Continuity in law of some additive functionals of bifractional Brownian motion. Issue 4 (19th May 2019)

• Record Type: Journal Article
• Authors: Ait Ouahra, M.; Ouahhabi, H.; Sghir, A.
• Is Part Of: Stochastics. Volume 91:Issue 4(2019)
• Page Start: 613
• Page End: 628
• Publication Date: 2019-05-19
• Subjects: Anisotropic Besov space -- continuity in law -- limit theorem -- tightness -- bifractional brownian motion -- fractional brownian motion -- local time -- fractional derivative -- Slowly varying function; Stochastic processes -- Periodicals; Probabilities -- Periodicals; 519.2
• DOI: 10.1080/17442508.2019.1568436
• Languages: English
• ISSNs: 1744-2508

Abstract

ABSTRACT: LetB H, K be a bifractional Brownain motion with indicesH ∈ ( 0, 1 ) andK ∈ ( 0, 1 ] . We prove the continuity in law, in some anisotropic Besov spaces, with respect to H and K . Our result generalizes those obtained by Jolis and Viles [ Continuity in law with respect to the Hurst parameter of the local time of the fractional Brownian motion, J. Theor. Probab. 20(2) (2007), pp. 133–152] of the fractional Brownian motion local time and gives a new result for the generalized fractional derivatives with kernel depending on slowly varying function of the local time ofB H, K . Notice that their result was generalized by Wu and Xiao [ Continuity in the Hurst index of the local times of anisotropic gaussian random fields, Stoch. Proc. Their Appl. 119 (2009), pp. 1823–1844] for wide class of anisotropic gaussian random fields satisfying some condition (A) which is not satisfied byB H, K . To prove our result, we use the decomposition in law ofB H, K given by Lei and Nualart [ A decomposition of the bifractional Brownian motion and some applications, Statist. Probab. Lett. 79 (2009), pp. 619–624]. Our result is also new in the space of continuous functions.