Exactly linearizing algebras for risk-sensitive filtering

A new approach to the solution of risk-sensitive control and estimation problems is developed. The particular application that is used as a vehicle for this development is the risk-sensitive filtering problem. The partial differential equation (PDE) associated with risk-sensitive filtering is a quasilinear, second-order PDE with a term which is quadratic in the gradient, The solution of this PDE forward in time propagates the filter conditional density. Although the PDE is nonlinear, the associated semi-group is linear over the log-plus algebra. This is used as a basis for a superposition principle and the development of a new class of numerical methods. In a second direction, the log-plus algebra is used to formulate the risk-sensitive filter problem in an entirely new light so that it takes the form of a standard filter via a generalized definition of what it means to be a probability measure