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arxiv: 2605.12775 · v1 · submitted 2026-05-12 · 🧮 math.OC

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· Lean Theorem

Indefinite Stochastic LQ Optimal Control for Jump-Diffusion Systems with Random Coefficients

Xinyu Ma , Qingxin Meng
Authors on Pith no claims yet

Pith reviewed 2026-05-14 20:01 UTC · model grok-4.3

classification 🧮 math.OC
keywords stochastic LQ controljump-diffusion systemsrandom coefficientsindefinite quadratic coststochastic Riccati equationoptimal feedback controlportfolio optimization
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The pith

Under a uniform convexity condition, indefinite stochastic LQ optimal controls exist for jump-diffusion systems with random coefficients and admit closed-loop feedback representations.

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

The paper establishes existence and uniqueness of open-loop optimal controls for indefinite stochastic linear-quadratic problems in jump-diffusion systems with random coefficients. It constructs an algebraic inverse flow from the zero-control base system, extracts the semimartingale kernel of the value function, and proves this kernel satisfies a generalized stochastic Riccati equation with jumps. A uniform convexity condition ensures the matrix N(t) is uniformly positive definite, which yields an exact closed-loop feedback form for the optimal control. This matters for extending classical indefinite LQ theory to systems where the control enters the jump dynamics, without relaxation methods or extra invertibility assumptions on the state. In a financial portfolio application with a jump-diffusion asset and negative terminal wealth weight, the condition reduces to an explicit inequality on risk aversion, volatility, jump magnitude, and risk-free rate that marks where optimal strategies exist.

Core claim

By constructing an algebraic inverse flow from the zero-control base system, the semimartingale kernel of the value function is extracted and shown to satisfy a generalized stochastic Riccati equation with jumps (SREJ). Under the uniform convexity condition, the existence and uniqueness of open-loop optimal controls for any initial pair is established, with the associated matrix N(t) being uniformly positive definite, which yields an exact closed-loop feedback representation of the optimal control via the SREJ. This approach accommodates the general case where the control enters the jump part (F ≠ 0) and requires no relaxation techniques or additional invertibility assumptions.

What carries the argument

The generalized stochastic Riccati equation with jumps (SREJ) whose solution yields the uniformly positive definite matrix N(t) used to construct the feedback representation of the optimal control.

If this is right

  • Optimal open-loop controls exist and are unique for any initial pair when the uniform convexity condition holds.
  • The optimal control admits an exact closed-loop feedback representation expressed via the solution of the SREJ.
  • In the financial portfolio problem, an optimal strategy exists precisely when the explicit inequality among risk aversion, volatility, jump magnitude, and risk-free rate is satisfied.
  • The results apply directly to systems in which the control affects the jump dynamics without requiring compensator methods or state-process invertibility assumptions.

Where Pith is reading between the lines

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

  • The algebraic inverse-flow construction could be adapted to test existence in related indefinite problems with regime switches or delayed jumps.
  • Numerical integration of the SREJ on simulated paths would allow direct verification of the closed-loop cost reduction in the portfolio example.
  • The parametric boundary in the finance application supplies a concrete test: simulate asset paths with parameters inside versus outside the inequality and check whether terminal wealth variance is minimized only inside the region.

Load-bearing premise

The uniform convexity condition must hold globally for all paths so that the matrix N(t) remains uniformly positive definite.

What would settle it

A sample path where the uniform convexity condition fails for some time interval, such that no optimal control exists for a given initial pair or the SREJ solution ceases to be positive definite.

read the original abstract

This paper studies indefinite stochastic linear-quadratic (LQ) optimal control for jump-diffusion systems with random coefficients. We construct an algebraic inverse flow from the zero-control base system, extract the semimartingale kernel of the value function, and prove that it satisfies a generalized stochastic Riccati equation with jumps (SREJ). Under a uniform convexity condition, we establish the existence and uniqueness of open-loop optimal controls for any initial pair and show that the associated matrix $\mathscr{N}(t)$ is uniformly positive definite, yielding an exact closed-loop feedback representation of the optimal control via the SREJ. A distinguishing feature of our approach is that it requires neither relaxation techniques (as in the compensator method) nor additional invertibility assumptions on the optimal state process, and it accommodates the general case where the control enters the jump part ($F \neq 0$). As an application, we analyze a financial portfolio problem with a jump-diffusion risky asset whose excess return is zero, where the investor minimizes a cost functional with a negative terminal wealth weight. The uniform convexity condition reduces to an explicit inequality among the risk aversion coefficient, volatility, jump magnitude, and risk-free rate, thereby delineating the parametric region in which an optimal strategy exists. These results extend classical indefinite LQ theory to jump-diffusion systems with random coefficients.

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 / 3 minor

Summary. The paper develops a theory for indefinite stochastic linear-quadratic optimal control of jump-diffusion systems with random coefficients. It constructs an algebraic inverse flow from the zero-control base system, extracts the semimartingale kernel of the value function, and verifies that this kernel satisfies a generalized stochastic Riccati equation with jumps (SREJ). Under a uniform convexity condition, the authors prove existence and uniqueness of open-loop optimal controls for arbitrary initial data and show that the associated matrix N(t) is uniformly positive definite, which yields an exact closed-loop feedback representation of the optimal control. The approach avoids relaxation techniques and extra invertibility assumptions on the optimal trajectory, accommodates the case F ≠ 0, and is illustrated by a financial portfolio problem in which the convexity condition reduces to an explicit scalar inequality.

Significance. If the central derivations hold, the work provides a direct and assumption-light extension of indefinite LQ theory to jump-diffusion systems with random coefficients. The explicit construction via inverse flow and the closed-loop representation without relaxation or trajectory-dependent invertibility are technically attractive features. The financial application demonstrates how the abstract uniform-convexity hypothesis translates into concrete parametric conditions, which may be useful for applications in stochastic control with jumps.

major comments (2)
  1. [§3] §3 (construction of the inverse flow and kernel extraction): the verification that the extracted semimartingale kernel satisfies the generalized SREJ is load-bearing for the entire theory. The outline in the abstract is coherent, but the explicit Itô calculus steps for the jump terms (especially when F ≠ 0) must be written out in full to confirm that no hidden invertibility or compensator assumptions are used.
  2. [§4] §4 (uniform convexity and positive-definiteness of N(t)): the claim that N(t) remains uniformly positive definite for all t relies on the global validity of the uniform convexity condition. It is not immediately clear from the stated argument how pathwise failures on a null set are ruled out when coefficients are random; a short additional estimate or localization argument would strengthen this step.
minor comments (3)
  1. Notation inconsistency: the matrix is written as script-N(t) in the abstract but as N(t) in the body; adopt a single symbol throughout.
  2. In the financial example, the derivation of the scalar inequality from the uniform convexity condition should be cross-referenced to the precise theorem or proposition number.
  3. A brief comparison paragraph with existing compensator-based approaches for jump LQ problems would help readers situate the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive evaluation, and constructive suggestions. We address each major comment below and will incorporate the requested clarifications in the revised manuscript.

read point-by-point responses

  1. Referee: [§3] §3 (construction of the inverse flow and kernel extraction): the verification that the extracted semimartingale kernel satisfies the generalized SREJ is load-bearing for the entire theory. The outline in the abstract is coherent, but the explicit Itô calculus steps for the jump terms (especially when F ≠ 0) must be written out in full to confirm that no hidden invertibility or compensator assumptions are used.

    Authors: We agree that a fully expanded verification of the Itô formula for the semimartingale kernel, with all jump terms written explicitly when F ≠ 0, is necessary for rigor. The current manuscript presents the main steps of the derivation but condenses some intermediate calculations. In the revision we will insert the complete Itô expansion in Section 3, explicitly computing the continuous and discontinuous parts, the compensator integrals, and the resulting drift and diffusion coefficients of the SREJ. This will confirm that the derivation relies only on the stated integrability and adaptedness assumptions, without any additional invertibility or compensator hypotheses. revision: yes

  2. Referee: [§4] §4 (uniform convexity and positive-definiteness of N(t)): the claim that N(t) remains uniformly positive definite for all t relies on the global validity of the uniform convexity condition. It is not immediately clear from the stated argument how pathwise failures on a null set are ruled out when coefficients are random; a short additional estimate or localization argument would strengthen this step.

    Authors: We appreciate this observation. The uniform convexity condition is imposed pathwise almost surely and uniformly in time, with all coefficients being progressively measurable. To make the pathwise validity explicit, we will add a short localization argument in Section 4: introduce a sequence of stopping times that exhaust the set where the uniform-convexity inequality holds, apply the argument on each localized interval, and pass to the limit using the uniform lower bound on the quadratic form. This yields that N(t) is uniformly positive definite for all t almost surely, without altering the main existence and uniqueness statements. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained from zero-control base system

full rationale

The paper constructs an algebraic inverse flow from the zero-control base system, extracts the semimartingale kernel of the value function, and directly verifies that this kernel satisfies the generalized SREJ. Uniform convexity is then applied to obtain uniform positive-definiteness of N(t) and the closed-loop representation. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the argument is stated to hold for F ≠ 0 and random coefficients without relaxation or extra invertibility assumptions on the optimal trajectory. The financial example merely specializes the convexity condition to an explicit scalar inequality, which is consistent with the general claim rather than presupposing it.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard results from stochastic calculus for jump-diffusions and existence theory for backward stochastic differential equations; no new free parameters or invented entities are introduced beyond the model primitives.

axioms (2)
  • standard math Existence and uniqueness of solutions to the controlled jump-diffusion SDE under standard Lipschitz and linear growth conditions
    Invoked implicitly to define the state process and admissible controls.
  • domain assumption Well-posedness of the generalized stochastic Riccati equation with jumps under the uniform convexity condition
    Central to extracting the semimartingale kernel and proving positivity of N(t).

pith-pipeline@v0.9.0 · 5535 in / 1384 out tokens · 50474 ms · 2026-05-14T20:01:07.220149+00:00 · methodology

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