Estimation of Parameters for Diffusion Processes with Jumps from Discrete Observations

In this paper, we consider a multidimensional diffusion process with jumps whose jump term is driven by a compound Poisson process. Let a(x,θ) be a drift coefficient, b(x,σ) be a diffusion coefficient respectively, and the jump term is driven by a Poisson random measure p. We assume that its intensity measure q^θ has a finite total mass. The aim of this paper is estimating the parameter α = (θ,σ) from some discrete data. We can observe n + 1 data at t_iⁿ = ih_n, . We suppose h_n → 0, nh_n → ∞, nh_n² → 0. Final version 20 December 2004     

On Uniform Laws of Large Numbers for Ergodic Diffusions and Consistency of Estimators

Consider a regular diffusion process X with finite speed measure m. Denote the normalized speed measure by μ. We prove that the uniform law of large numbers holds if the class has an envelope function that is μ-integrable, or if is bounded in L _p(μ) for some p>1. In contrast with uniform laws of large numbers for i.i.d. random variables, we do not need conditions on the ‘size’ of the class in terms of bracketing or covering numbers. The result is a consequence of a number of asymptotic properties of diffusion local time that we derive. We apply our abstract results to improve consistency results for the local time estimator (LTE) and to prove consistency for a class of simple M-estimators. This revised version was published online in June 2006 with corrections to the Cover Date.