Embedding adaptive JLQG into LQ martingale with a completely observable stochastic control matrix

With jump linear quadratic Gaussian (JLQG) control one refers to the control under quadratic performance criterion of a linear Gaussian system, the coefficients of which are completely observable, while they are jumping according to a finite state Markov process. With adaptive JLQG one refers to the more complicated situation that the finite state process is only partially observable. Although many practically applicable results have been developed, JLQG and adaptive JLQG control are lagging behind those for LQG and adaptive LQG. The aim of this paper is to help improving the situation through introducing an exact transformation which embeds adaptive JLQG control into one of LQ martingale (LQM) control with a completely observable stochastic control matrix. By LQM control we mean the control of a martingale driven linear system under a quadratic performance criterion. With the LQM transformation the adaptive JLQG control can be studied within the framework of robust or minimax control without the need for the usual approach of averaging or approximating the adaptive JLQG dynamics. To show the effectiveness of our transformation, we use it to characterize the open-loop-optimal feedback policy for adaptive JLQG control