Martingale Analysis for Assets with Discontinuous Returns

The equivalent martingale measure approach is applied to a financial market subject to jump-diffusion uncertainty. The uncertainty in the market is caused by a multidimensional Brownian motion process and a multidimensional point process of jumps admitting stochastic intensity. Under a boundedness condition on the relative risk premium on jumps, an equivalent risk-neutral probability measure is identified and is used to construct hedging portfolios for consumption processes and contingent claims. These are then applied to problems of utility maximization.