Asymptotic equivalence of discretely observed diffusion processes and their Euler scheme: small variance case
This paper establishes the global asymptotic equivalence, in the sense of the Le Cam \(\Delta \)-distance, between scalar diffusion models with unknown drift function and small variance on the one side, and nonparametric autoregressive models on the other side. The time horizon \(T\) is kept fixed and both the cases of discrete and continuous observation of the path are treated. We allow non constant diffusion coefficient, bounded but possibly tending to zero. The asymptotic equivalences are established by constructing explicit equivalence mappings.
KeywordsNonparametric experiments Deficiency distance Asymptotic equivalence Diffusion processes Autoregression
Mathematics Subject ClassificationPrimary 62B15 Secondary 62G20, 60G5
I would like to thank Valentine Genon-Catalot for several interesting discussions, especially in suggesting to taking into account the relation between diffusion processes with small variance and deterministic limits. Also, I would like to give a special thank to Pierre Étoré, with whom a lot of hours were spent discussing different approaches to the proof of Lemma 4.10. More generally, I am very grateful for all the time he has invested in supervising this project.
- Carter AV ‘(2009) Asymptotially sufficient statistis in nonparametric regression ex-periments with correlated noise. J Probab Stat , 19, Article ID 275308Google Scholar
- Freidlin MI, Wentzell AD (2012) Random perturbations of dynami al systems. Grundlehren der Mathematishen Wissenshaften [FundamentalPriniples of Mathemati al Sienes. Translated from the 1979 Russian originalby Joseph Szücs. 3rd edn., Springer, Heidelberg, vol 260. pp xxviii+458Google Scholar
- Guy R (2013) Inférence dans le cadre des maladies transmissibles par des processus de diffusion. Ph.D thesisGoogle Scholar
- Kutoyants YA (1984) On nonparametric estimation of trend coefficients in a diffusion process. Stat Control Stoch Process 230:250Google Scholar
- Mariucci E (2014) Asymptotic equivalence for inhomogeneous jump diffusion processes and white noise. arXiv: 1405.0480