A differential calculus for random fields is developed and combined with the S-transform to obtain an explicit strong solution of the Cauchy problem
du(t,x) = (Lu + cu)(t,x)dt + ∑i=1mhiu(t,x)dYit
u(0,x) = u0(x), x∈ℝd
Here L is a linear second order elliptic operator, hi and c are real functions, and Yti = ∫0t ψi(s)ds + Wit, whereWt is a Brownian motion. An application of the solution to nonlinear filtering and mathematical finance is also considered.