Dynamic programming for semi-Markov modulated SDEs. (3rd August 2022)

• Record Type: Journal Article
• Authors: Azevedo, N.; Pinheiro, D.; Pinheiro, S.
• Is Part Of: Optimization. Volume 71:Number 8(2022)
• Page Start: 2315
• Page End: 2342
• Publication Date: 2022-08-03
• Subjects: Stochastic optimal control -- dynamic programming -- semi-Markov processes; 49L20 -- 60K15 -- 90C39 -- 91G80; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2020.1839072
• Languages: English
• ISSNs: 0233-1934

Abstract

ABSTRACT: We consider a stochastic optimal control problem with state variable dynamics described by a stochastic differential equation of diffusive type modulated by a semi-Markov process with a finite state space. The time horizon is both deterministic and finite. Within such setup, we provide a detailed proof of the dynamic programming principle and use it to characterize the value function as a viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation. We illustrate our results with an application to Mathematical Finance: the generalization of Merton's optimal consumption-investment problem to financial markets with semi-Markov switching.