Stochastic differential game, Esscher transform and general equilibrium under a Markovian regime-switching Lévy model

In this paper, we discuss three different approaches to select an equivalent martingale measure for the valuation of contingent claims under a Markovian regime-switching Lévy model. These approaches are the game theoretic approach, the Esscher transformation approach and the general equilibrium approach. We employ the dynamic programming principle to derive the optimal strategies and the value functions in the stochastic differential game and the general equilibrium approaches, each of which lead to an equivalent martingale measure. We also compare equivalent martingale measures chosen by the three approaches. Under certain conditions, the equivalent martingale measures chosen by the stochastic differential game and the Esscher transformation approaches coincide. If the equity premium is in its equilibrium state, the equivalent martingale measures chosen by the Esscher transformation and the general equilibrium approaches are identical.