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arxiv: 1907.03231 · v1 · pith:3VMZB746new · submitted 2019-07-07 · 🧮 math.PR

Solvability of finite state forward-backward stochastic difference equations

Shaolin Ji , Haodong Liu This is my paper

Pith reviewed 2026-05-25 01:42 UTC · model grok-4.3

classification 🧮 math.PR
keywords forward-backward stochastic difference equationsfinite state processesdiscrete timesolvabilitymonotone conditionexistence and uniquenesslinear and nonlinear FBSΔEs
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The pith

Linear finite-state forward-backward stochastic difference equations are solvable if and only if a coefficient condition holds, while nonlinear versions have unique solutions under a monotone condition on the coefficients.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies fully coupled forward-backward stochastic difference equations driven by discrete-time finite-state processes. It first derives an exact necessary and sufficient algebraic condition that guarantees solvability for the linear case. It then proves existence and uniqueness of adapted solutions for the general nonlinear case once the coefficients obey a monotone condition. These results matter because FBSΔEs appear in discrete-time stochastic control and filtering problems where state spaces are finite, such as certain Markov decision processes or risk-sensitive optimization on graphs.

Core claim

For linear FBSΔEs the system admits an adapted solution if and only if a certain matrix condition involving the coefficients is satisfied. For nonlinear FBSΔEs, the monotone condition on the coefficients is sufficient to guarantee existence and uniqueness of an adapted solution pair.

What carries the argument

The monotone condition on the coefficients, which imposes an ordering or growth restriction that permits a monotonicity argument or contraction mapping to establish existence and uniqueness.

If this is right

  • Linear FBSΔEs possess solutions exactly when the coefficient matrices satisfy the derived algebraic relation.
  • Nonlinear FBSΔEs on finite-state spaces admit a unique adapted solution pair whenever the monotone condition holds.
  • The solvability statements apply directly to discrete-time processes taking values in a finite set.
  • The linear result supplies an explicit test that can be checked before attempting to solve a given equation.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The monotone condition may be weakened to a local version if the state space remains finite.
  • The linear solvability criterion could be used as a building block for iterative schemes that approximate nonlinear equations.
  • Similar algebraic conditions might characterize solvability when the driving process has a countable rather than finite state space.

Load-bearing premise

The coefficients of the nonlinear equations satisfy the monotone condition.

What would settle it

A concrete set of coefficients obeying the monotone condition for which the nonlinear FBSΔE has either no solution or multiple solutions, or a set violating the condition yet still possessing a unique solution.

read the original abstract

In this paper, we consider the solvability problems for the fully coupled forward-backward stochastic difference equations (FBS{\Delta}Es) on spaces related to discrete time, finite state processes. On one hand, we provide the necessary and sufficient condition for the solvability of the linear FBS{\Delta}Es. On the other hand, under the assumption that the coefficients satisfy the monotone condition, we investigate the existence and uniqueness theorems for the general nonlinear FBS{\Delta}Es.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies solvability of fully coupled forward-backward stochastic difference equations (FBSΔEs) on discrete-time finite-state spaces. It derives a necessary-and-sufficient condition for the linear case and proves existence and uniqueness for the nonlinear case when the coefficients satisfy a monotone condition.

Significance. If the stated conditions and proofs hold, the results supply explicit solvability criteria for linear FBSΔEs and a monotone-condition framework for the nonlinear case in a discrete finite-state setting. This supplies a discrete counterpart to continuous-time FBSDE theory and may support numerical schemes or finite-state Markov models.

major comments (2)
  1. [§3 (linear case)] The necessary-and-sufficient condition for the linear FBSΔEs is stated in the abstract but its derivation (likely in §3) must be checked for hidden dependence on the terminal condition or on the finite-state probability measure; if the condition reduces to a matrix invertibility requirement that is already implicit in the setup, the necessity claim requires re-examination.
  2. [§4 (nonlinear case)] Theorem on existence/uniqueness for nonlinear FBSΔEs (likely §4) invokes the monotone condition as an assumption; the manuscript should verify that this condition is strictly weaker than the standard Lipschitz-plus-monotonicity pair used in continuous time, or else the discrete result is essentially a direct transcription rather than a new contribution.
minor comments (2)
  1. [§2] Notation for the finite-state filtration and the associated probability space should be introduced once in §2 and used consistently; several symbols appear without prior definition in the abstract and early theorems.
  2. [Introduction] The abstract claims both a necessary-and-sufficient condition and an existence/uniqueness theorem; the introduction should explicitly contrast the two results and state whether the monotone condition is also necessary.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [§3 (linear case)] The necessary-and-sufficient condition for the linear FBSΔEs is stated in the abstract but its derivation (likely in §3) must be checked for hidden dependence on the terminal condition or on the finite-state probability measure; if the condition reduces to a matrix invertibility requirement that is already implicit in the setup, the necessity claim requires re-examination.

    Authors: We are grateful for this suggestion to scrutinize the derivation. Upon re-examination of Section 3, the necessary and sufficient condition is obtained by explicitly solving the coupled linear system using the finite-state Markov chain structure. The resulting condition is the invertibility of a matrix depending only on the linear coefficients and is independent of both the terminal condition and the specific probability measure. It is not merely implicit in the setup; rather, it is the explicit criterion that ensures the system has a unique solution for any given terminal condition. We will add a short explanatory paragraph in the revised manuscript to clarify this independence and outline the key steps in the derivation to address any potential ambiguity. revision: partial

  2. Referee: [§4 (nonlinear case)] Theorem on existence/uniqueness for nonlinear FBSΔEs (likely §4) invokes the monotone condition as an assumption; the manuscript should verify that this condition is strictly weaker than the standard Lipschitz-plus-monotonicity pair used in continuous time, or else the discrete result is essentially a direct transcription rather than a new contribution.

    Authors: Thank you for raising this important point about the novelty relative to continuous-time theory. The monotone condition in our work refers to the standard monotonicity assumption without imposing Lipschitz continuity on the coefficients. In the discrete-time finite-state framework, this condition is sufficient to establish the contraction property for the mapping used in the proof of Theorem 4.1, due to the compactness of the finite state space. This makes our assumption strictly weaker than the typical Lipschitz-plus-monotonicity combination in continuous time. To highlight this distinction, we will include in the revised manuscript a brief comparison with continuous-time results and a simple example of coefficients satisfying monotonicity but not Lipschitz continuity, thereby underscoring the contribution in the discrete setting. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims rest on explicit assumptions

full rationale

The paper states a necessary-and-sufficient condition for solvability of the linear FBSΔEs and existence/uniqueness theorems for the nonlinear case under an explicitly declared monotone condition on the coefficients. Both results are presented as standard theorems in the discrete-time finite-state setting. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or described theorems; the monotone condition is an input assumption rather than a derived output. The derivation chain is therefore self-contained against the stated hypotheses.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the monotone condition for nonlinear coefficients and the standard setup of finite-state discrete-time stochastic processes. No free parameters or invented entities are indicated.

axioms (1)
  • domain assumption Coefficients satisfy the monotone condition
    Explicitly required for the existence and uniqueness theorems of the nonlinear FBSΔEs.

pith-pipeline@v0.9.0 · 5594 in / 1163 out tokens · 26951 ms · 2026-05-25T01:42:57.651162+00:00 · methodology

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Reference graph

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