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arxiv: 2605.04275 · v1 · submitted 2026-05-05 · 🧮 math.OC

Stochastic Optimal Linear Quadratic Controls with A Recursive Cost Functional in Infinite Horizon

Lin Li , Jiongmin Yong This is my paper

Pith reviewed 2026-05-08 17:16 UTC · model grok-4.3

classification 🧮 math.OC
keywords stochastic linear quadratic controlrecursive cost functionalinfinite horizonweighted L2-stabilizabilitybackward stochastic differential equationsalgebraic Riccati equationopen-loop solvabilityclosed-loop solvability
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The pith

A weighted L2-stabilizability condition equates recursive-cost stochastic LQ problems to classical LQ problems over infinite horizons.

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

This paper tackles the well-posedness issue for backward stochastic differential equations in L1 over infinite time when solving stochastic linear quadratic control problems that feature recursive cost functionals. It introduces a weighted L2-stabilizability notion for the linear dynamics and shows that this condition makes the recursive-cost problem equivalent to a standard LQ problem. The equivalence then lets existing open-loop and closed-loop solvability results carry over directly in the form of forward-backward stochastic differential equations and algebraic Riccati equations. The same reduction applies to the nonhomogeneous case.

Core claim

Under the weighted L2-stabilizability condition, the infinite-horizon stochastic LQ problem with recursive cost functional becomes equivalent to a classical stochastic LQ problem. This equivalence allows all known solvability criteria for open-loop controls, expressed via forward-backward stochastic differential equations, and for closed-loop controls, expressed via algebraic Riccati equations, to be transferred to the recursive setting.

What carries the argument

The weighted L2-stabilizability condition on the linear system, which guarantees well-posedness of the associated BSDE in L1 and produces the equivalence to a classical LQ problem.

If this is right

  • Open-loop solvability of the recursive problem is equivalent to the solvability of a suitable forward-backward stochastic differential equation.
  • Closed-loop solvability is characterized by the existence of a solution to an algebraic Riccati equation.
  • The same equivalence and solvability results hold for nonhomogeneous linear systems.
  • All classical LQ results on existence, uniqueness, and representation of optimal controls translate directly to the recursive-cost setting.

Where Pith is reading between the lines

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

  • Numerical methods developed for classical infinite-horizon LQ problems could be reused for recursive-cost versions once the stabilizability condition is verified.
  • The stabilizability notion may extend to other infinite-horizon stochastic control problems whose costs involve recursive or nonstandard structures.
  • Limits of the infinite-horizon results as the horizon tends to a large finite value could recover corresponding finite-horizon statements for recursive costs.

Load-bearing premise

The linear system must satisfy the weighted L2-stabilizability condition to guarantee that the backward stochastic differential equation remains well-posed in L1 over infinite time.

What would settle it

An explicit linear system that meets the weighted L2-stabilizability condition yet whose associated BSDE fails to admit a unique L1 solution or whose recursive-cost optimal control problem fails to reduce to a classical LQ problem.

read the original abstract

This paper is concerned with a stochastic linear quadratic (LQ, for short) control problem with a recursive cost functional in an infinite horizon. A main difficult is well-posedness of the BSDE in $L^1$ and in infinite horizon. A notion of weighted $L^2$-stabilizability is introduced and characterized, which will lead to an equivalence of the optimal control problem having recursive cost functional with a classical LQ problem. Then all the results of classical problems for open-loop and closed-loop solvability of such an LQ problem can be translated, in terms of the solvability of a forward-backward stochastic differential equation and that of algebraic Riccati equation. Finally, the nonhomogeneous is discussed.

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 a stochastic linear-quadratic control problem with recursive cost functional over an infinite horizon. It introduces a notion of weighted L²-stabilizability, characterizes it, and uses it to establish equivalence between the recursive-cost problem and a classical LQ problem. This equivalence permits translation of open-loop and closed-loop solvability results via forward-backward SDEs and algebraic Riccati equations; the non-homogeneous case is also treated.

Significance. If the central equivalence holds, the work supplies a systematic route for importing classical LQ solvability criteria into the recursive-cost setting, which is useful for infinite-horizon problems where standard Lipschitz or uniform-integrability conditions fail. The weighted stabilizability notion is a concrete technical contribution that directly targets the L¹ well-posedness obstacle for the infinite-horizon BSDE.

major comments (2)
  1. [Introduction and BSDE well-posedness section] The claim that weighted L²-stabilizability alone guarantees a unique L¹ solution to the infinite-horizon BSDE (the key step enabling the equivalence) is stated in the abstract and introduction but lacks an explicit a-priori estimate or uniform-integrability argument in the provided text. The driver induced by the recursive cost satisfies only linear growth; without a concrete bound showing that the weighting produces the required integrability over [0,∞), the L¹ well-posedness step remains unverified and blocks translation of the classical results.
  2. [Equivalence theorem and non-homogeneous case] The equivalence between the recursive-cost LQ problem and the classical LQ problem is asserted once weighted L²-stabilizability holds, yet the manuscript supplies no explicit list of coefficient conditions (e.g., on the weighting function or on the linear system matrices) under which the FBSDE and ARE results carry over verbatim. Without these conditions, it is unclear whether the open-loop/closed-loop solvability statements remain valid for the non-homogeneous case discussed at the end.
minor comments (2)
  1. [Section 2] Notation for the weighting function and the precise definition of weighted L²-stabilizability should be introduced with a numbered definition rather than inline text.
  2. [Abstract and Section 4] The abstract mentions 'all the results of classical problems' can be translated; a short table or explicit list of which classical theorems are invoked would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and insightful comments on our manuscript. The points raised regarding the explicit verification of L¹ well-posedness for the infinite-horizon BSDE and the precise coefficient conditions for the equivalence results are helpful. We address each major comment below and will incorporate clarifications in the revised version.

read point-by-point responses

  1. Referee: [Introduction and BSDE well-posedness section] The claim that weighted L²-stabilizability alone guarantees a unique L¹ solution to the infinite-horizon BSDE (the key step enabling the equivalence) is stated in the abstract and introduction but lacks an explicit a-priori estimate or uniform-integrability argument in the provided text. The driver induced by the recursive cost satisfies only linear growth; without a concrete bound showing that the weighting produces the required integrability over [0,∞), the L¹ well-posedness step remains unverified and blocks translation of the classical results.

    Authors: We appreciate the referee's observation. The well-posedness result (Theorem 3.2) proceeds by using the weighted L²-stabilizability condition to construct a comparison process whose L¹ integrability follows from the exponential weighting and the linear growth of the driver; the argument is completed via a standard localization and Fatou-type passage to the limit. Nevertheless, we agree that an explicit a-priori estimate would make the integrability step more transparent. In the revision we will insert a dedicated lemma (new Lemma 3.1) that derives the uniform L¹ bound directly from the weighting function and the stabilizability assumption, thereby rendering the L¹ well-posedness fully self-contained. revision: yes

  2. Referee: [Equivalence theorem and non-homogeneous case] The equivalence between the recursive-cost LQ problem and the classical LQ problem is asserted once weighted L²-stabilizability holds, yet the manuscript supplies no explicit list of coefficient conditions (e.g., on the weighting function or on the linear system matrices) under which the FBSDE and ARE results carry over verbatim. Without these conditions, it is unclear whether the open-loop/closed-loop solvability statements remain valid for the non-homogeneous case discussed at the end.

    Authors: The equivalence (Theorem 4.1) is stated under the standing assumptions of the paper (bounded measurable coefficients for the linear system, square-integrable inhomogeneous terms, and the weighting function belonging to the admissible class introduced in Definition 2.3). These are precisely the conditions that guarantee the classical FBSDE and ARE results apply verbatim once weighted stabilizability is verified. For the non-homogeneous case the same coefficient hypotheses are used, with the additional requirement that the inhomogeneous processes lie in L². To eliminate any ambiguity we will add an explicit remark (new Remark 4.2) that enumerates these conditions and confirms that the open-loop and closed-loop solvability statements transfer unchanged. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via new stabilizability notion.

full rationale

The paper introduces and characterizes a weighted L2-stabilizability condition for the underlying linear system, then uses it to establish equivalence between the recursive-cost LQ problem and the classical LQ problem. This equivalence permits direct translation of open-loop/closed-loop solvability results in terms of FBSDE and ARE solvability. The L1 well-posedness of the infinite-horizon BSDE is asserted to follow from the stabilizability condition, but the derivation chain does not reduce any central claim to a fitted parameter, self-definition, or self-citation load-bearing step. No equations or steps in the provided abstract and description exhibit the result being equivalent to its inputs by construction. The approach is therefore independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on standard existence results for BSDEs and linear stochastic systems plus the newly introduced stabilizability notion; no free parameters or invented physical entities appear.

axioms (2)
  • standard math Existence and uniqueness of solutions to linear BSDEs under suitable integrability conditions
    Invoked to guarantee well-posedness once weighted L2-stabilizability holds.
  • domain assumption Standard linear growth and Lipschitz conditions on the system coefficients
    Required for the stochastic differential equation and cost functional to be well-defined.
invented entities (1)
  • Weighted L2-stabilizability no independent evidence
    purpose: To ensure well-posedness of the recursive BSDE and establish equivalence to classical LQ
    New concept defined and characterized in the paper; no independent evidence outside the derivation is provided.

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Works this paper leans on

27 extracted references · 27 canonical work pages

  1. [1]

    Briand and Y

    P. Briand and Y. Hu,BSDE with quadratic growth and unbounded terminal value, arXiv:math.PR/0504002 v1 1 Apr 2005

    work page arXiv 2005

  2. [2]

    Duffie and L

    D. Duffie and L. G. Epstein,Stochastic differential utility,Econometrica,60 (1992), no. 2, 353-394

    work page 1992

  3. [3]

    Duffie and L

    D. Duffie and L. G. Epstein,Asset pricing with stochastic differential utility,Review Financial Studies, 5 (1992), 411-436

    work page 1992

  4. [4]

    El Karoui, S

    N. El Karoui, S. Peng, and M. C. Quenez,Backward stochastic differential equations in finance,Math. Finance,7 (1997), 1–71

    work page 1997

  5. [5]

    Fan,Bounded solutions,L p (p >1) solutions andL 1 solutions for one dimensional BSDEs under general assumptions,Stoch

    S. Fan,Bounded solutions,L p (p >1) solutions andL 1 solutions for one dimensional BSDEs under general assumptions,Stoch. Proc. Appl.,126 (2016), 1511–1552

    work page 2016

  6. [6]

    Fan and D

    S. Fan and D. Liu,A class of BSDE with integrable parameters,Stat. Probab. Letters,80 (2010), 2024–2031

    work page 2010

  7. [7]

    Fuhrman and G

    M. Fuhrman and G. Tessitore,Infinite horizon backward stochastic differential equations and elliptic equations in Hilbert spaces,Ann. Probab.,32 (2004), 607–660

    work page 2004

  8. [8]

    Klimsiak and M

    T. Klimsiak and M. Rzymowski,A priori estimates for multidimensional BSDEs with integrable data, Electron. Commun. Probab.,29 (2024), no.34

    work page 2024

  9. [9]

    Lazrak,Generalized stochastic differential utility and preference for information,Ann

    A. Lazrak,Generalized stochastic differential utility and preference for information,Ann. Appl. Probab., 14 (2004), 2149–2175

    work page 2004

  10. [10]

    Lazrak and M

    A. Lazrak and M. C. Quenez,A generalized stochastic differential utility,Math. Oper. Res.,28 (2003), 154–180

    work page 2003

  11. [11]

    Li and J

    L. Li and J. Yong,Stochastic optimal linear quadratic controls with a recursive dost functional,Evol. Equa Control Th.,to appear. arXiv preprint arXiv:2601.21748, 2026

    work page arXiv 2026

  12. [12]

    S. Luo, X. Li, and Q. Wei,Infinite time horizon stochastic recursive control problems with jumps: dynamic programming and stochastic verification theorems,. SIAM J. Control Optim.,63 (2025) 796– 821

    work page 2025

  13. [13]

    J. Ma, P. Protter, and J. Yong,Solving forward-backward stochastic differential equations explicitly — a four step scheme.,Probab. Theory Related Fields,98 (1994), 339–359

    work page 1994

  14. [14]

    Ma and J

    J. Ma and J. Yong,Forward-Backward Stochastic Differential Equations and Their Applications,Lecture Notes in Math., Vol. 1702, Springer-Verlag, 1999

    work page 1999

  15. [15]

    Mou and J

    L. Mou and J. Yong,Two-person zero-sum linear quadratic stochasrtic differential games by a Hilbert space method,J. Ind. Manag. Optim.,2 (2006), 95–117

    work page 2006

  16. [16]

    Peng,A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation,Stoch

    S. Peng,A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation,Stoch. Stoch. Rep.,38 (1992), 119–134

    work page 1992

  17. [17]

    Peng,Backward SDE and relatedg-expectation,Pitman Res

    S. Peng,Backward SDE and relatedg-expectation,Pitman Res. Notes Math.Ser. 364, Longman, Harlow, 1997, 141–159

    work page 1997

  18. [18]

    Peng and Y

    S. Peng and Y. Shi,Infinite horizon forward-backward stochastic differential equations,Stoh. Proc. Appl.,85 (2000), 75–92

    work page 2000

  19. [19]

    Penrose,A generalized inverse of matrices,Proc

    R. Penrose,A generalized inverse of matrices,Proc. Camb. Phil. Soc.,52 (1955), 17–19. 22

    work page 1955

  20. [20]

    Sun and J

    J. Sun and J. Yong,Stochastic linear quadratic optimal control problems in infinite horizon,Appl. Math. Optim.,78 (2018), 145–183

    work page 2018

  21. [21]

    Sun and J

    J. Sun and J. Yong,Stochastic Linear-Quadratic Optimal Control Theory: Open-Loop and Closed-Loop Solutions,Springer, 2020

    work page 2020

  22. [22]

    Wei and Z

    Q. Wei and Z. Yu,Time-inconsistent recursive zero-sum stochastic differential games,Math. Control & Related Fields,8 (2018),

    work page 2018

  23. [23]

    Yin,On solutions of a class of infinite horizon FBSDEs,Stat.& Probab

    J. Yin,On solutions of a class of infinite horizon FBSDEs,Stat.& Probab. Lett.,78 (2008), 2412–2419

    work page 2008

  24. [24]

    Yong,Remarks on some short rate term structure models,J

    J. Yong,Remarks on some short rate term structure models,J. Indust. Management Optim.,2 (2006), 119–134

    work page 2006

  25. [25]

    Yong,A stochastic linear quadratic optimal control problem with generalized expectation,Stoch

    J. Yong,A stochastic linear quadratic optimal control problem with generalized expectation,Stoch. Anal. Appl.,26 (2008), 1136–1160

    work page 2008

  26. [26]

    Yong and X

    J. Yong and X. Y. Zhou,Stochastic controls: Hamiltonian systems and HJB equations,Springer, 1999

    work page 1999

  27. [27]

    Yu,Infinite horizon jump-diffusion forward-backward stochastic differential equations and their appli- cation to backward linear-quadratic problem,ESAIM: Control, Optim

    Z. Yu,Infinite horizon jump-diffusion forward-backward stochastic differential equations and their appli- cation to backward linear-quadratic problem,ESAIM: Control, Optim. Cal. Var.,23 (2017), 1331–1359. 23

    work page 2017