On necessary and sufficient conditions for near-optimal singular stochastic controls

Abstract

This paper is concerned with necessary and sufficient conditions for near-optimal singular stochastic controls for systems driven by a nonlinear stochastic differential equations (SDEs in short). The proof of our result is based on Ekeland's variational principle and some dilecate estimates of the state and adjoint processes. This result is a generalization of Zhou's stochastic maximum principle for near-optimaity to singular control problem.

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

“…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.…”

“…Stochastic maximum principle for optimal control problems of forward backward systems involving impulse controls has been studied in Wu and Zhang [3,12]. The stochastic maximum principle for singular control was considered by many authors, see for instance [1,2,[4][5][6][7][8][9][10]. The first version of maximum principle for singular stochastic control problems was obtained by Cadenillas and Haussmann [9].…”

“…In Dufour and Miller [10], the authors derived stochastic maximum principle where the singular part has a linear form. The necessary and sufficient conditions for near-optimal singular control was obtained by Hafayed Abbas and Veverka [5]. For this type of problem, the reader may consult the papers by Haussmann and Suo [4] and the list of references therein.…”

On near-optimal necessary and sufficient conditions for forward-backward stochastic systems with jumps, with applications to finance

“…Indeed, optimal controls may not even exist in many situations, while near-optimal controls always exist. Various kinds of near-optimal control problems have been investigated in [6], [10], [12], [11], [13], [16], [18], [31], [32], [33]. In an interesting paper, Zhou [33] established second-order necessary as well as sufficient conditions for near-optimal stochastic controls for controlled diffusion, where the coefficients were assumed to be twice continuously differentiable.…”

“…In an interesting paper, Zhou [33] established second-order necessary as well as sufficient conditions for near-optimal stochastic controls for controlled diffusion, where the coefficients were assumed to be twice continuously differentiable. However, in Hafayed, Abbas and Veverka [11], the authors extended Zhou's maximum principle [33] to singular stochastic control. The near-optimal control problem for systems governed by Volterra integral equations has been studied in Pan and Teo [18].…”