Unconstrained recursive importance sampling (Q988764)

The basic problems in numerical probability is to optimize the computation by a Monte Carlo simulation of real quantities through a probabilistic representations as an expectation. In the present paper an unconstrained stochastic approximation method for finding the optimal change of measure to reduce the variance of Monte Carlo simulations is proposed. The scalar or process parameters are selected by a classical Robbins-Monro procedure without projection or truncation. The convergence for a large class of multidimensional distributions as well as for diffusion processes is proved. The efficiency of the developed algorithm is illustrated. In the Introduction the basic notions, statements, algorithms, results on the paradigm of finite-dimensional and infinite-dimensional settings are recalled. The classical procedure of the Robbins-Monro algorithm is presented. In Section 2 the translation for log-concave probability distributions and the Esscher transform are investigated. The finite-dimensional setting is focused. In Section 2.1 the main result of the paper -- Theorem 1 -- is presented. This theorem is a slight extension of the Robbins-Monro convergence theorem. A program how to apply Theorem 1 is realized. In Section 2.2 Theorem 1 is applied for revision of the Gaussian distribution. In Section 2.3 a self-controlled variant of the algorithms is applied to the case of a too dissymmetric functions. In Theorem 2, under the given assumptions, a recursive procedure is defined, and the distribution and the convergence of the constructed random variables are shown. In Section 2.4 the parametrized exponential change of measure, or the Esscher transform, is considered as an approach to design an importance sampling procedure. In Theorem 3 also, under the given assumptions, a recursive procedure is defined, and the distribution and the convergence of the constructed random variables are shown. In Section 3 the functional version of the presented algorithm is introduced. This approach is based on the Girsanov theorem. The \(d\)-dimensional Itô process as a solution to a stochastic differential equation is considered. In Theorem 4 an algorithm for a construction of a recursive sequence is presented and its convergence is shown. In Section 4 some comments on the practical implementations of the importance sampling by means of a translation in a finite-dimensional setting are presented. In Section 4.1 a purely adaptive approach is considered to reduce the variance. In Section 4.2 the weak rate of convergence is discussed in terms of the central limit theorem. In Section 4.3 the problem of an extension to more general sets of parameters is briefly considered. In section 5 some numerical experiments are carried out. In Section 5.1 both variance reducing approaches to proceed a recursive importance sampling are compared. The translation case for the Robbins-Monro procedure and the Esscher transform are used and graphically illustrated. In Section 5.2 the Down and In Call option is used to find the solution of a given diffusion equation. Three different basis on \(L^2\) -- the Legendre polynomials, the Karhunen-Loève basis and the Haar basis are used to present the solution of the \(k\)-Scholes model. The obtained results are compared. The paper finishes with an appendix, where the proof of Theorem 1 is presented.