Bernoulli

A central limit theorem for normalized functions of the increments of a diffusion process, in the presence of round-off errors

Sylvain Delattre and Jean Jacod

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Abstract

Let X be a one-dimensional diffusion process. For each n 1 we have a round-off level α n >0 and we consider the rounded-off value X t ( α) =α[X t/α] . We are interested in the asymptotic behaviour of the processes U (n,φ) t=1 n 1 i[nt]φ(X ( i-1)/n ( α),n (X i /n ( α)-X ( i-1)/n ( α))) as n goes to + : under suitable assumptions on φ , and when the sequence α n goes to a limit β [0,) , we prove the convergence of U (n,φ) to a limiting process in probability (for the local uniform topology), and an associated central limit theorem. This is motivated mainly by statistical problems in which one wishes to estimate a parameter occurring in the diffusion coefficient, when the diffusion process is observed at times i /n and is subject to rounding off at some level α n which is 'small' but not 'very small'.

Article information

Source
Bernoulli, Volume 3, Number 1 (1997), 1-28.

Dates
First available in Project Euclid: 4 May 2007

Permanent link to this document
https://projecteuclid.org/euclid.bj/1178291930

Mathematical Reviews number (MathSciNet)
MR1466543

Zentralblatt MATH identifier
0882.60017

Keywords
functional limit theorems round-off errors

Citation

Delattre, Sylvain; Jacod, Jean. A central limit theorem for normalized functions of the increments of a diffusion process, in the presence of round-off errors. Bernoulli 3 (1997), no. 1, 1--28. https://projecteuclid.org/euclid.bj/1178291930


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