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Risk‐sensitive maximum principle for stochastic optimal control of mean‐field type Markov regime‐switching jump‐diffusion systems
Author(s) -
Moon Jun
Publication year - 2021
Publication title -
international journal of robust and nonlinear control
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.361
H-Index - 106
eISSN - 1099-1239
pISSN - 1049-8923
DOI - 10.1002/rnc.5358
Subject(s) - mathematics , maximum principle , jump diffusion , convexity , optimal control , mean field theory , separation principle , stochastic differential equation , markov process , mathematical optimization , jump , nonlinear system , statistics , physics , quantum mechanics , financial economics , economics , state observer
We consider the risk‐sensitive optimal control problem for mean‐field type Markov regime‐switching jump‐diffusion systems driven by Brownian motions and Poisson jumps with (Markovian) switching coefficients. The system is coupled with its mean‐filed term, that is, the expected value of the state process, and the objective functional is of the risk‐sensitive type. Our problem is closely related to the mean‐field type robust optimization problem for a general class of stochastic jump systems due to the inherent feature of the risk‐sensitive objective functional. By establishing the logarithmic transformations of the associated equivalent singular risk‐neutral control problem, we obtain the risk‐sensitive maximum principle type necessary and sufficient conditions for optimality, where the sufficient condition requires an additional convexity assumption. The risk‐sensitive maximum principle in this article is characterized as the variational inequality, together with the first‐ and second‐order (mean‐field type) adjoint processes as well as the auxiliary first‐order adjoint process. Unlike the risk‐neutral and mean‐field free cases, the additional adjoint equation is induced due to the mean‐field coupling term and the risk‐sensitive logarithmic transformation. We apply the risk‐sensitive maximum principle of this article to the risk‐sensitive linear‐quadratic problem, for which an explicit optimal solution is obtained.