Section: New Results
Stochastic analysis and applications
In this section we present our results on issues which are more abstract than the preceding ones and, at first glance, might appear decorrelated from our applied studies. However most of them are originally motivated by modelling problems, or technical difficulties to overcome in order to analyze in full generality stochastic numerical methods or properties of stochastic models.
Penalization of diffusion processes
Keywords : normalized exponential weights, limiting laws, Wiener measure, Sturm-Liouville equation, Ray-Knight's theorems, rate of convergence, Bessel processes, penalization, enlargement of filtration, maximum, minimum, local time, down-crossings.
Jointly with M. Yor (Université Paris 6), P. Vallois and B. Roynette have continued their study of the penalization of diffusion processes [Oops!] , [Oops!] , [Oops!] , [Oops!] . The penalization procedure allows one to modify a given process (for example a Brownian motion or a Bessel processes) in order to get a new path property (for instance boundedness or a different behavior in a neighborhood of 0). This procedure can be very interesting in modelling to go further than the approach based on diffusions.
Approximation schemes for the local time
In [Oops!] , [Oops!] , P. Vallois and B. Bérard-Bergery (Université Nancy 1) have proposed new approximation schemes related to the local time process of the standard Brownian motion. Some rates of convergence have been exhibited explicitly.