zbMATH — the first resource for mathematics

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.

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G35 Signal detection and filtering (aspects of stochastic processes)
93E11 Filtering in stochastic control theory
Full Text: DOI
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.