Obstacle problems for parabolic SDEs with Hölder continuous diffusion: From weak to strong solutions

The article focuses on the qualitative analysis of the following stochastic variational inequalitydu(t)+A(t,u(t))dt+∂IK(t)(u(t))dt∋g(t,u(t))dW(t), considered in a Gelfand-Lions triple space setup V⊂H⊂V⋆. We study the existence and uniqueness of a strong solution under the assumption of Hölder continuity for the diffusion coefficient of our obstacle problem. Imposing some weaker assumptions on the barriers, we provide the existence of an weak variational solution for the multivalued problem. Moreover, the asymptotic behavior of the solution and a maximum principle are provided.