t\quad for\quad t\in [0,T]$ are considered, where $$(y_ t)$$ is a continuous scalar process whose distribution is absolutely continuous with respect to Wiener measure.
Among approximations to $$x_ T$$ that depend on $$(y_ t)$$ only at the discretization points, $$t=0,h,2h,...,T$$, the conditional mean $$E(x_ T| y_ 0,...,y_ T)$$ is asymptotically optimal in the sense that it minimizes all symmetrical conditional moments of the error. A conditional central limit theorem is derived for this minimal error, and a finite difference formula is developed which yields approximations with the same asymptotically optimal properties. This formula is necessarily more complex than the familiar Mil’shtejn scheme [G. N. Mil’shtejn, Teor. Veroyatn. Primen 19, 583-588 (1974; Zbl 0314.60039)]. The latter has the maximum order of convergence but its error, considered as a power series in the discretization parameter h, does not have the minimal leading coefficient.
The results generalize for a special class of equations with multi- dimensional forcing terms.

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60G35 Signal detection and filtering (aspects of stochastic processes) 93E11 Filtering in stochastic control theory
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