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

Cattiaux, Patrick; Léonard, Christian

Ann. Inst. Henri Poincaré, Probab. Stat. 30, No. 1, 83-132 (1994).

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

Keywords:

relative entropy; Föllmer measure; Girsanov transformation; Kullback information; large deviations; measure valued empirical process; conservative diffusions