Nonlinear semigroup for the unnormalized conditional density (Q1063570)

The paper is concerned with the following partially observable control problems; the state process \(X_ t\) and the observation process \(Y_ t\) are governed by stochastic differential equations \[ dX_ t=b(X_ t,U_ t)dt+\sigma (X_ t)dW_ t,\quad dY_ t=h(X_ t)dt+d\tilde W_ t \] where \(U_ t\) is an admissible control. The author shows the regularity of the unnormalized conditional probability density, under some mild conditions, and constructs a nonlinear semigroup whose generator is related to the Mortensen equation.