Fractal and Stochastic Process

点击翻译按钮获取中文摘要

In this research, we studied among all a deterministic version of the Ito calculus.Deterministic Brownian motions are stochastic processes with noncorrelated, stationary and strictly ergodic increments having 0-entropy and 0-expectation. The self-similarity of order 1/2 follows from these properties. Such processes have a lot of variety and have different properties. It is not the case of the Brownian motion where the process is characterized as a process with stationary and independent increments with 0-expectation and standard variance.Among the deterministic Brownian motions, the simplest one is the N-process (N_t ; t∈R). We consider a process Y_t=H (N_t, t), where the function H(x, s) is twice continuously differentible in x and once continuously differentible in s and H_x(x, s)>0. The function H is consisered completely unknown except for these properties. We want to predict the value Y^c from the observation Y_J : ={Y_t ; t∈J}, where J=[a, b] and a<b< c. We proved that there exists a estimator Y_c such that<<numerical formula>>as c↓b with the following C (b) as the constant in O ( ) :<<numerical formula>>