<p>This paper is concerned with an optimal control problem for stochastic system with regime switching and mean-field interactions in an infinite horizon. The discounted framework is adopted to ensure the stability of the state equation and the well-posedness of the cost functional. By choosing an appropriate discount factor, we first, as a preliminary, establish the global solvability for infinite horizon conditional mean-field (forward and backward) stochastic differential equations with Markov chains and the asymptotic property of their solutions when time goes to infinity. Then, we prove a sufficient stochastic maximum principle for the infinite horizon optimal control problem by means of a dual approach under some convexity condition of the associated Hamiltonian function. Finally, the maximum principle is applied to solve a cash flow management problem of an insurance firm, which turns out to be a linear quadratic optimal control problem. An explicit optimal premium policy and the minimum cost are obtained based on two algebraic Riccati equations and an additional linear equation. Numerical experiments are reported to illustrate the theoretical results.</p>

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Sufficient stochastic maximum principle for mean-field control problems with regime switching in an infinite horizon

  • Na Li,
  • Siyu Lv,
  • Zhen Wu,
  • Jie Xiong

摘要

This paper is concerned with an optimal control problem for stochastic system with regime switching and mean-field interactions in an infinite horizon. The discounted framework is adopted to ensure the stability of the state equation and the well-posedness of the cost functional. By choosing an appropriate discount factor, we first, as a preliminary, establish the global solvability for infinite horizon conditional mean-field (forward and backward) stochastic differential equations with Markov chains and the asymptotic property of their solutions when time goes to infinity. Then, we prove a sufficient stochastic maximum principle for the infinite horizon optimal control problem by means of a dual approach under some convexity condition of the associated Hamiltonian function. Finally, the maximum principle is applied to solve a cash flow management problem of an insurance firm, which turns out to be a linear quadratic optimal control problem. An explicit optimal premium policy and the minimum cost are obtained based on two algebraic Riccati equations and an additional linear equation. Numerical experiments are reported to illustrate the theoretical results.