## Optimization of weighted Monte Carlo methods with respect to auxiliary variables.(Russian, English)Zbl 1052.65003

Sib. Mat. Zh. 45, No. 2, 399-409 (2004); translation in Sib. Math. J. 45, No. 2, 331-340 (2004).
The authors consider the problems whose mathematical model is determined by some Markov chain terminating with probability one, see, e.g., [S. M. Ermakov and G. A. Mikhajlov, Statistical simulation. (Russian). Moscow: Nauka (1982; Zbl 0599.65001)]. Moreover, they have to estimate linear functionals of a solution to an integral equation of the second kind with the corresponding substochastic kernel and free term. To construct weighted modifications of numerical statistical models, they supplement the coordinates of the phase space with auxiliary variables whose random values functionally define the transitions in the initial chain.
Having implemented each auxiliary random variable, the authors multiply the weight by the ratio of the corresponding densities of the initial and numerically modeled distributions. They solve the minimization problem for the variances of estimators of linear functionals by choosing the modeled distribution of the first auxiliary random variable.

### MSC:

 65C05 Monte Carlo methods 45K05 Integro-partial differential equations 65R20 Numerical methods for integral equations

Zbl 0599.65001
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