Optimality necessary conditions in singular stochastic control problems with nonsmooth data

Abstract

The present paper studies the stochastic maximum principle in singular optimal control, where the state is governed by a stochastic differential equation with nonsmooth coefficients, allowing both classical control and singular control. The proof of the main result is based on the approximation of the initial problem, by a sequence of control problems with smooth coefficients. We, then apply Ekeland's variational principle for this approximating sequence of control problems, in order to establish necessary conditions satisfied by a sequence of near optimal controls. Finally, we prove the convergence of the scheme, using Krylov's inequality in the nondegenerate case and the Bouleau-Hirsch flow property in the degenerate one. The adjoint process obtained is given by means of distributional derivatives of the coefficients.

Singular Stochastic Control and Optimal Stopping with Partial Information of Itô--Lévy Processes

“…This was extended by Boetius and Kohlmann [7], and subsequently extended further by Benth and Reikvam [6], to more general continuous diffusions. More recently, maximum principles for singular stochastic control problems have been studied in [1,2,3,4]. None of these papers deal with jumps in the state dynamics and none of them deal with partial information control.…”

Irreversible Capital Accumulation with Economic Impact

“…Given a capacity expansion processes ζ ∈ Z, we consider the capacity process Y defined by (2) and the economic indicator process X given by (4)- (5). Using Itô's formula, we can verify that…”

“…Since then, the area has attracted considerable interest in the literature. Apart from references that we have discussed in the context of capacity expansion models, Bahlali et al [5] Haussmann and Suo [27,28] [46] and Zhu [49], provide an alphabetically ordered list of further contributions. In the references discussed above, the controlled process affects the state dynamics in a purely additive way: the change of the state process due to control action does not depend on the state process itself.…”

A stochastic maximum principle for mixed regular-singular control problems via Malliavin calculus

“…Similar techniques have been used by Anderson [1] and Bahlali et al [6], to study the stochastic maximum principle for relaxed-singular controls. The case of systems with non smooth coefficients has been treated by Bahlali et al [4], where the classical derivatives are replaced by the generalized ones in the definition of adjoint processes. See also the recent paper by ∅ksendal and Sulem [29], where Malliavin calculus techniques have been used to define the adjoint process.…”