Approximate formulas for the evaluation of the mathematical expectation of functionals from the solution to the linear Skorohod equation

Abstract

This paper is devoted to the construction of approximate formulas for calculating the mathematical expectation of nonlinear functionals from the solution to the linear Skorohod stochastic differential equation with a random initial condition. To calculate the mathematical expectations of nonlinear functionals from random processes, functional analogs of quadrature formulas have been developed, based on the requirement of their accuracy for functional polynomials of a given degree. Most often, formulas are constructed that are exact for polynomials of the third degree [1–9], which are used to obtain an initial approximation and in combination with approximations of the original random process. In the latter case, they are usually also exact for polynomials of a given degree and are called compound formulas. However, in the case of processes specified in the form of compound functions from other random processes the constructed functional quadrature formulas, as a rule, have great computational complexity and cannot be used for computer implementation. This is exactly what happens in the case of functionals from the solutions of stochastic equations. In [1, 2], the approaches to solving this problem were considered for some types of Ito equations in martingales. The solution of the problem is simplified in the cases when the solution of the stochastic equation is found in explicit form: the corresponding approximations were obtained in the cases of the linear equations of Ito, Ito – Levy and Skorohod in [3–11]. In [7, 8, 11], functional quadrature formulas were constructed that are exact for the approximations of the expansions of the solutions in terms of orthonormal functional polynomials and in terms of multiple stochastic integrals. This work is devoted to the approximate calculation of the mathematical expectations of nonlinear functionals from the solution of the linear Skorokhod equation with a leading Wiener process and a random initial condition. A new approach to the construction of quadrature formulas, exact for functional polynomials of the third degree, based on the use of multiple Stieltjes integrals over functions of bounded variation in the sense of Hardy – Krause, is proposed. A composite approximate formula is also constructed, which is exact for second-order functional polynomials, converging to the exact expectation value, based on a combination of the obtained quadrature formula and an approximation of the leading Wiener process. The test examples illustrating the application of the obtained formulas are considered.