Indefinite Stochastic Linear-Quadratic Optimal Control Problems with Random Coefficients and Poisson Jumps: Closed-Loop Representation of Open-Loop Optimal Controls

Jiaqiang Wen, Jie Xiong, Kai Ding, Xin Zhang

Authors on Pith no claims yet

Pith reviewed 2026-05-14 18:00 UTC · model grok-4.3

classification 🧮 math.OC

keywords stochastic linear-quadratic controlindefinite weighting matricesPoisson jumpsstochastic Riccati equationclosed-loop representationrandom coefficientsfinite-horizon optimal control

0 comments

The pith

Under a uniform convexity condition, the stochastic Riccati equation with Poisson jumps admits a unique strongly regular solution that gives open-loop optimal controls a closed-loop representation.

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

This paper studies finite-horizon stochastic linear-quadratic optimal control problems featuring random coefficients, Poisson jumps, and possibly indefinite weighting matrices. It proves that when the cost functional meets a uniform convexity condition, the associated stochastic Riccati equation with jumps possesses a unique strongly regular solution. This solution is built from the stochastic value flow through a small-interval localization approach, avoiding reliance on global matrix representations or nonsingularity assumptions on jump multipliers. Consequently, the open-loop optimal control can be expressed in closed-loop form. The paper also derives sufficient conditions for uniform convexity and provides examples with indefinite weights and nonzero jump martingale terms in the equation.

Core claim

Under the uniform convexity condition on the cost functional, the stochastic Riccati equation with jumps admits a unique strongly regular solution constructed from the stochastic value flow by small-interval localization, implying that the open-loop optimal control admits a closed-loop representation without using a global P = Y X^{-1} form or requiring nonsingularity of the jump multiplier I_n + E.

What carries the argument

The stochastic Riccati equation (SRE) with jumps, built from the stochastic value flow using small-interval localization to achieve strong regularity.

Load-bearing premise

The cost functional satisfies a uniform convexity condition.

What would settle it

Constructing a specific instance of the problem where uniform convexity holds but no strongly regular solution to the SRE exists, or where the derived closed-loop control fails to be optimal.

read the original abstract

This paper studies finite-horizon stochastic linear-quadratic optimal control problems with random coefficients and Poisson jumps, where the weighting matrices may be random and indefinite. Under a uniform convexity condition on the cost functional, we prove that the associated stochastic Riccati equation (SRE) with jumps admits a unique strongly regular solution. As a consequence, the open-loop optimal control admits a closed-loop representation. The proof does not rely on a global representation of the form $P=\mathbf Y\mathbf X^{-1}$ or on any nonsingularity condition on the jump multiplier $I_n+E$ in the state equation. Instead, we construct $P$ from the stochastic value flow, and derive the strong regularity of the Riccati solution by a small-interval localization method. In addition, sufficient conditions are obtained for uniform convexity, and examples are presented to illustrate indefinite terminal and control weighting matrices and a nonzero jump martingale component in the SRE.

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.

Desk Editor's Note private letter to a colleague

The paper gives a workable route to closed-loop representations in indefinite stochastic LQ problems with random coefficients and jumps by building the Riccati solution from the value flow and proving regularity locally on small intervals.

read the letter

The main takeaway is that this paper develops a way to represent open-loop optimal controls in closed-loop form for indefinite stochastic LQ problems with random coefficients and Poisson jumps. They do this by constructing the Riccati solution P directly from the stochastic value flow and proving its strong regularity using a small-interval localization technique, which avoids the standard global P = Y X^{-1} and any nonsingularity requirement on the jump multiplier. This approach works under a uniform convexity condition on the cost functional. They derive sufficient conditions for that convexity and give examples with indefinite weights and nonzero jump terms in the SRE. The construction looks internally consistent and addresses cases where previous methods fail due to the random and indefinite nature of the coefficients. The soft spot is that the result stands or falls with the uniform convexity assumption. While they provide some sufficient conditions, verifying it in practice could be challenging, and the localization method's robustness for general jump processes would benefit from more explicit error bounds or numerical checks. This work is for researchers in stochastic optimal control, particularly those interested in jump-diffusion models and indefinite costs in applications like finance. A reader familiar with Riccati equations in stochastic settings would get value from the new proof strategy. It deserves serious peer review to examine the detailed derivations and assess the generality of the localization argument.

Referee Report

0 major / 3 minor

Summary. The manuscript studies finite-horizon stochastic linear-quadratic optimal control problems with random coefficients and Poisson jumps, allowing indefinite random weighting matrices. Under a uniform convexity condition on the cost functional, it proves that the associated stochastic Riccati equation (SRE) with jumps admits a unique strongly regular solution. Consequently, the open-loop optimal control admits a closed-loop representation. The proof constructs P directly from the stochastic value flow and establishes strong regularity via small-interval localization, without using a global representation of the form P = Y X^{-1} or requiring nonsingularity of the jump multiplier I_n + E. Sufficient conditions for uniform convexity are derived and examples with indefinite weights and nonzero jump martingale terms are provided.

Significance. If the central result holds, the work advances the theory of indefinite stochastic LQ control with jumps and random coefficients by providing a closed-loop representation under weaker structural assumptions than those relying on global matrix inverses or nonsingularity conditions. The direct construction of P from the value flow and the localization technique for regularity are notable strengths that could apply to related stochastic optimization problems. The derivation of sufficient convexity conditions and the explicit examples further enhance the contribution.

minor comments (3)

[Introduction] The definition and precise meaning of 'strongly regular solution' for the SRE should be stated explicitly in the introduction or §2, as it is central to the main theorem but only referenced in the abstract.
[Examples] In the examples section, the verification that the uniform convexity condition holds should include explicit numerical checks or bounds on the relevant quadratic forms rather than relying solely on qualitative arguments.
Notation for the Poisson jump measure and the associated martingale term in the SRE could be standardized with a brief comparison to standard references to improve readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests primarily on the domain assumption of uniform convexity of the cost functional; no free parameters are fitted and no new entities are postulated.

axioms (1)

domain assumption Uniform convexity condition on the cost functional
This condition is invoked to guarantee existence and uniqueness of the strongly regular SRE solution.

pith-pipeline@v0.9.0 · 5476 in / 1235 out tokens · 63261 ms · 2026-05-14T18:00:24.991475+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

[1]

Ait Rami, J

M. Ait Rami, J. B. Moore, and X. Y. Zhou. Indefinite stochastic linear quadratic control and generalized differential Riccati equation.SIAM Journal on Control and Optimization, 40:1296–1311, 2002

work page 2002
[2]

M. Athans. The role and use of the stochastic linear-quadratic-gaussian problem in control system design.IEEE Transactions on Automatic Control, 16(6):529–552, 1971

work page 1971
[3]

Bensoussan

A. Bensoussan. Lectures on stochastic control. InNonlinear Filtering and Stochastic Control, volume 972 ofLecture Notes in Mathematics, pages 1–62. Springer, 1982

work page 1982
[4]

J.-M. Bismut. Linear quadratic optimal stochastic control with random coefficients.SIAM Journal on Control and Optimization, 14:419–444, 1976

work page 1976
[5]

J.-M. Bismut. Contrˆ ole des syst` emes lin´ eaires quadratiques: Applications de l’int´ egrale stochastique. InS´ eminaire de Probabilit´ es, XII, volume 649 ofLecture Notes in Mathematics, pages 180–264. Springer, 1978

work page 1978
[6]

S. Chen, X. Li, and X. Y. Zhou. Stochastic linear quadratic regulators with indefinite control weight costs.SIAM Journal on Control and Optimization, 36:1685–1702, 1998

work page 1998
[7]

Chen and X

S. Chen and X. Y. Zhou. Stochastic linear quadratic regulators with indefinite control weight costs. II.SIAM Journal on Control and Optimization, 39:1065–1081, 2000

work page 2000
[8]

M. H. A. Davis.Linear Estimation and Stochastic Control. Chapman and Hall, London, 1977. 32

work page 1977
[9]

El Karoui

N. El Karoui. Les aspects probabilistes du contrˆ ole stochastique. In ´Ecole d’ ´Et´ e de Probabilit´ es de Saint-Flour IX–1979, volume 876 ofLecture Notes in Mathematics, pages 73–238. Springer, 1981

work page 1979
[10]

Hu and X

Y. Hu and X. Y. Zhou. Indefinite stochastic Riccati equations.SIAM Journal on Control and Optimization, 42:123– 137, 2003

work page 2003
[11]

Li and S

J. Li and S. Peng. Stochastic optimization theory of backward stochastic differential equations with jumps and viscosity solutions of Hamilton–Jacobi–Bellman equations.Nonlinear Analysis: Theory, Methods & Applications, 70(4):1776–1796, 2009

work page 2009
[12]

N. Li, Z. Wu, and Z. Yu. Indefinite stochastic linear-quadratic optimal control problems with random jumps and related stochastic Riccati equations.Science China Mathematics, 61:563–576, 2018

work page 2018
[13]

A. E. B. Lim and X. Y. Zhou. Stochastic optimal LQR control with integral quadratic constraints and indefinite control weights.IEEE Transactions on Automatic Control, 44:1359–1369, 1999

work page 1999
[14]

Q. Meng, Y. Dong, Y. Shen, and S. Tang. Optimal controls of stochastic differential equations with jumps and random coefficients: Stochastic Hamilton–Jacobi–Bellman equations with jumps.Applied Mathematics & Optimization, 87(3), 2023

work page 2023
[15]

B. P. Molinari. The time-invariant linear-quadratic optimal control problem.Automatica, 13:347–357, 1977

work page 1977
[16]

S. Peng. Open problems on backward stochastic differential equations. In S. Chen, X. Li, J. Yong, and X. Y. Zhou, editors,Control of Distributed Parameter and Stochastic Systems, pages 265–273. Springer, Boston, MA, 1999

work page 1999
[17]

Protter.Stochastic Integration and Differential Equations, volume 21 ofStochastic Modelling and Applied Probability

Philip E. Protter.Stochastic Integration and Differential Equations, volume 21 ofStochastic Modelling and Applied Probability. Springer, 2nd edition, 2005

work page 2005
[18]

J. Sun, X. Li, and J. Yong. Open-loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems.SIAM Journal on Control and Optimization, 54:2274–2308, 2016

work page 2016
[19]

J. Sun, J. Xiong, and J. Yong. Indefinite stochastic linear-quadratic optimal control problems with random coefficients: Closed-loop representation of open-loop optimal controls.The Annals of Applied Probability, 31(1):460–499, 2021

work page 2021
[20]

Sun and J

J. Sun and J. Yong. Linear quadratic stochastic differential games: Open-loop and closed-loop saddle points.SIAM Journal on Control and Optimization, 52:4082–4121, 2014

work page 2014
[21]

S. Tang. General linear quadratic optimal stochastic control problems with random coefficients: Linear stochastic Hamilton systems and backward stochastic Riccati equations.SIAM Journal on Control and Optimization, 42:53–75, 2003

work page 2003
[22]

S. Tang. Dynamic programming for general linear quadratic optimal stochastic control with random coefficients.SIAM Journal on Control and Optimization, 53(2):1082–1106, 2015

work page 2015
[23]

Tang and X

S. Tang and X. Li. Necessary conditions for optimal control of stochastic systems with random jumps.SIAM Journal on Control and Optimization, 32(5):1447–1475, 1994

work page 1994
[24]

J. Wen, X. Li, J. Xiong, and X. Zhang. Stochastic linear-quadratic optimal control problems with random coefficients and Markovian regime switching system.SIAM Journal on Control and Optimization, 61(2):949–979, 2023

work page 2023
[25]

W. M. Wonham. On a matrix Riccati equation of stochastic control.SIAM Journal on Control, 6:681–697, 1968

work page 1968
[26]

Yong and X

J. Yong and X. Y. Zhou.Stochastic Controls: Hamiltonian Systems and HJB Equations, volume 43. Springer-Verlag, New York, 1999

work page 1999
[27]

Zhang, Y

F. Zhang, Y. Dong, and Q. Meng. Backward stochastic Riccati equation with jumps associated with stochastic linear quadratic optimal control with jumps and random coefficients.SIAM Journal on Control and Optimization, 58(1):393–424, 2020

work page 2020
[28]

Zhang, X

X. Zhang, X. Li, and J. Xiong. Open-loop and closed-loop solvabilities for stochastic linear quadratic optimal control problems of Markovian regime switching system.ESAIM: Control, Optimisation and Calculus of Variations, 27:69, 2021. 33

work page 2021