Method of power series for stochastic equations with analytical coefficients (Q1974343)

The aim of the paper is to propose a general analytical theory of stochastic equations with diffusion and drift that are analytical in a variable space. The obtained results are based on the analogy between the deterministic and the stochastic case, and using the fact that the Cauchy-Kovalevskaya theorem can effectively be applied to represent the solution of an integro-differential equation (with analytical coefficients) as a power series in an initial condition. Thus, the author investigates a stochastic analog of the Cauchy-Kovalevskaya theorem for some special hypotheses: (1) stochastic equations in a Hilbert space with linear diffusion and drift, which is analytical in a small neighbourhood of zero; (2) scalar equations with diffusion and drift which are analytical and, possibly, nonlinear in a neighbourhood of zero. Techniques of formal power series are used to establish these new results.