Martingale representation for contingent claims with regime switching

We derive a martingale representation for a contingent claim under a Markov-modulated version of the Black-Scholes economy. The martingale representation for the price of the claim is established with respect to an equivalent martingale measure chosen by the Esscher transform. Under some differentiability conditions for the coefficients of the price processes, we shall identify explicitly the integrands in the martingale representation using stochastic flows. We shall introduce a zero-coupon bond to minimize the residual risk due to incomplete hedging.