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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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References:
[1] Barnett S., Matrix Methods in Stability Theory (1970) · Zbl 0243.93017
[2] Bartle R. G., The Elements of Real Analysis (1976) · Zbl 0309.26003
[3] Clark J. M. C., Stochastic Differential Systems-Filtering and Control (1980)
[4] Clark J. M. C., Advances in Filtering and Optimal Stochastic Control (1982)
[5] Friedman A., Stochastic Differential Equations and Applications 1 (1975) · Zbl 0323.60056
[6] DOI: 10.1007/BF02846028 · Zbl 0053.40901
[7] Milshtein G. N., Theory Prob. Appl 19 pp 557– (1974)
[8] Newton N. J., Discrete Approximations for Markov Chain Filters (1984)
[9] Newton N. J., Theory and Applications of Nonlinear Control Systems pp 555– (1986)
[10] Talay D., Filtering and Control of Random Processes (1984) · Zbl 0542.93077
[11] Wagner W., Approximation of ltd Differential Equations (1978)
[12] DOI: 10.1007/BF00536382 · Zbl 0164.19201
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