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Representation of asymptotic values for nonexpansive stochastic control systems

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  • Li, Juan
  • Zhao, Nana

Abstract

In ergodic stochastic problems the limit of the value function Vλ of the associated discounted cost functional with infinite time horizon is studied, when the discounted factor λ tends to zero. These problems have been well studied in the literature and the used assumptions guarantee that the value function λVλ converges uniformly to a constant as λ→0. The objective of this work consists in studying these problems under the assumption, namely, the nonexpansivity assumption, under which the limit function is not necessarily constant. Our discussion goes beyond the case of the stochastic control problem with infinite time horizon and discusses also Vλ given by a Hamilton–Jacobi–Bellman equation of second order which is not necessarily associated with a stochastic control problem. On the other hand, the stochastic control case generalizes considerably earlier works by considering cost functionals defined through a backward stochastic differential equation with infinite time horizon and we give an explicit representation formula for the limit of λVλ, as λ→0.

Suggested Citation

  • Li, Juan & Zhao, Nana, 2019. "Representation of asymptotic values for nonexpansive stochastic control systems," Stochastic Processes and their Applications, Elsevier, vol. 129(2), pages 634-673.
    • Handle: RePEc:eee:spapps:v:129:y:2019:i:2:p:634-673
    DOI: 10.1016/j.spa.2018.03.015
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    References listed on IDEAS

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    1. Richou, Adrien, 2009. "Ergodic BSDEs and related PDEs with Neumann boundary conditions," Stochastic Processes and their Applications, Elsevier, vol. 119(9), pages 2945-2969, September.
    2. Debussche, Arnaud & Hu, Ying & Tessitore, Gianmario, 2011. "Ergodic BSDEs under weak dissipative assumptions," Stochastic Processes and their Applications, Elsevier, vol. 121(3), pages 407-426, March.
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