Fully Coupled Nonlinear Forward-Backward Stochastic Difference Equations: Spectral Contraction and Infinite-Horizon Maximum Principle
Abstract
This paper develops an explicit spectral-contraction approach to studying fully coupled nonlinear forward--backward stochastic difference equations on infinite horizon and their applications to stochastic control. The estimates for this fully coupled system are assembled into an explicit two-dimensional nonnegative matrix. Its spectral radius yields an explicit sufficient discount threshold, and a corresponding equivalent weighted product norm is constructed to establish the contraction property. Moreover, for an infinite-horizon control problem with convex control constraints and an accumulated discounted running cost, we derive a Pontryagin-type stochastic maximum principle, its equivalent pointwise normal-cone formulation, and a verification theorem. Finally, a recursive risk-adjusted portfolio example is given to demonstrate the applications of our theoretical results, and a projection-type sufficient optimality condition for this example is derived.
Related
- Stability and Periodicity Performance of Solutions of Nonlinear Difference Equations
- Analytical Methods for Solving Non-Autonomous Higher-Order Nonlinear Difference Equations
- Existence and Nonexistence of Solutions of Certain type of Nonlinear Differential and Differential-Difference Equations
- Explicit meromorphic solutions of certain types of nonlinear differential-difference equations
- On the Pontryagin Maximum Principle for Rough Stochastic Differential Equations
BibTeX
@misc{chen2026fully,
title = {{Fully Coupled Nonlinear Forward-Backward Stochastic Difference Equations: Spectral Contraction and Infinite-Horizon Maximum Principle}},
author = {Rui Chen and Qi Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11354},
eprint = {2609.11354},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11354},
}
RIS
TY - UNPB
TI - Fully Coupled Nonlinear Forward-Backward Stochastic Difference Equations: Spectral Contraction and Infinite-Horizon Maximum Principle
AU - Rui Chen
AU - Qi Zhang
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11354
UR - https://arxiv.org/abs/2609.11354
AB - This paper develops an explicit spectral-contraction approach to studying fully coupled nonlinear forward--backward stochastic difference equations on infinite horizon and their applications to stochastic control. The estimates for this fully coupled system are assembled into an explicit two-dimensional nonnegative matrix. Its spectral radius yields an explicit sufficient discount threshold, and a corresponding equivalent weighted product norm is constructed to establish the contraction property. Moreover, for an infinite-horizon control problem with convex control constraints and an accumulated discounted running cost, we derive a Pontryagin-type stochastic maximum principle, its equivalent pointwise normal-cone formulation, and a verification theorem. Finally, a recursive risk-adjusted portfolio example is given to demonstrate the applications of our theoretical results, and a projection-type sufficient optimality condition for this example is derived.
ER -