With jump linear quadratic Gaussian (JLQG) control, one refers to the control under a 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 linear quadratic Gaussian (LQG) and adaptive LQG, Tile aim of this paper is to help improve the situation by introducing an exact transformation which embeds adaptive JLQG control into LQM (linear quadratic Martingale) 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 he studied within the framework of robust or minimax control without the need for the usual approach of averaging or approximating the adaptive JLQG dynamic. To show the effectiveness of our transformation, it is used to characterize the open-loop-optimal feedback (OLOF) policy for adaptive JLQG control.