Stochastic differential games with random coefficients and state constraints: associated stochastic Hamilton-Jacobi-Bellman-Isaacs equations with Neumann boundary conditions
Abstract
We study a stochastic differential game in which the state process is governed by a reflected stochastic differential equation (RSDE) and the recursive payoff is defined by a generalized backward stochastic differential equation (GBSDE). Unlike the Markovian models in \cite{Li08sicon,Fan}, all coefficients may depend directly on ω, so the value functions are random fields. We show that the lower and upper value functions are viscosity solutions of stochastic Hamilton--Jacobi--Bellman--Isaacs (HJBI) equations with nonlinear Neumann boundary conditions, thereby extending the framework of \cite{Qiu22AAP} to reflected state constraints. Under weak boundary regularity, we derive estimates for RSDEs and GBSDEs, prove regularity of the value functions, and establish the dynamic programming principle (DPP) as well as the existence of viscosity solutions. Finally, under additional assumptions in a one-dimensional linear setting, we prove a comparison theorem and the uniqueness of bounded (Ly,0)-viscosity solutions, where Ly is the constant coefficient of y in the running generator. To prove the comparison theorem, we homogenize the boundary condition and construct smooth adapted comparison fields by solving deterministic uniformly parabolic Neumann problems in finite dimensions.
Related
- Pathwise uniqueness for degenerate stochastic differential equations with Hölder continuous coefficients
- Backward stochastic partial differential equations with unbounded random coefficients
- Multidimensional Infinite-Horizon Markovian Backward Stochastic Differential Equations with Continuous Coefficients and Nonzero-Sum Differential Games
- A New Modifications of the Milstein and Euler Methods and the Taylor-Ito Expansion for Ito Stochastic Differential Equations with Drift and Diffusion Coefficients Having Non-Differentiable or Singular Time-Dependent Multipliers
- Infinite-Horizon Stochastic Recursive Differential Games with Jumps: Nonautonomous Coefficients and Nonzero Terminal Values
BibTeX
@misc{buckdahn2026stochastic,
title = {{Stochastic differential games with random coefficients and state constraints: associated stochastic Hamilton-Jacobi-Bellman-Isaacs equations with Neumann boundary conditions}},
author = {Rainer Buckdahn and Juan Li and Yanwei Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28062},
eprint = {2610.28062},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28062},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Stochastic differential games with random coefficients and state constraints: associated stochastic Hamilton-Jacobi-Bellman-Isaacs equations with Neumann boundary conditions
AU - Rainer Buckdahn
AU - Juan Li
AU - Yanwei Li
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28062
UR - https://arxiv.org/abs/2610.28062
AB - We study a stochastic differential game in which the state process is governed by a reflected stochastic differential equation (RSDE) and the recursive payoff is defined by a generalized backward stochastic differential equation (GBSDE). Unlike the Markovian models in \cite{Li08sicon,Fan}, all coefficients may depend directly on ω, so the value functions are random fields. We show that the lower and upper value functions are viscosity solutions of stochastic Hamilton--Jacobi--Bellman--Isaacs (HJBI) equations with nonlinear Neumann boundary conditions, thereby extending the framework of \cite{Qiu22AAP} to reflected state constraints. Under weak boundary regularity, we derive estimates for RSDEs and GBSDEs, prove regularity of the value functions, and establish the dynamic programming principle (DPP) as well as the existence of viscosity solutions. Finally, under additional assumptions in a one-dimensional linear setting, we prove a comparison theorem and the uniqueness of bounded (L_y,0)-viscosity solutions, where L_y is the constant coefficient of y in the running generator. To prove the comparison theorem, we homogenize the boundary condition and construct smooth adapted comparison fields by solving deterministic uniformly parabolic Neumann problems in finite dimensions.
ER -