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Indefinite stochastic LQ controls with Markovian jumps in a finite time horizon

This paper is concerned with a stochastic linear–quadratic (LQ) control problem over a finite time horizon with Markovian jumps in the problem parameters. The problem is indefinite in that the cost weighting matrices for the state and control are allowed to be indefinite. A system of coupled generalized (differential) Riccati equations (CGREs) is introduced to cope with the indefiniteness of the problem. Specifically, it is proved that the solvability of the CGREs is sufficient for the well-posedness of the stochastic LQ problem. Moreover, it is shown that the solvability of the CGREs is necessary for the well-posedness of the stochastic LQ problem and the existence of optimal (feedback/open-loop) controls via the dynamic programming approach. An example is presented to illustrate the results established.