Sub-Infinite Horizon Stochastic Linear-Quadratic Optimal Control Problems and Delayed Backward Riccati Equations

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 $T>0$. 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.