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Let M be a random measure and L be an elliptic pseudo-differential operator on Rd. We study the solution of the stochastic problem LX = M, X(O) = O when some homogeneity and integrability conditions are assumed. If M is a Gaussian measure the process X belongs to the class of Elliptic Gaussian Processes which has already been studied. Here the law of M is not necessarily Gaussian. We characterize the solutions X which are self-similar and with stationary increments in terms of the driving mcasure...
Let {bH(t), t∈ℝ} be the fractional brownian motion with parameter 0<H<1. When 1/2<H, we consider diffusion equations of the type X(t)=c+∫0tσ(X(u)) dbH(u)+∫0tμ(X(u)) du. In different particular models where σ(x)=σ or σ(x)=σ
x and μ(x)=μ or μ(x)=μ
x, we propose a central limit theorem for estimators of H and of σ based on regression methods. Then we give tests of the hypothesis on σ for these models. We also consider functional estimation on σ(⋅)...
The integrated brownian motion is sometimes known as the Langevin process. Lachal studied several excursion laws induced by the latter. Here we follow a different point of view developed by Pitman for general stationary processes. We first construct a stationary Langevin process and then determine explicitly its stationary excursion measure. This is then used to provide new descriptions of Itô’s excursion measure of the Langevin process reflected at a completely inelastic boundary, which has been...
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