This paper deals with the dynamic exponential utility indifferent value process of a contingent claim. We regard the risky asset pricing process as a semimartingale with general jumps based on a random measure, where the GKW (Galtchouk-Kunita-Watanabe) decomposition of the martingale part of the exponential utility indifferent value process's Doob-Meyer decomposition satisfies the Jacod decomposition. A backward stochastic differential equation (BSDE) is established from the fact that the exponential utility indifferent value process is a martingale for the optimal investment strategy under the minimal entropy measure. The uniqueness of the solution to the equation is proved by converting the BSDE to bounded mean oscillation (BMO) martingales under a new probability measure. The division of the equation's generator into [ΔA = 0] and [ΔA≠ 0] proves the existence of an optimal investment strategy. As a result, the exponential utility indifference value process of the contingent claim is the solution of the BSDE.