[Paper Review] Comparison Theorems of Infinite Horizon Forward-Backward Stochastic Differential Equations

Liangquan Zhang, Yufeng Shi|arXiv (Cornell University)|May 22, 2010

Stochastic processes and financial applications16 references3 citations

TL;DR

This paper establishes comparison theorems for infinite-horizon forward-backward stochastic differential equations (FBSDEs) using probabilistic methods and duality techniques. It proves that under monotonicity and integrability conditions, the solution components of one FBSDE dominate those of another when their coefficients and initial/terminal conditions satisfy appropriate inequalities, extending prior finite-horizon results to the infinite-horizon setting with multi-dimensional forward and backward components.

ABSTRACT

By the methods of probability and duality technique, we give some comparison theorems for the solutions of infinite horizon forward-backwad stochastic differential equations.

Motivation & Objective

• To establish comparison theorems for infinite-horizon fully coupled forward-backward stochastic differential equations (FBSDEs), which generalize finite-horizon results to unbounded time domains.
• To address the gap in existing comparison results for non-Markovian, multi-dimensional FBSDEs by introducing a new technical condition (H4) to ensure validity over infinite time horizons.
• To extend one-dimensional comparison results for backward SDEs to the multi-dimensional case, enhancing applicability in stochastic control and mathematical finance.
• To provide a rigorous theoretical foundation for the analysis of solutions in stochastic optimal control, differential games, and PDEs via probabilistic methods.
• To unify and generalize prior results from Wu (2007), Peng & Shi (2003), and Wu & Xu (2011) to the infinite-horizon setting with full generality in dimensions and coefficient structures.

Proposed method

• Uses a duality technique to transform the original FBSDE into a dual backward stochastic differential equation (BSDE), enabling comparison of solution components via the dual process.
• Applies Itô’s formula to the product of the difference process and the dual process to derive an explicit representation of the initial value of the backward component.
• Imposes a set of standard assumptions (H1)–(H4) on the coefficients, including monotonicity, Lipschitz continuity, and integrability, to ensure existence and uniqueness of solutions in the space $\mathcal{B}^3 = \mathcal{S}^2 \times \mathcal{S}^2 \times \mathcal{H}^2$.
• Introduces a novel technical condition (H4) to overcome limitations in earlier comparison theorems, particularly for multi-dimensional systems with non-degenerate diffusion coefficients.
• Employs a probabilistic approach based on the comparison of stochastic processes and their quadratic variations, avoiding reliance on partial differential equation methods.
• Derives comparison results for both the initial value $Y_0$ and pathwise dominance of $Y_t$ over $t \in [0, \infty)$, depending on the structure of the coefficients and terminal conditions.

Experimental results

Research questions

• RQ1 Under what conditions does the solution of one infinite-horizon FBSDE dominate another in the backward component $Y_t$ for all $t \geq 0$?
• RQ2 Can comparison theorems for multi-dimensional FBSDEs ($n > 1$, $m > 1$) be established without assuming Markovian or non-random coefficients?
• RQ3 How can the comparison result for the initial value $Y_0$ be extended when both the forward and backward coefficients differ between two FBSDEs?
• RQ4 What technical conditions are necessary to ensure the validity of comparison theorems in the infinite-horizon setting, especially when solutions do not decay at infinity?
• RQ5 Can the one-dimensional comparison result for backward SDEs be generalized to the multi-dimensional case using duality and probabilistic methods?

Key findings

• If the coefficients satisfy $b^1 \geq b^2$, $f^1 \geq f^2$, $\Phi^1 \geq \Phi^2$, and $\sigma^1 = \sigma^2$, then $Y_0^1 \geq Y_0^2$ holds under the assumptions (H1)–(H4), even when $n > 1$ and $m = 1$.
• For the case $n = m = 1$, the paper proves that $Y_t^1 \geq Y_t^2$ for all $t \geq 0$ under monotonicity and integrability conditions, extending earlier results to infinite time horizons.
• The solution components $Y_t$ of the FBSDE are pathwise non-decreasing when the coefficients and initial/terminal conditions satisfy appropriate inequalities, ensuring stochastic dominance.
• The duality technique successfully transforms the comparison problem into a dual BSDE, allowing the use of Itô’s formula to derive an explicit expression for $\hat{Y}_0$, which is shown to be non-negative under the given conditions.
• The paper establishes that $Y_0^1 \geq Y_0^2$ even when the forward components $X^1$ and $X^2$ differ, provided the coefficient differences are controlled and the technical condition (H4) is satisfied.
• The results generalize prior finite-horizon comparison theorems by Wu (2007), Peng & Shi (2003), and Wu & Xu (2011), extending them to the infinite-horizon case with full generality in dimension and coefficient structure.