The stochastic reflection problem with multiplicative noise

This paper addresses the existence and uniqueness of strong solutions to the stochastic variational inequality dX−ΔXdt+F(t,ξ,X)dt+β(X)dt∋∑k=1mXμkdβk(t)+f(t)dt in a bounded domain O⊂RdO⊂Rd with Dirichlet homogeneous conditions. Here β(r)=0β(r)=0 for r>0r>0, β(0)=]−∞,0]β(0)=]−∞,0], β(r)≠0β(r)≠0 for r<0r<0. An application to the one-phase Stefan problem with a stochastic heat source is given. One studies also the corresponding stochastic parabolic equation with Signorini boundary conditions ∂X∂ν+β(X)∋0 on ∂O∂O.