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Dynamic programming for semi-Markov modulated SDEs

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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.… Click to show full 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.

Keywords: modulated sdes; programming semi; markov modulated; dynamic programming; semi markov

Journal Title: Optimization
Year Published: 2020

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