[Paper Review] Nonlocal Fully Nonlinear Parabolic Differential Equations Arising in Time-Inconsistent Problems

Lei Qian, Chi Seng Pun|arXiv (Cornell University)|Oct 8, 2021

Stability and Controllability of Differential Equations4 citations

TL;DR

This paper establishes the well-posedness—existence, uniqueness, and stability—of nonlocal fully nonlinear parabolic PDEs arising in time-inconsistent stochastic control problems, using tailored Banach spaces and fixed-point arguments. It further links these PDEs to a flow of second-order forward-backward stochastic differential equations (2FBSDEs), providing a new Feynman–Kac-type representation for such systems.

ABSTRACT

We prove the well-posedness results, i.e. existence, uniqueness, and stability, of the solutions to a class of nonlocal fully nonlinear parabolic partial differential equations (PDEs), where there is an external time parameter $t$ on top of the temporal and spatial variables $(s,y)$ and thus the problem could be considered as a flow of equations. The nonlocality comes from the dependence on the unknown function and its first- and second-order derivatives evaluated at not only the local point $(t,s,y)$ but also at the diagonal line of the time domain $(s,s,y)$. Such equations arise from time-inconsistent problems in game theory or behavioural economics, where the observations and preferences are (reference-)time-dependent. To address the open problem of the well-posedness of the corresponding nonlocal PDEs (or the time-inconsistent problems), we first study the linearized version of the nonlocal PDEs with an innovative construction of appropriate norms and Banach spaces and contraction mappings over which. With fixed-point arguments, we obtain the well-posedness of nonlocal linear PDEs and establish a Schauder-type prior estimate for the solutions. Then, by the linearization method, we analogously establish the well-posedness under the fully nonlinear case. Moreover, we reveal that the solution of a nonlocal fully nonlinear parabolic PDE is an adapted solution to a flow of second-order forward-backward stochastic differential equations.

Motivation & Objective

• To resolve the open problem of well-posedness for nonlocal fully nonlinear parabolic PDEs arising in time-inconsistent stochastic control.
• To establish existence, uniqueness, and stability of solutions using novel function spaces and fixed-point techniques.
• To connect the nonlocal PDEs to a flow of second-order forward-backward stochastic differential equations (2FBSDEs).
• To generalize existing Feynman–Kac formulas by incorporating diagonal dependence and time-parameterized dynamics.
• To extend the theoretical foundation for behavioral economics and stochastic control under reference-dependent preferences.

Proposed method

• Constructs specialized Banach spaces and norms tailored to handle nonlocality and time-parameter dependence in the PDE.
• Applies contraction mapping principles in these spaces to prove well-posedness of the linearized PDE version.
• Derives a Schauder-type a priori estimate to control solution behavior in terms of nonhomogeneous terms and initial data.
• Uses linearization and fixed-point iteration to extend results to the fully nonlinear case.
• Establishes a stochastic representation by applying Itô’s formula to the PDE solution, linking it to a flow of 2FBSDEs.
• Demonstrates that the solution of the nonlocal PDE is an adapted solution to the derived flow of 2FBSDEs, generalizing classical Feynman–Kac formulas.

Experimental results

Research questions

• RQ1 Can the well-posedness of nonlocal fully nonlinear parabolic PDEs with diagonal dependence be established under general conditions?
• RQ2 How can appropriate function spaces and norms be constructed to handle the nonlocal structure and time-parameter dependence?
• RQ3 What is the connection between such nonlocal PDEs and a flow of second-order forward-backward stochastic differential equations?
• RQ4 Can a generalized Feynman–Kac formula be derived for systems involving diagonal terms of the solution and its derivatives?
• RQ5 How does the solution of the nonlocal PDE relate to subgame perfect equilibria in time-inconsistent stochastic control problems?

Key findings

• The well-posedness of the nonlocal fully nonlinear parabolic PDE is established via linearization and Banach’s fixed-point theorem in appropriately constructed Banach spaces.
• A Schauder-type a priori estimate is derived, quantifying the solution’s dependence on the initial condition and nonhomogeneous term.
• The solution of the nonlocal PDE is shown to be an adapted solution to a flow of second-order forward-backward stochastic differential equations (2FBSDEs).
• The system of 2FBSDEs (53) generalizes classical 2FBSDEs by incorporating diagonal terms of the solution and its derivatives in a nonlinear fashion.
• The PDE-FBSDE connection provides a new Feynman–Kac-type representation for flows of 2FBSDEs, extending classical results.
• The results significantly advance the theory of time-inconsistent stochastic control by enabling the inclusion of control in the diffusion coefficient and solving equilibrium HJB equations.