Stochastic equations in the space of formal series: Convergence of solution series

We consider stochastic equations in the space of formal series (an analogue of power series in a Hilbert space with no requirements of convergence). Such an equation is treated as a countable system of linear equations in a Hilbert space. The solution is defined as formal series, whose components solve the given system. The existence and the uniqueness of solution is proved. The main result of this paper is a local convergence of the solution (the convergence of the solution series over a random time interval) in some particular cases. We apply this result to study classical stochastic equations in a Hilbert space with analytical coefficients, whose solutions exist only within a random time interval.