{"ID":6267087,"CreatedAt":"2026-07-10T01:11:38.759438437Z","UpdatedAt":"2026-07-13T01:02:08.706470581Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.08225","arxiv_id":"2607.08225","title":"Sub-Infinite Horizon Stochastic Linear-Quadratic Optimal Control Problems and Delayed Backward Riccati Equations","abstract":"In this paper, we investigate a class of so-called sub-infinite horizon stochastic linear-quadratic optimal control problems, in which the initial time $t$ is arbitrarily taken from $[0,\\infty)$ and the running cost is defined over $[t,t+T]$ for a given $T\u003e0$. The optimal control of this type of problem can be obtained by standard methods; however, it is shown that the resulting optimal control is generally time-inconsistent. Thus, instead of seeking an optimal control, which is time-inconsistent, we aim to find a time-consistent, locally optimal, and time-invariant equilibrium strategy, by introducing a new and very interesting type of Riccati equation. Its main feature is that the generator depends on a delay term of the unknown. In other words, this Riccati equation is a backward ordinary differential equation (ODE) with delay, which is equivalent to a forward ODE with advanced terms. Such an equation is essentially a Fredholm integral equation, whose solvability is challenging. We overcome the difficulty by deriving a sharp a priori estimate and applying the Leray--Schauder fixed point theorem. To this end, we establish a comparison theorem between two matrix-valued nonlinear algebraic equations. The convergence behavior of the solution to the Riccati equation as $T\\to\\infty$ is also provided.","short_abstract":"In this paper, we investigate a class of so-called sub-infinite horizon stochastic linear-quadratic optimal control problems, in which the initial time $t$ is arbitrarily taken from $[0,\\infty)$ and the running cost is defined over $[t,t+T]$ for a given $T\u003e0$. The optimal control of this type of problem can be obtained...","url_abs":"https://arxiv.org/abs/2607.08225","url_pdf":"https://arxiv.org/pdf/2607.08225v1","authors":"[\"Yutao Chen\",\"Hongwei Lou\",\"Hanxiao Wang\"]","published":"2026-07-09T08:18:09Z","proceeding":"math.OC","tasks":"[\"math.OC\"]","methods":"[]","has_code":false}