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Projecting the Fokker-Planck Equation onto a finite dimensional exponential family
Damiano Brigo1, Giovanni Pistone

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.
1:  Department of Mathematics [Imperial College London]
Nonlinear diffusions – Fokker-Planck equation – finite dimensional families – exponential families – stochastic differential equations – Fisher metric – differential geometry and statistics – convergence.