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

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 $L<sup>1$ and in infinite horizon. A notion of weighted $L<sup>2$-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.

• The paper extends infinite-horizon BSDE theory to drivers with weighted-$L^1$ integrability, establishing finiteness when $p_2>1+\Gamma^2/(2\Theta)$ and addressing a gap beyond standard $L^2$ assumptions.
• The paper characterizes weighted $L^2$-stability and stabilizability through Lyapunov inequalities and transforms the recursive-cost problem into an equivalent classical stochastic LQ problem with modified system coefficients.
• The paper derives open-loop solvability through an infinite-horizon FBSDE and closed-loop solvability through a weighted-stabilizing algebraic Riccati equation, while identifying direct solvability of the weighted correction equation as an open issue.

Problem formulation and the $L^1$ obstruction

The paper studies a stochastic linear-quadratic (LQ) optimal control problem in an infinite horizon where the cost functional is recursive in the sense of Duffie–Epstein stochastic differential utility: the cost is the value $Y(0)$ of a linear BSDE driven by the running cost $f(s,X(s),u(s))$ along the state-control pair of the system $[A,C;B,D]$:

$$dY(s)=[\Theta Y(s)+\Gamma Z(s)-f(s,X(s),u(s))]ds+Z(s)dW(s),\qquad \Theta,\Gamma\ge 0.$$

The central technical difficulty is well-posedness. Since $u(\cdot)\in L^2$, the driver $f$ is merely $L^1$ in time and probability, and the paper works with an infinite horizon. The authors note that finite-horizon BSDEs with $L^1$ data are already delicate, and that existing infinite-horizon results (Peng–Shi, Yin, Sun–Yong, Fuhrman–Tessitore, and extensions to jumps by Yu, Wei–Yu, Luo–Li–Wei) all require at least $L^p$ integrability, $Y(0)$0. To the authors' knowledge, no prior work treats infinite-horizon BSDEs with merely $Y(0)$1 drivers, so the paper first develops this theory.

The key device is a Doléans exponential discount weight $Y(0)$2 solving $Y(0)$3, $Y(0)$4, i.e., $Y(0)$5. A finite-horizon representation gives $Y(0)$6, which motivates weighted spaces $Y(0)$7 in which integrability is measured against $Y(0)$8. The limit $Y(0)$9 is well-defined provided $f(s,X(s),u(s))$0 or $f(s,X(s),u(s))$1 is weighted-$f(s,X(s),u(s))$2 integrable; the class of admissible drivers is dense in the weighted $f(s,X(s),u(s))$3 space but is not itself a vector space, a structural point the authors state explicitly. For $f(s,X(s),u(s))$4, the weighted integral is finite whenever

$f(s,X(s),u(s))$5

which generalizes the classical stability condition $f(s,X(s),u(s))$6 (the case $f(s,X(s),u(s))$7) of Sun–Yong. Thus the paper extends infinite-horizon $f(s,X(s),u(s))$8 BSDE solvability to the full range $f(s,X(s),u(s))$9.

Weighted $[A,C;B,D]$0-stabilizability

For the cost functional to be finite, the state and control must be integrable against $[A,C;B,D]$1. The paper introduces weighted $[A,C;B,D]$2-stability of $[A,C;B,D]$3 (the uncontrolled state is in $[A,C;B,D]$4) and weighted $[A,C;B,D]$5-stabilizability of $[A,C;B,D]$6 (existence of $[A,C;B,D]$7 with $[A,C;B,D]$8 weighted $[A,C;B,D]$9-stable). The main characterization is a Lyapunov inequality: $$dY(s)=[\Theta Y(s)+\Gamma Z(s)-f(s,X(s),u(s))]ds+Z(s)dW(s),\qquad \Theta,\Gamma\ge 0.$$0 is weighted $$dY(s)=[\Theta Y(s)+\Gamma Z(s)-f(s,X(s),u(s))]ds+Z(s)dW(s),\qquad \Theta,\Gamma\ge 0.$$1-stable if and only if there exists $$dY(s)=[\Theta Y(s)+\Gamma Z(s)-f(s,X(s),u(s))]ds+Z(s)dW(s),\qquad \Theta,\Gamma\ge 0.$$2 with

$$dY(s)=[\Theta Y(s)+\Gamma Z(s)-f(s,X(s),u(s))]ds+Z(s)dW(s),\qquad \Theta,\Gamma\ge 0.$$3

with a constructive proof via a finite-horizon ODE in $$dY(s)=[\Theta Y(s)+\Gamma Z(s)-f(s,X(s),u(s))]ds+Z(s)dW(s),\qquad \Theta,\Gamma\ge 0.$$4 and passage to the limit, yielding the explicit solution $$dY(s)=[\Theta Y(s)+\Gamma Z(s)-f(s,X(s),u(s))]ds+Z(s)dW(s),\qquad \Theta,\Gamma\ge 0.$$5. The practical consequence is a checkable criterion: weighted $$dY(s)=[\Theta Y(s)+\Gamma Z(s)-f(s,X(s),u(s))]ds+Z(s)dW(s),\qquad \Theta,\Gamma\ge 0.$$6-stability of $$dY(s)=[\Theta Y(s)+\Gamma Z(s)-f(s,X(s),u(s))]ds+Z(s)dW(s),\qquad \Theta,\Gamma\ge 0.$$7 is equivalent to ordinary $$dY(s)=[\Theta Y(s)+\Gamma Z(s)-f(s,X(s),u(s))]ds+Z(s)dW(s),\qquad \Theta,\Gamma\ge 0.$$8-stability of the modified pair $$dY(s)=[\Theta Y(s)+\Gamma Z(s)-f(s,X(s),u(s))]ds+Z(s)dW(s),\qquad \Theta,\Gamma\ge 0.$$9, and analogously for stabilizability of $u(\cdot)\in L^2$0 versus $u(\cdot)\in L^2$1. A companion estimate for the nonhomogeneous system gives a uniform bound $u(\cdot)\in L^2$2.

Under hypothesis (H4) — weighted $u(\cdot)\in L^2$3-stabilizability with $u(\cdot)\in L^2$4 a stabilizer, $u(\cdot)\in L^2$5, $u(\cdot)\in L^2$6, and $u(\cdot)\in L^2$7 — the admissible set $u(\cdot)\in L^2$8 (controls for which the recursive cost is well-defined) coincides with $u(\cdot)\in L^2$9, and the paper formulates Problem (LQ) with open-loop and closed-loop solvability notions.

Equivalence with a classical LQ problem

The core structural result is a change of variables. Setting $f$0 and $f$1, the weighted system transforms into a classical (unweighted) controlled system $f$2 with $f$3, $f$4, $f$5, $f$6, and the recursive cost becomes an ordinary integral cost with transformed (random) inhomogeneities $f$7. Hence Problem (LQ) with recursive cost is equivalent to Problem (LQ̃), a classical stochastic LQ problem on the infinite horizon, and weighted stabilizability of the original system is equivalent to $f$8-stabilizability of the transformed one. This equivalence is the paper's central simplification: all classical results transfer.

Translated solvability characterizations

Applying the Sun–Yong theory to (LQ̃) and translating back yields two characterizations. Open-loop solvability: a state-control pair $f$9 is open-loop optimal if and only if the FBSDE

$L^1$0

with $L^1$1, admits an adapted solution satisfying the stationary condition $L^1$2. Closed-loop solvability: Problem (LQ) is closed-loop solvable if and only if the modified algebraic Riccati equation

$L^1$3

admits a weighted $L^1$4-stabilizing solution $L^1$5, with the optimal feedback gain $L^1$6 and the correction process $L^1$7, where $L^1$8 solves a weighted backward ODE. A notable feature is that although $L^1$9 and $L^1$0 are deterministic, their weighted transforms $L^1$1 are random, so the $L^1$2-system is genuinely stochastic in the transformed problem while its translation back has deterministic coefficients.

Extension to nonhomogeneous dynamics

For systems with deterministic additive drift and diffusion terms $L^1$3, the decomposition $L^1$4 splits the cost into a part depending on $L^1$5 and an additive term independent of both; the latter can be dropped, reducing the problem to the homogeneous case already treated. This reduction relies on the determinism of $L^1$6 and $L^1$7.

Limitations and open questions

Several caveats are stated by the authors. The admissible-driver class $L^1$8 is not a vector space, so the well-posedness framework is inherently one-sided (requiring $L^1$9 or $L^p$0 to be weighted integrable). The analysis is carried out for one-dimensional Brownian motion, with the multi-dimensional case asserted to be similar but not detailed. Most notably, the authors concede in a remark that they are not able to directly prove solvability of the weighted $L^p$1-equation by $L^p$2 itself; closed-loop solvability is established only via the transformed process $L^p$3, so a direct well-posedness proof of the weighted correction equation remains open. The extension section treats only deterministic nonhomogeneities; random $L^p$4 are not covered.

Conclusion

The paper extends infinite-horizon BSDE theory to drivers that are merely weighted-$L^p$5 integrable, introduces and fully characterizes weighted $L^p$6-stabilizability via Lyapunov equations and an equivalent classical stabilizability condition, and proves an exact equivalence between stochastic LQ control with recursive cost and classical stochastic LQ control. Through this equivalence, open-loop and closed-loop solvability of the recursive-cost problem are characterized respectively by an FBSDE and by a modified algebraic Riccati equation, transferring the entire classical theory to the recursive setting.