Dynamic Programming and a Verification Theorem for the Recursive Stochastic Control Problem of Jump-Diffusion Models With Random Coefficients

Jun Moon, Tamer Basar

Research output: Contribution to journalArticlepeer-review

Abstract

In this article, we consider the stochastic optimal control problem for (forward) stochastic differential equations (SDEs) with jump diffusions and random coefficients under a recursive-type objective functional captured by a backward SDE (BSDE). Due to the jump-diffusion process with random coefficients in both the constraint (forward SDE) and the recursive BSDE objective functional, the associated Hamilton-Jacobi-Bellman equation (HJBE) is an integro-type second-order nonlinear stochastic PDE driven by both Brownian motion and (compensated) Poisson process, which we call the integro-type stochastic HJBE (ISHJBE) with jump diffusions. We first prove the dynamic programming principle for the value function using the backward semigroup associated with the recursive objective functional and the precise estimates of BSDEs, by which the continuity of the value function is also shown. Then we establish a verification theorem, which provides a sufficient condition of optimality and characterizes the value function using the (stochastic) solution of the ISHJBE with jump diffusions. Under suitable assumptions, we show the existence and uniqueness of the weak solution to the ISHJBE via the Sobolev space technique. Finally, we apply the verification theorem to the general indefinite linear-quadratic problem and the utility maximization problem; for both problems, explicit optimal solutions are characterized by solving the corresponding ISHJBE with jump diffusions.

Original languageEnglish (US)
Pages (from-to)6474-6488
Number of pages15
JournalIEEE Transactions on Automatic Control
Volume67
Issue number12
DOIs
StatePublished - Dec 1 2022
Externally publishedYes

Keywords

  • Dynamic programming
  • Dynamical systems
  • Forward and backward stochastic differential equations with jump diffusions
  • integro-type stochastic PDE
  • Mathematical models
  • Moon
  • Optimal control
  • random coefficients
  • Stochastic processes
  • verification theorem
  • Viscosity
  • Forward and backward stochastic differential equations (BSDE) with jump diffusions

ASJC Scopus subject areas

  • Electrical and Electronic Engineering
  • Control and Systems Engineering
  • Computer Science Applications

Fingerprint

Dive into the research topics of 'Dynamic Programming and a Verification Theorem for the Recursive Stochastic Control Problem of Jump-Diffusion Models With Random Coefficients'. Together they form a unique fingerprint.

Cite this