Stochastic variational inequalities with oblique subgradients

In this paper we will study the existence and uniqueness of the solution for the stochastic variational inequality with oblique subgradients of the following form: { d X t + H ( X t ) ∂ φ ( X t ) ( d t ) ∋ f ( t , X t ) d t + g ( t , X t ) d B t , t > 0 , X 0 = x ∈ Dom ( φ ) ¯ . Here, the mixture between the monotonicity property of the subdifferential operator ∂ φ and the Lipschitz property of the matrix mapping X ⟼ H ( X ) leads to stronger difficulties in comparison to the classical case of stochastic variational inequalities. The existence result is based on a deterministic approach: a differential system with singular input is first analyzed.