Projecting the Fokker-Planck Equation onto a finite dimensional exponential family

Damiano Brigo (1) , Giovanni Pistone

1 Department of Mathematics [Imperial College London]

In the present paper we discuss problems concerning evolutions of densities related to Ito diffusions in the framework of the statistical exponential manifold. We develop a rigorous approach to the problem, and we particularize it to the orthogonal projection of the evolution of the density of a diffusion process onto a finite dimensional exponential manifold. It has been shown by D. Brigo (1996) that the projected evolution can always be interpreted as the evolution of the density of a different diffusion process. We give also a compactness result when the dimension of the exponential family increases, as a first step towards a convergence result to be investigated in the future. The infinite dimensional exponential manifold structure introduced by G. Pistone and C. Sempi is used and some examples are given.

Mots clés: Nonlinear diffusions; Fokker-Planck equation; finite dimensional families; exponential families; stochastic differential equations; Fisher metric; differential geometry and statistics; convergence.; convergence

Domaines: Probabilités [math.PR]

Damiano Brigo, Giovanni Pistone. Projecting the Fokker-Planck Equation onto a finite dimensional exponential family. 1996. ⟨hal-00351455⟩