  A typical approach to tackle stochastic control problems with partial
observation is to separate the control and estimation tasks. However, it is
well known that this separation generally fails to deliver an actual optimal
solution for risk-sensitive control problems. This paper investigates the
separability of a general class of risk-sensitive investment management
problems when a finite-dimensional filter exists. We show that the
corresponding separated problem, where instead of the unobserved quantities,
one considers their conditional filter distribution given the observations, is
strictly equivalent to the original control problem. We widen the applicability
of the so-called Modified Zakai Equation (MZE) for the study of the separated
problem and prove that the MZE simplifies to a PDE in our approach.
Furthermore, we derive criteria for separability. We do not solve the separated
control problem but note that the existence of a finite-dimensional filter
leads to a finite state space for the separated problem. Hence, the difficulty
is equivalent to solving a complete observation risk-sensitive problem. Our
results have implications for existing risk-sensitive investment management
models with partial observations in that they establish their separability.
Their implications for future research on new applications is mainly to provide
conditions to ensure separability.
