An asymptotically efficient difference formula for solving stochastic differential equations.

*(English)*Zbl 0618.60053Time 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.

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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