| Abstract: |
| This paper addresses the solvability of a broad class of nonlocal second-order backward stochastic differential equations featuring two temporal parameters or backward stochastic Volterra integral equations (2BSVIEs). These equations arise in the characterization of equilibrium strategies and corresponding value functions for time-inconsistent (TIC) stochastic control problems, where agents` present- or state-biased preferences violate Bellman`s principle of optimality. In such contexts, our formulation extends the scope of existing work by allowing both the drift and volatility of the underlying state process to be controllable, and considering objective functionals that depend on both the initial time and state. The comprehensive nature of our 2BSVIE framework requires moving away from a purely probabilistic approach for demonstrating solvability, directing us instead towards an analytical method grounded in partial differential equations (PDEs). Specifically, we employ a continuity method and Banach`s fixed-point arguments within custom-designed Banach spaces to establish the well-posedness and regularity of solutions for a class of PDEs with nonlocality in both time and space (nPDEs). Subsequently, we derive a Feynman--Kac-type formula using It\^{o}`s lemma to establish a relationship between the solutions of the 2BSVIEs and the nPDEs, thereby proving the solvability of the general 2BSVIEs. These solvability results significantly advance the understanding of long-standing open problems in equilibrium Hamilton-Jacobi-Bellman (HJB) equations and TIC controls. Finally, we present two globally solvable financial examples. |
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