The stochastic maximum principle in optimal control of singular diffusions with non linear coefficients

Abstract

We consider a stochastic control problem of a non linear system in which the variable control has two components, the first being absolutely continuous and the second singular. We assume a convex state constraint, a non convex cost criterion and we allow the absolutely continuous component of the control to enter both the drift and diffusion coefficients. The maximum principle is established by using mainly a convex perturbation on a given optimal control. This result generalizes at the same time the result obtained by Cadellinas-Haussman as well as that obtained by Bensoussan.

The Relationship between the Stochastic Maximum Principle and the Dynamic Programming in Singular Control of Jump Diffusions

“…Finally, using these theoretical results, we solve explicitly an example, on optimal harvesting strategy for a geometric Brownian motion, with jumps. Note that our results improve those in [1,2] to the jump diffusion setting. Moreover we generalize results in [3,4], by allowing 2 International Journal of Stochastic Analysis both classical and singular controls, at least in the complete information setting.…”

“…Let be a classical solution of (81) with the terminal condition (82), such that for some constants 1 …”

“…The first result that covers singular control problems was obtained by Cadenillas and Haussmann [18], in which they consider linear dynamics, convex cost criterion, and convex state constraints. A first-order weak stochastic maximum principle was developed via convex perturbations method for both absolutely continuous and singular components by Bahlali and Chala [1]. The second-order stochastic maximum principle for nonlinear SDEs with a controlled diffusion matrix was obtained by Bahlali and Mezerdi [19], extending the Peng maximum principle [20] to singular control problems.…”

Singular mean-field optimal control for forward-backward stochastic systems and applications to finance

“…Proof of duality relation (11). By applying integration by parts formula to * (t)x 1 (t), and since x 1 (0) = 0 we get…”

“…The stochastic singular control problems have received considerable research attention in recent years due to wide applicability in a number of different areas, see for instance [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] . In most classical cases, the optimal singular control problem was investigated through dynamic programming principle.…”