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

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.

• The paper develops a general solution theory for indefinite stochastic LQ control with Brownian and Poisson noise, random coefficients, and controls entering diffusion and jump terms.
• It proves that uniform convexity alone guarantees positivity of the key feedback matrix, existence and uniqueness of the generalized stochastic Riccati equation with jumps, and a unique optimal closed-loop control.
• The inverse-flow method avoids state-invertibility and relaxation assumptions, while a portfolio example shows how jumps and stochastic market conditions generate intertemporal hedging demand under explicit convexity bounds.

Problem and setting

The paper studies indefinite stochastic linear-quadratic (LQ) optimal control for jump-diffusion systems with random coefficients. The state obeys a controlled SDE driven by a Brownian motion and an independent compensated Poisson random measure, with all coefficients $A,B,C,D,E,F$ predictable and uniformly bounded; the control may enter both the diffusion part ($D\neq 0$) and the jump part ($F\neq 0$). The cost functional carries a terminal weight $G$, running weights $(Q,S,R)$, none of which is assumed positive (semi-)definite. For each initial pair $(t,\xi)$ the problem is to attain the essential infimum of the conditional cost $J(t,\xi;u)$ over square-integrable predictable controls.

The central difficulty in the indefinite setting is that classical positive-definiteness arguments no longer certify solvability of the associated stochastic Riccati equation or invertibility of the matrix needed for feedback synthesis. Prior work handled this either via stopping-time/invertibility arguments on the optimal state process (valid only for purely continuous paths), via relaxed compensator methods that introduce auxiliary processes not expressible through the original coefficients, or by postulating positive definiteness of a key matrix as a hypothesis. This paper develops an approach free of all three devices.

Main contributions

Algebraic inverse flow from the zero-control base system. Under the natural non-singularity condition $\det(I+E(t,e))\ge\delta>0$, the fundamental solution $\Phi$ of the uncontrolled system is invertible, and its inverse $\Psi=\Phi^{-1}$ satisfies an explicit SDEP. This yields a variation-of-constants representation of the state and, crucially, a stochastic flow of homeomorphisms: for almost every $D\neq 0$0, the map $D\neq 0$1 is a continuous bijection of $D\neq 0$2. Because jump-diffusion paths are càdlàg rather than continuous, the left-continuity-based stopping time argument of Sun–Xiong–Yong does not extend to this setting; the inverse-flow construction replaces it entirely, making any invertibility assumption on the optimal state process unnecessary.

Semimartingale structure of the value kernel. Under the uniform convexity condition (UC)—namely $D\neq 0$3 for some $D\neq 0$4—the value function admits a quadratic representation $D\neq 0$5 with $D\neq 0$6 essentially bounded, $D\neq 0$7. The dynamic programming principle holds, and $D\neq 0$8 possesses an RCLL semimartingale modification

$D\neq 0$9

with finite-variation part pathwise $F\neq 0$0-integrable, martingale parts $F\neq 0$1-integrable, and integrated variations possessing moments of every order. These regularity properties are what allow the jump-induced terms in the subsequent Riccati analysis to be controlled.

Solvability of the generalized SREJ and uniform positivity of $F\neq 0$2. Defining

$F\neq 0$3

together with $F\neq 0$4 and $F\neq 0$5 built from the system coefficients and $F\neq 0$6, the paper proves—rather than assumes—that $F\neq 0$7 a.e., a.s., under the sole uniform convexity condition. The proof proceeds by showing the drift rate of the submartingale $F\neq 0$8 is nonnegative, converting this into a pointwise inequality via the change-of-variables induced by the inverse flow, then applying a spike-variation argument with a composite control (constant spike followed by the optimal continuation). A technically important point is the use of a Bochner-integral version of the Lebesgue differentiation theorem: a naive scalar LDT would fail because the test variable depends on the point at which the limit is taken, creating a measure-theoretic circularity; the Bochner argument extracts a single null set valid for all test directions simultaneously. With $F\neq 0$9 established, minimizing the quadratic drift yields the feedback minimizer $G$0, and vanishing of the drift along optimal trajectories identifies $G$1, i.e., the triple $G$2 solves the generalized stochastic Riccati equation with jumps (SREJ)

$G$3

Existence holds for general random coefficients with no restriction on $G$4; uniqueness follows by comparing two solutions through their closed-loop systems and invoking uniqueness of the Doob–Meyer decomposition. A consequence is that the uniform positivity of $G$5, which Moon–Chung assumed as a hypothesis and which Zhang–Dong–Meng obtained only under positive-definite cost weights, is here derived from uniform convexity alone.

Verification theorem and feedback synthesis. Given the SREJ solution, the feedback control $G$6 with $G$7 is shown to be the unique open-loop optimal control for every initial pair, and $G$8. Well-posedness of the closed-loop SDEP requires care because $G$9 is generally only square-integrable, not bounded; the paper invokes a Gal'chuk-type existence result whose pathwise integrability conditions are verified directly, replacing the stronger essential boundedness used elsewhere. Optimality is proved by Itô's formula on stopped intervals $(Q,S,R)$0, with dominated convergence and Fatou's lemma handling the localization limit—a necessary device since arbitrary admissible controls need not have bounded moments of all orders. Uniqueness follows because equality in the cost bound forces $(Q,S,R)$1, and $(Q,S,R)$2 forces the closed-loop relation identically.

Financial application

The framework is illustrated by a portfolio problem in which a risky asset has zero excess return,

$(Q,S,R)$3

and the investor minimizes $(Q,S,R)$4. The negative terminal weight makes the problem indefinite. Using the Itô isometry and independence of the driving noises, uniform convexity reduces exactly to the explicit parametric inequality

$(Q,S,R)$5

which in the time-homogeneous case becomes $(Q,S,R)$6. When it fails, the cost functional is unbounded below—the investor would take arbitrarily large positions—and no optimum exists; when it holds, the verification theorem yields existence, uniqueness, and the feedback form $(Q,S,R)$7, where $(Q,S,R)$8 is expressed entirely through the scalar SREJ coefficients. In the deterministic-coefficient special case the SREJ collapses to $(Q,S,R)$9, $(t,\xi)$0, giving $(t,\xi)$1, zero feedback gain, and hence $(t,\xi)$2: with no risk premium and perfectly foreseeable opportunities there is neither speculative nor hedging motive. In the genuinely stochastic case, however, $(t,\xi)$3 and $(t,\xi)$4 are generically nonzero, so the gain is nonzero even though the asset carries no expected excess return. The resulting position is a pure intertemporal hedging demand in Merton's sense: the investor trades against adverse shifts in volatility, jump intensity, and interest rates. Notably, jump-induced hedging can arise from $(t,\xi)$5 alone even when the diffusion martingale component vanishes, and the $(t,\xi)$6-dependent term inside $(t,\xi)$7 shows that jumps affect the convexity of the problem itself, not merely the optimal position.

Limitations and open questions

Several restrictions should be noted. First, the theory rests on the uniform convexity condition (UC); while the financial example shows it reduces to a checkable inequality in that case, the paper states plainly that UC is not expressible in closed form for general systems, so verifiability in broader classes remains unresolved. Second, the non-singularity assumption $(t,\xi)$8 is essential to the inverse-flow construction; whether the results survive degenerate jump maps is not addressed. Third, the analysis is confined to the finite-horizon case with square-integrable controls and a single Poisson measure with finite characteristic measure; infinite-horizon extensions and regime-switching couplings are identified but not treated. Finally, the paper proposes rather than develops numerical schemes for the SREJ (e.g., discretization of BSDEs with jumps), leaving computational implementation open.

Conclusion

The paper delivers a self-contained theory of indefinite stochastic LQ control for jump-diffusion systems with random coefficients: existence and uniqueness of open-loop optimal controls, proven (not assumed) uniform positivity $(t,\xi)$9, unique solvability of the generalized SREJ for general coefficients including $J(t,\xi;u)$0, and exact closed-loop feedback synthesis—all without relaxation techniques or invertibility assumptions on the optimal state. The key methodological move is the algebraic inverse flow built from the zero-control base system, which sidesteps the left-continuity obstruction created by jumps. The zero-excess-return portfolio example demonstrates that the abstract uniform convexity hypothesis translates into explicit parametric conditions and isolates pure hedging demand as the economic content of the indefinite formulation.