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

被引:67
作者
Delattre, S
Jacod, J
机构
[1] Laboratoire de ProbabiliteÂs (CNRS URA 224), Universite Pierre et Marie Curie (Paris-6), 4 place Jussieu Tour 56, Paris
关键词
functional limit theorems; round-off errors; stochastic differential equations;
D O I
10.2307/3318650
中图分类号
O21 [概率论与数理统计]; C8 [统计学];
学科分类号
020208 ; 070103 ; 0714 ;
摘要
Let X be a one-dimensional diffusion process. For each n greater than or equal to 1 we have a round-off level alpha(n) > 0 and we consider the rounded-off value X-t((alpha n)) = alpha(n)[X-t/alpha(n)]. We are interested in the asymptotic behaviour of the process U(n,phi)(t) = 1/n Sigma(1 less than or equal to i less than or equal to[nt])phi(X(i-1/n)((alpha n)),root n(X-i/n((alpha n)) - X(i-1/n)((alpha n)) - X(i-1/n)((alpha n))) as n goes to +infinity: under suitable assumptions on phi, and when the sequence alpha(n) root n goes to a limit beta is an element of [0, infinity), we prove the convergence of U(n,phi) 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 alpha(n) which is 'small' but not 'very small'.
引用
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页码:1 / 28
页数:28
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