COMPUTATIONAL ALGORITHMS FOR DISTRIBUTIONS OF INTEGRAL QUADRATIC FUNCTIONALS, DETERMINED BY SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS

Abstract

Path integration is one of the effective methods of modern theoretical physics and applied mathematics. It is well known that from the known constructions of path integrals only path integrals with respect to the Gaussian measure are taken. The development of computational methods and tools makes it possible to successfully solve a variety of problems. When considering real Markov processes, the result of taking the corresponding path integrals over the Gaussian measure contains root expressions. As a rule, these expressions are Laplace transformants of the desired distributions of values of integral functionals. To obtain the distributions themselves, it is necessary to perform the inverse Laplace transform, in other words, to find the value of the corresponding Fourier integral on the Riemann surface. Due to the two-valued nature of these root expressions, it is impossible to determine the correct sign of the resulting radicals using computer tools. This in turn leads to the need to develop analytical methods focused on the research stage preceding the numerical one. The paper presents the results of the analytical determination of typical path integrals. The procedure for taking a path integral of quadratic form with respect to the amplitude of the solution of a stochastic differential equation is described in detail. The meaningful meaning of this functional is that it describes the average power of a normal process relative to a finite observation interval – the solution of a stochastic differential equation. The paper presents results devoted specifically to the analytical and numerical aspects of obtaining physical and applied dependencies in problems, part of which is the need for statistical averaging in the functional space of solutions of the stochastic differential solution used. As results, dependencies are given that describe the probabilistic properties of the integral functionals under consideration.