Abstract
We give a generalized measurable-predictable solution to a higher-order stochastic parabolic initial-value problem in terms of the further generalized Hermite polynomials. Condition and estimates on the existence and uniqueness of the solution are given. We prove the upper second moment growth bound estimate for the solution and consequently show that the second moment of the solution grows exponentially in time with respect to the parameter lambda at the precise rate of 2 + c(3)lambda(2)Lip(sigma)(2), c(3) > 0 and; c(3)Lip(sigma)(2) as the noise level increases.