Abstract
This paper is concerned with a stochastic linear-quadratic (LQ) optimal control problem on infinite time horizon, with regime switching, random coefficients, and cone control constraint. To tackle the problem, two new extended stochastic Riccati equations (ESREs) on infinite time horizon are introduced. The existence of the nonnegative solutions, in both standard and singular cases, is proved through a sequence of ESREs on finite time horizon. Based on this result and some approximation techniques, we obtain the optimal state feedback control and optimal value for the stochastic LQ problem explicitly. Finally, we apply these results to solve a lifetime portfolio selection problem of tracking a given wealth level with regime switching and portfolio constraint.
Highlights
Linear-quadratic (LQ) optimal control is one of the most important research topics in control theory
We study a stochastic LQ problem on infinite time horizon with regime switching and random coefficients, where the control variable has to be constrained in a cone
Approximated by a sequence of backward stochastic differential equations (BSDEs) on finite time horizon, we prove that the two systems of extended stochastic Riccati equations (ESREs) admit nonnegative solutions
Summary
Linear-quadratic (LQ) optimal control is one of the most important research topics in control theory. Chen and Zhou [3] addressed the conic stabilizability of the controlled stochastic differential equations with cone constraints, and solved the corresponding stochastic LQ problem on infinite horizon via stabilizing solutions of two related ESREs. Li et al [11] studied a stochastic LQ problem with Markovian jumps on infinite time horizon. We study a stochastic LQ problem on infinite time horizon with regime switching and random coefficients, where the control variable has to be constrained in a cone. To solve the control problem, we introduce two systems of BSDEs termed extended stochastic Riccati equations (ESREs) on infinite time horizon. To the best of our knowledge, this is the first paper concerning stochastic LQ problem on infinite time horizon with random coefficients and singular control weighting matrices.
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