A multidimensional Wiener process is controlled by an additive process of bounded variation. A convex nonnegative function measures the cost associated with the position of the state process, and the cost of controlling is proportional to the displacement induced. We minimize a limiting time-average expected (ergodic) criterion. Under reasonable assumptions, we prove that the optimal discounted cost converges to the optimal ergodic cost. Moreover, under some additional conditions there exists a convex Lipschitz continuous function solution to the corresponding Hamilton-Jacobi-Bellman equation which provides an optimal stationary feedback control.
Numerical Analysis and Computation | Probability
Menaldi, J.-L., Robin, M. & Taksar, M.I. Math. Control Signal Systems (1992) 5: 93. doi: 10.1007/BF01211978