Application of functional integrals to stochastic equations

Abstract

Representing a probability density function (PDF) and other quantities describing a solution of stochastic differential equations by a functional integral is considered in this paper. Methods for the approximate evaluation of the arising functional integrals are presented. Onsager–Machlup functionals are used to represent PDF by a functional integral. Using these functionals the expression for PDF on a small time interval Δt can be written. This expression is true up to terms having an order higher than one relative to Δt. A method for the approximate evaluation of the arising functional integrals is considered. This method is based on expanding the action along the classical path. As an example the application of the proposed method to evaluate some quantities to solve the equation for the Cox–Ingersol–Ross type model is considered.