On Nonlinear SDE’S whose Densities Evolve in a Finite—Dimensional Family

Abstract

This paper moves from the differential geometric approach to nonlinear filtering as developed by Brigo, Hanzon and LeGland [MIL] and Brigo [BR2]. We consider the projection in Fisher metric of the density evolution of a diffusion process onto an exponential manifold. We examine the projected evolution and interpret it as the density evolution of a different diffusion process. This result leads to the following consequence: Given an arbitrary diffusion coefficient and an arbitrary exponential family, one can define a drift such that the density of the resulting diffusion process evolves in the prescribed (for example Gaussian) exponential family. We construct also nonlinear SDEs with prescribed stationary exponential density. We shortly introduce a possible financial interpretation of this result. Furthermore we present some hints on how, for particular models, convergence of the original density towards an invariant distribution implies existence of a finite dimensional exponential family for which the projected density converges to the same distribution. Finally, we use the results proven on diffusion processes to derive existence results for filtering problems. Given a prescribed (possibly nonlinear) diffusion coefficient for the state equation, a prescribed (possibly nonlinear) observation function and a partially prescribed exponential family, one can define a drift for the state equation such that the resulting nonlinear filtering problem has a solution which is finite dimensional and which stays on the prescribed exponential family.