Continuity of the Value Function for Stochastic Sparse Optimal Control

In this paper, we investigate the continuity of the value function for a stochastic sparse optimal control. The most common method to solve stochastic optimal control problems is the dynamic programming. Specifically, if the value function is smooth, it satisfies the associated Hamilton-Jacobi-Bellman (HJB) equation. However, in general, the value function for our problem is not differentiable because of the nonsmoothness of the L0 cost functional. Instead, we can expect that the value function is a viscosity solution to the HJB equation. This paper shows the continuity of our value function as a first step for showing that the value function is a viscosity solution.