Market Viability and Martingale Measures under Partial Information.

Summary We consider a financial market model with a single risky asset whose price process evolves according to a general jump-diffusion with locally bounded coefficients and where market participants have only access to a partial information flow $(\E_t)_{t\geq0}\subseteq(\F_t)_{t\geq0}$. For any utility function, we prove that the partial information financial market is locally viable, in the sense that the problem of maximizing the expected utility of terminal wealth has a solution up to a stopping time, if and only if the marginal utility of the terminal wealth is the density of a partial information equivalent martingale measure (PIEMM). This equivalence result is proved in a constructive way by relying on maximum principles for stochastic control under partial information. We then show that the financial market is globally viable if and only if there exists a partial information local martingale deflator (PILMD), which can be explicitly constructed. In the case of bounded coefficients, the latter turns out to be the density process of a global PIEMM. We illustrate our results by means of an explicit example.