A Level-Set Approach for Stochastic Optimal Control Problems Under Controlled-Loss Constraints
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A Level-Set Approach for Stochastic Optimal Control Problems Under Controlled-Loss Constraints Géraldine Bouveret1
· Athena Picarelli2
Received: 13 January 2020 / Accepted: 18 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We study a family of optimal control problems under a set of controlled-loss constraints holding at different deterministic dates. The characterization of the associated value function by a Hamilton–Jacobi–Bellman equation usually calls for strong assumptions on the dynamics of the processes involved and the set of constraints. To treat this problem in the absence of those assumptions, we first convert it into a state-constrained stochastic target problem and then solve the latter by a level-set approach. With this approach, state constraints are managed through an exact penalization technique. Keywords Hamilton–Jacobi–Bellman equations · Viscosity solutions · Optimal control · Expectation constraints Mathematics Subject Classification 93E20 · 49L20 · 49L25 · 35K55
1 Introduction Under general assumptions, the value function associated with unconstrained stochastic optimal control problems can be characterized as the unique continuous viscosity solution of a second-order Hamilton–Jacobi–Bellman (HJB) equation (see, e.g., [1,2]). However, the characterization of the value function associated with stochastic optimal control problems involving state constraints still raises challenges. Such problems
Communicated by Kok Lay Teo.
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Géraldine Bouveret [email protected] Athena Picarelli [email protected]
1
School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore
2
Department of Economics, Università di Verona, Verona, Italy
123
Journal of Optimization Theory and Applications
arise in many applications and are the object of our study. In particular, we focus on state constraints holding in expectation and on a set of deterministic dates. Constraints of this type often involve loss functions and are referred in the literature as controlled-loss constraints. In the finance literature, optimization under risk-measure constraints has been at the cornerstone of modern portfolio selection theory since the pioneering work [3]. We refer the interested reader to [4,5] for an exposition of the different models that have emerged in portfolio selection and their solution methods, and to [6–8] for additional examples of risk-measure constrained portfolio selection problems. In particular, [4] presents a comprehensive analysis of utility-deviation-risk portfolio selection problems. In that study, a deviation-risk-measure term, designed as the expected value of a function of the spread between the underlying portfolio and its mean at the terminal date, appears in the objective function as a penalization to the expected utility. The authors thus derive, under a complete market setting, the necessary and sufficient conditions for optimality through the derivation of a primitive static problem, the so-ca
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