On calculation of moments of the solutions to one class of linear Skorohod SDE on Poisson space

Abstract

The known representation of the solution of the linear stochastic differential equation of Skorohod on Poisson space with random coefficients and an initial condition contains as an unknown parameter a family of transformations of the probability space of the leading random process determined by the solution of the integral stochastic equation. In this paper, we consider cases when the solution of this integral equation can be found in explicit form. Explicit solutions are obtained in two cases in the class of linear Skorohod equations on Poisson space with random coefficients and an initial condition linearly dependent on the time of the first jump of the leading process. The first three moments of the solution of the original SDEs are estimated and a numerical example is given. The obtained formulas for calculating the moments of the solution of Skorohod SDE with the leading Poisson process can be used in constructing approximate formulas for calculating the mathematical expectations of nonlinear functionals of the solution, similar to those considered earlier for Skorohod equations with the leading Wiener process.