Existence and convergence results for infinite dimensional nonlinear stochastic equations with multiplicative noise

The solution X n to a nonlinear stochastic differential equation of the form d X n ( t ) + A n ( t ) X n ( t ) d t − 1 2 ∑ j = 1 N ( B j n ( t ) ) 2 X n ( t ) d t = ∑ j = 1 N B j n ( t ) X n ( t ) d β j n ( t ) + f n ( t ) d t , X n ( 0 ) = x , where β j n is a regular approximation of a Brownian motion β j , B j n ( t ) is a family of linear continuous operators from V to H strongly convergent to B j ( t ) , A n ( t ) → A ( t ) , { A n ( t ) } is a family of maximal monotone nonlinear operators of subgradient type from V to V ′ , is convergent to the solution to the stochastic differential equation d X ( t ) + A ( t ) X ( t ) d t − 1 2 ∑ j = 1 N B j 2 ( t ) X ( t ) d t = ∑ j = 1 N B j ( t ) X ( t ) d β j ( t ) + f ( t ) d t , X ( 0 ) = x . Here V ⊂ H ≅ H ′ ⊂ V ′ where V is a reflexive Banach space with dual V ′ and H is a Hilbert space. These results can be reformulated in terms of Stratonovich stochastic equation d Y ( t ) + A ( t ) Y ( t ) d t = ∑ j = 1 N B j ( t ) Y ( t ) ∘ d β j ( t ) + f ( t ) d t .