Minimization of the dispersion of estimate by collisions in problems with elements of alternating signs. (Russian) Zbl 0542.65092

Bulavskij, Yu. V.

Zh. Vychisl. Mat. Mat. Fiz. 24, No. 1, 148-152 (1984).

We consider the integral equation \[ (1)\quad \phi(x)=\int_{x}k(x,x')\phi(x')dx'+h(x), \] briefly \(\phi =K\phi +h,\) where \(X\subset {\mathbb{R}}^ n\), \(h\in L_{\infty}(X)\) and K is a linear operator in \(L_{\infty}(X)\). For each \(x_ 0\in X\) one can approximate \(\phi(x_ 0)\) by the Monte Carlo method

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. Let \(x_ 0,x_ 1,...,x_ n,..\). be the trajectory of a random process

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in X with transient probability densities \(p_ n(x_ 0,...,x_{n-1},x)=(1-p(x_{n- 1}))r_ n(x_ 0,...,x_{n-1},x).\) We define \(Q_ 0=1\), \(Q_ n=0\) for \(p(x_{n-1})=1\) and \(Q_ n=Q_{n-1}k(x_{n-1},x_ n)/p_ n(x_ 0,...,x_{n-1},x_ n)\) for \(p(x_{n-1})<1\), (\(n\geq 1)\). Under certain assumptions \((2)\quad \xi_{x_ 0}=\sum^{\infty}_{k=0}Q_ kh(x_ k)\) is an unbiased estimate of \(\phi(x_ 0)\). This procedure may be generalized to a nonlinear equation \[ (3)\quad \phi(x)=\int_{x}...\int_{x}k(x,y_ 1,...,y_ m)\prod^{m}_{i=1}\phi(y_ i)dy_ i+h(x) \] [cf. S. M. Ermakov, ibid. 13, 564-573 (1973; Zbl 0266.65083)]. For fixed p(x), \(x\in X\) there arises the problem to find \(r_ n(x_ 0,...,x_{n- 1},x)\) (\(n\geq 1)\) such that the dispersion of the estimate (2) is minimal uniformly with respect to \(x_ 0\). Cf. G. A. Mikhajlov [Nonlinear theory of optimization of statistical simulation for the solution of integral equations of second kind, ibid. 21, 1435-1444 (1981)] has solved this problem for nonnegative k and h. The solution for general k and h (in (1) and (3)) is given in the present paper.

Reviewer: V.Veselý

MSC:

65R20	Numerical methods for integral equations
65C05	Monte Carlo methods
45G10	Other nonlinear integral equations

Keywords:

minimal dispersion; trajectory of a random process

Citations:

Zbl 0266.65083