A Limit Law for Functionals of Multiple Independent Fractional Brownian Motions
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
A LIMIT LAW FOR FUNCTIONALS OF MULTIPLE INDEPENDENT FRACTIONAL BROWNIAN MOTIONS∗
{[)
Qian YU (
School of Statistics, East China Normal University, Shanghai 200241, China E-mail : [email protected] Abstract Let B = {B H (t)}t≥0 be a d-dimensional fractional Brownian motion with Hurst parameter H ∈ (0, 1). Consider the functionals of k independent d-dimensional fractional Brownian motions 1 √ n
Z
ent1 0
···
Z
entk 0
f (B H,1 (s1 ) + · · · + B H,k (sk ))ds1 · · · dsk ,
where the Hurst index H = k/d. Using the method of moments, we prove the limit law and extending a result by Xu [19] of the case k = 1. It can also be regarded as a fractional generalization of Biane [3] in the case of Brownian motion. Key words
Limit theorem; fractional Brownian motion; method of moments; chaining argument
2010 MR Subject Classification
1
60F17; 60G15; 60G22
Introduction
Let B = {B H (t)}t≥0 be a fractional Brownian motion (fBm) with Hurst parameter H ∈ (0, 1). A stochastic calculus with respect to it has been intensively developed (see, for example, Biagini et al. [2], Nualart [9]). It is a central Gaussian process with B H (0) = 0 and the covariance function 1 2H E B H (t)B H (s) = t + s2H − |t − s|2H 2
for all t, s > 0. This process was first introduced by Kolmogorov and studied by Mandelbrot and Van Ness [8], where a stochastic integral representation in terms of a standard Brownian motion was established: q i 2HΓ( 32 − H) Z h H− 1 H− 1 B H (t) = (t − s)+ 2 − (−s)+ 2 dB(s), 1 Γ(H + 2 )Γ(2 − 2H) R ∗ Received
December 12, 2018; revised October 14, 2019. Q. Yu is partially supported by ECNU Academic Innovation Promotion Program for Excellent Doctoral Students (YBNLTS2019-010) and the Scientific Research Innovation Program for Doctoral Students in Faculty of Economics and Management (2018FEM-BCKYB014).
No.3
Q. Yu: LIMIT LAW FOR FRACTIONAL BROWNIAN MOTIONS
735
where u+ = max{u, 0}, and B(s) is standard Brownian motion. For H = 12 , B H coincides with the standard Brownian motion B, but B H is neither a semimartingale nor a Markov process unless H = 21 . On the basis of sufficient study of fBm, the research technology of the fBm is gradually mature, and many results about Brownian motion can be extended to the fBm, especially some classical limit theories. In this article, we will consider the limit law for functionals of d-dimensional fBm. Let B H (t) = (B H1 (t), · · · , B Hd (t)), t ≥ 0 be a d-dimensional fBm with Hurst index H in (0, 1). Let B H,1 , B H,2 , · · · , B H,k be k independent copies of B H with H = k/d. If k = 1, the local time of fBm B H does not exist. This is called the critical case. In this condition, Xu [19] considered the limit law. That is, for any bounded and integrable R function f : Rd → R and Rd f (x)dx = 0, 1 √ n
Z
ent
0
L
f (B H (s))ds −→ cf,d
p l(M −1 (t))η
as n tends to infinity, where cf,d is a constant depend on f and d, l(t) is the local time at 0 of a Brownian motion
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