Abstract
In the present paper we derive, via a backward induction technique, an ad hoc maximum principle for an optimal control problem with multiple random terminal times. We thus apply the aforementioned result to the case of a linear quadratic controller, providing solutions for the optimal control in terms of Riccati backward SDE with random terminal time.
Highlights
In the last decades stochastic optimal control theory has received an increasing attention by the mathematical community, in connection with several concrete applications, spanning from industry to finance, from biology to crowd dyamics, etc
A Department of Computer Science, University of Verona, Strada le Grazie, 15, Verona, 37134, Italy E-mail addresses: francescogiuseppe.cordoni@univr.it (Francesco Cordoni), luca.dipersio@univr.it (Luca Di Persio) methods via the Hamilton-Jacobi-Bellman (HJB) equation, and methods based on the maximum principle via backward stochastic differential equations (BSDEs), see, e.g., [19, 33, 37] In particular BSDEs’ methods have proved to be adapted for a large set of stochastic optimal control (SOC)-problems, as reported, e.g., in [34]
After having derived the main result, i.e. the aforementioned maximum principle, we will consider the particular case of a linear–quadratic control problem, that is we assume the underlying dynamics to be linear in both the state variable and the control, with quadratic costs to be minimized
Summary
In the last decades stochastic optimal control theory has received an increasing attention by the mathematical community, in connection with several concrete applications, spanning from industry to finance, from biology to crowd dyamics, etc. After having derived the main result, i.e. the aforementioned maximum principle, we will consider the particular case of a linear–quadratic control problem, that is we assume the underlying dynamics to be linear in both the state variable and the control, with quadratic costs to be minimized Such type of problems have been widely studied both from a theoretical and practical point of view since they often allow to obtain closed form solution for the optimal control. The paper is organized as follows: in Section 2 we introduce the general setting, clarifying main assumptions; Section 2.1 is devoted to the proof of the necessary maximum principle, whereas in Section 2.2 we will prove the sufficient maxim principle; at last, in Section 3, we apply previous results to the case of a linear–quadratic control problems deriving the global solution by an interative scheme to solve a system of Riccati BSDEs
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