On the Existence of Weak Variational Solutions to Stochastic Differential Equations

We study the existence of weak variational solutions in a Gelfand triplet of real separable Hilbert spaces, under continuity, growth, and coercivity conditions on the coefficients of the stochastic differential equation. The laws of finite dimensional approximations are proved to weakly converge to the limit which is identified as a weak solution. The solution is an H– valued continuous process in L2(Ω, C([0, T ], H)) ∩ L2([0, T ] × Ω, V ). Under the assumption of monotonicity the solution is strong and unique.