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Minimization of the Kullback information of diffusion processes. (English) Zbl 0790.60032

Summary: We compute an explicit expression for the rate function of large deviations for the measure valued empirical process \(\overline X^ N(t)={1\over N}\sum^ N_{i=1}\delta_{X_ i(t)}\), where the \(X_ i\)’s are independent copies of a diffusion process in \(\mathbb{R}^ d\). This is done by minimizing the relative entropy (Kullback information) of a probability measure \(Q\) with respect to the law \(P\) of \(X_ i\) when all marginals of \(Q\) are fixed. The finiteness of the rate function is connected with the existence of conservative diffusions, with a general diffusion matrix. These diffusion processes are constructed in very general cases.

MSC:

60F10 Large deviations
60J60 Diffusion processes
60G44 Martingales with continuous parameter
60H05 Stochastic integrals
60J57 Multiplicative functionals and Markov processes
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