Stochastic control of a partially observed linear stochastic system with an exponential-of-integral performance index

The stochastic control problem with linear stochastic differential equations driven by Brownian motion processes and as cost functional the exponential of a quadratic form is considered. The solution consists of a linear control law and of a linear stochastic differential equation. The latter has the same structure as the Kalman filter but depends explicitly on the cost functional. The separation property does not hold in general for the solution to this problem.