A Minimality Property of the Minimal Martingale Measure

Let X be a continuous adapted process for which there exists an equivalent local martingale measure (ELMM). The minimal martingale measure P̂ is the unique ELMM for X with the property that local P -martingales strongly orthogonal to the P -martingale part of X are also local P̂ -martingales. We prove that if P̂ exists, it minimizes the reverse relative entropy H(P |Q) over all ELMMs Q for X. A counterexample shows that the assumption of continuity cannot be dropped.