Approximation des trajectoires et temps local des diffusions. (Approximation of trajectories and local times of diffusions)
For a diffusion process X and its regularizations either by convolution with an approximate identity or by polygonal interpolation, this paper studies convergence of the normalized upcrossing processes of the regularizations to the local time process of the diffusion. The principal part of the paper is concerned with studying the joint density of \(X_{\epsilon}(t)\), \(X^ 1_{\epsilon}(t)\), where \(X_{\epsilon}\) is the regularization obtained by convolution and X is expressible as a function of Brownian motion. A clever conditioning argument is used to address this joint density. Bounds on the behaviour of the joint density are used to provide answers to the question about upcrossings and local time.
scientific article; zbMATH DE number 4104096