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

The solution XnXn to a nonlinear stochastic differential equation of the form dXn(t)+An(t)Xn(t)dt−12∑j=1N(Bjn(t))2Xn(t)dt=∑j=1NBjn(t)Xn(t)dβjn(t)+fn(t)dt, Xn(0)=xXn(0)=x, where βjn is a regular approximation of a Brownian motion βjβj, Bjn(t) is a family of linear continuous operators from VV to HH strongly convergent to Bj(t)Bj(t), An(t)→A(t)An(t)→A(t), {An(t)}{An(t)} is a family of maximal monotone nonlinear operators of subgradient type from VV to V′V′, is convergent to the solution to the stochastic differential equation dX(t)+A(t)X(t)dt−12∑j=1NBj2(t)X(t)dt=∑j=1NBj(t)X(t)dβj(t)+f(t)dt, X(0)=xX(0)=x. Here V⊂H≅H′⊂V′V⊂H≅H′⊂V′ where VV is a reflexive Banach space with dual V′V′ and HH is a Hilbert space. These results can be reformulated in terms of Stratonovich stochastic equation dY(t)+A(t)Y(t)dt=∑j=1NBj(t)Y(t)∘dβj(t)+f(t)dt.