An averaging principle for nonlinear parabolic PDEs via reflected FBSDEs driven by G-Brownian motion
Mengyao Hou
TL;DR
This work develops an averaging principle for a class of forward-backward SDEs with reflection driven by $G$-Brownian motion, addressing a singular perturbation problem for fully nonlinear PDEs with a lower obstacle. The authors introduce a penalization-based approach to transform the reflected $G$-BSDE into a general $G$-BSDE, establish uniform a priori bounds, and relate the stochastic system to a viscosity solution via a nonlinear Feynman–Kac framework. Under the averaging condition on the nonlinear generator (H5), they prove that the scaled value functions $u^ ext{ε}(t,x)$ converge to the unique viscosity solution $ar{u}$ of the averaged obstacle PDE $iglackslashackslash ext{min}(- rac{\partial ar{u}}{\partial t}-ar{F}, ar{u}-S)=0 igrackslash$, with terminal data $ar{u}(T,x)=oldsymbol{ar{ }(x)}$. The results extend the averaging principle to reflected $G$-FBSDEs and provide a probabilistic characterization of the limit via obstacle problems for fully nonlinear parabolic PDEs, with implications for models under volatility uncertainty.
Abstract
In this paper, we are concerned with the averaging problem for a class of forward-backward stochastic differential equations with reflection driven by G-Brownian motion (reflected G-FBSDEs), which corresponds to the singular perturbation problem of a kind of fully nonlinear partial differential equations (PDEs) with a lower obstacle. The reflection keeps the solution above a given stochastic process. By the use of the nonlinear stochastic techniques and viscosity solution methods, we prove that the limit distribution of solution is the unique viscosity solution of an obstacle problem for a fully nonlinear parabolic PDEs.