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 $dX_n(t)+A_n(t)X_n(t),dt-\tfrac12\sum_{j=1}^N(B_j^n(t))^2X_n(t),dt=\sum_{j=1}^N B_j^n(t)X_n(t)d\beta_j^n(t)+f_n(t),dt$,$X_n(0)=x$,where$\beta_j^n$ is a regular approximation of a Brownian motion ${\beta }_j$, $B_j^n(t)$ is a family of linear continuous operators from $V$ to $H$ strongly convergent to $B_j(t)$, $A_n(t)\to A(t)$, $\{A_n(t)\}$ is a family of maximal monotone nonlinear operators of subgradient type from $V$ to $V^{\mathrm{\prime }}$, is convergent to the solution to the stochastic differential equation $dX(t)+A(t)X(t),dt-\frac12\sum_{j=1}^NB_j^2(t)X(t),dt=\sum_{j=1}^NB_j(t)X(t),d\beta_j(t)+f(t) ,dt$,$X(0)=x$\mathrm{.}Here$V\subset H\cong H’\subset V’$where$V$ is a reflexive Banach space with dual $V^{\mathrm{\prime }}$ and $H$ is a Hilbert space. These results can be reformulated in terms of Stratonovich stochastic equation $dY(t)+A(t)Y(t)dt={\sum }_{j=1}^N{B}_j(t)Y(t)\circ d{\beta }_j(t)+f(t)dt$.