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Equilibrium master equations for time-inconsistent problems with distribution dependent rewards

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  • Zongxia Liang
  • Fengyi Yuan

Abstract

We provide a unified approach to find equilibrium solutions for time-inconsistent problems with distribution dependent rewards, which are important to the study of behavioral finance and economics. Our approach is based on {\it equilibrium master equation}, a non-local partial differential equation on Wasserstein space. We refine the classical notion of derivatives with respect to distribution and establish It$\hato$'s formula in the sense of such refined derivatives. Our approach is inspired by theories of Mckean-Vlasov stochastic control and mean field games, but is significantly different from both in that: we prohibit marginal distribution of state to be an input of closed loop control; we solve the best reaction to individual selves in an intra-person game instead of the best reaction to large populations as in mean field games. As applications, we reexamine the dynamic portfolio choice problem with rank dependent utility based on the proposed novel approach. We also recover the celebrated extended HJB equation when the reward of the problem has a nonlinear function of expectation while reformulating and weakening the assumptions needed. Most importantly, we provide a procedure to find an equilibrium solution of a dynamic mean-ES portfolio choice problem, which is completely new to the literature.

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  • Zongxia Liang & Fengyi Yuan, 2021. "Equilibrium master equations for time-inconsistent problems with distribution dependent rewards," Papers 2112.14462, arXiv.org, revised Apr 2022.
    • Handle: RePEc:arx:papers:2112.14462
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    References listed on IDEAS

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    3. Ying Hu & Hanqing Jin & Xun Yu Zhou, 2021. "Consistent investment of sophisticated rank‐dependent utility agents in continuous time," Mathematical Finance, Wiley Blackwell, vol. 31(3), pages 1056-1095, July.
    4. Tomas Björk & Mariana Khapko & Agatha Murgoci, 2017. "On time-inconsistent stochastic control in continuous time," Finance and Stochastics, Springer, vol. 21(2), pages 331-360, April.
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    7. Tomas Björk & Agatha Murgoci & Xun Yu Zhou, 2014. "Mean–Variance Portfolio Optimization With State-Dependent Risk Aversion," Mathematical Finance, Wiley Blackwell, vol. 24(1), pages 1-24, January.
    8. Zongxia Liang & Fengyi Yuan, 2021. "Weak equilibria for time-inconsistent control: with applications to investment-withdrawal decisions," Papers 2105.06607, arXiv.org, revised Jun 2023.
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