Stochastic PDEs driven by G-Brownian motion and the associated Backward Doubly Stochastic Differential Equations
Laurent Denis, Jing Zhang
TL;DR
The paper advances the theory of stochastic PDEs under model uncertainty by establishing the well-posedness of quasilinear $G$-SPDEs and the associated GBDSDEs within the $G$-expectation framework. It develops the backward stochastic calculus for $G$-Brownian motion and a forward integration theory with respect to the Hunt-process martingale, culminating in a representation that links the $G$-SPDE to GBDSDEs. The main contributions are the existence/uniqueness proofs for weak solutions, a representation theorem in the product space, and a thorough construction of the doubly stochastic representation, with a discussion of the limitations for nonlinear operators and inter-$G$-Brownian dependence. These results provide a rigorous probabilistic interpretation of nonlinear $G$-SPDEs and lay groundwork for applications under uncertainty in finance and diffusion processes. Future work is suggested on extending to nonlinear operators and exploring dependent $G$-motions in more general settings.
Abstract
Our aim is to study the well-posedness of quasilinear stochastic partial differential equations driven by G-Brownian motion (GSPDEs for short) and the associated backward doubly stochastic differential equations (GBDSDEs for short). We first prove the existence and uniqueness of weak solution to GSPDEs by analytical approach, and then solve the corresponding GBDSDEs. Finally, the relation between GSPDEs and GBDSDEs is established.