On martingale measures for stochastic processes with independent increments

We consider a special semimartingale $X$ with independent increments and prove the existence and equivalence of a local martingale measure $\mathbf{P}^H$ for $X$, which minimizes the Hellinger process under the assumption that there exists an equivalent local martingale measure for $X$. This is done under the restriction of quasi-left-continuity and boundedness of jumps of $X$. Furthermore, we investigate the relation between the well-known minimal martingale measure $\mathbf{P}^{\min}$ and $\mathbf{P}^H$. It is shown that in a sense $\mathbf{P}^{\min}$ is an approximation of $\mathbf{P}^H$.