Some results on optimal control of stochastic systems with state chance constraints

We consider a continuous-time control problem with random initial condition and chance constraints. We solve this problem by discretizations and we prove that the discrete-time problem is convex and can be solved by the method of the logarithmic barrier function. Then we prove that when the discretization step goes to zero, the cost of the solutions of the discrete-time problems converge to the optimal cost of the continuous-time problem.