Defaultable asset management with incomplete information

We consider a market where asset prices could be affected by multiple defaults along with possible factors including frailty. The aim of an investor is to maximize the expected utility of the terminal wealth, based on the observed data of the underlying asset(s) and the default history up to the current time. The main purpose is then to determine the conditional intensity of the future defaults, given the observed stock prices and the past defaults, without using the so-called "density hypothesis". The problem is naturally reduced to a nonlinear filtering problem, for which the so-called the H-hypothesis is known to fail. We show that the problem can be solved dynamically via a system of Zakai equations for the conditional densities between and at consecutive defaults. A related BSDE with jumps that has quadratic growth in both continuous and jump martingale integrands will also be studied, as a by product of the utility optimization problem.