How do future expectations affect the financial sector? Expectations modeling, infinite horizon (controlled) random FBSDEs and stochastic viscosity solutions

TL;DR

This paper develops a mathematical framework for modeling how future expectations influence the financial sector using infinite horizon coupled FBSDEs and stochastic viscosity solutions, providing existence, uniqueness, and control principles.

Contribution

It introduces a novel approach connecting infinite horizon FBSDEs with stochastic viscosity solutions and establishes key properties and control principles under standard conditions.

Findings

Proved existence and uniqueness of solutions for the class of FBSDEs.

Connected FBSDEs with stochastic PDEs via a generalized four-step scheme.

Provided a stochastic maximum principle for optimal control applications.

Abstract

In this paper we study a class of infinite horizon fully coupled forward-backward stochastic differential equations (FBSDEs), that are stimulated by various continuous time future expectations models with random coefficients. Under standard Lipschitz and monotonicity conditions, and by means of the contraction mapping principle, we establish existence, uniqueness, a comparison property and dependence on a parameter of adapted solutions. Making further the connection with infinite horizon quasilinear backward stochastic partial differential equations (BSPDEs) via a generalization of the well known four-step-scheme, we are led to the notion of stationary stochastic viscosity solutions. A stochastic maximum principle for the optimal control problem of such FBSDEs is also provided as an application to this framework.