Representation of the bilinear system output by multiple stochastic integrals

Consideration was given to the scalar stochastic differential Ito equation whose drift and diffusion coefficients are affine functions of the phase coordinate. Its solution was represented in terms of the stochastic exponent which plays the part of the equation resolvent. Expansion of the stochastic exponent in series in the Hermit polynomials induces expansion of the solution of the bilinear equation in series in multiple stochastic integrals. Obtained were the moment characteristics of the stochastic integrals solving the problem of statistical analysis of the approximate solutions of the bilinear stochastic systems. A model of the financial $(B,S)$-market and its optimization in terms of a nonsquare criterion were considered by way of example of a bilinear system.