Joint time-state generalized semiconcavity of the value function of a jump diffusion optimal control problem

Abstract

We prove generalized semiconcavity results, jointly in time and state variables, for the value function of a stochastic finite horizon optimal control problem, where the evolution of the state variable is described by a general stochastic differential equation (SDE) of jump type. Assuming that terms comprising the SDE are \(C^1\)-smooth, and that running and terminal costs are semiconcave in generalized sense, we show that the value function is also semiconcave in generalized sense, estimating the semiconcavity modulus of the value function in terms of smoothness and generalized semiconcavity moduli of data. Of course, these translate into analogous regularity results for (viscosity) solutions of integro-differential Hamilton–Jacobi–Bellman equations due to their controllistic interpretation. This paper may be seen as a sequel to Feleqi (Dyn Games Appl 3(4):523–536, 2013), where we dealt with the generalized semiconcavity of the value function only in the state variable.

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Acknowledgements

I am much in debt to and thank an anonymous reviewer, whose extensive comments, corrections and suggestions helped me very much in improve the paper. I would like to thank also Prof. Piermarco Cannarsa and Prof. Martino Bardi for useful conversations and their advise.

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Correspondence to Ermal Feleqi.

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Feleqi, E. Joint time-state generalized semiconcavity of the value function of a jump diffusion optimal control problem. Nonlinear Differ. Equ. Appl. 26, 4 (2019). https://doi.org/10.1007/s00030-018-0550-6

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Mathematics Subject Classification

  • 35D10
  • 35E10
  • 60H30
  • 93E20

Keywords

  • Generalized semiconcavity
  • Value function
  • Optimal control
  • Jump diffusions
  • Partial integro-differential Hammilton–Jacobi–Bellman equations