THE LONG TIME BEHAVIOR OF THE FRACTIONAL ORNSTEIN-UHLENBECK PROCESS WITH LINEAR SELF-REPELLING DRIFT

doi:10.1007/s10473-024-0216-x

Abstract

Abstract: Let $B^{H} $ be a fractional Brownian motion with Hurst index $\frac{1}{2}\leq H< 1$. In this paper, we consider the equation (called the Ornstein-Uhlenbeck process with a linear self-repelling drift) $\begin{equation*} {\rm d}X_{t}^{H}={\rm d}B_{t}^{H}+\sigma X_t^{H}{\rm d}t+\nu {\rm d}t-\theta \left(\int_{0}^{t}(X_t^{H}-X_{s}^{H}){\rm d}s\right){\rm d}t, \end{equation*} $ where $\theta<0$, $\sigma,\nu \in \mathbb{R}$. The process is an analogue of {self-attracting} diffusion (Cranston, Le Jan. Math Ann, 1995, 303: 87-93). Our main aim is to study the large time behaviors of the process. We show that the solution $X^H$ diverges to infinity as $t$ tends to infinity, and obtain the speed at which the process $X^H$ diverges to infinity as $t$ tends to infinity.

Key words: fractional Brownian motion, stochastic difference equations, rate of convergence, asymptotic

CLC Number:

60G22

Xiaoyu XIA, Litan YAN, Qing YANG. THE LONG TIME BEHAVIOR OF THE FRACTIONAL ORNSTEIN-UHLENBECK PROCESS WITH LINEAR SELF-REPELLING DRIFT[J].Acta mathematica scientia,Series B, 2024, 44(2): 671-685.

Trendmd

References

[1] Benaïm M, Ledoux M, Raimond O. Self-interacting diffusions. Prob Theory Rel Fields, 2002, 122: 1-41
[2] Benaïm M, Ciotir I, Gauthier C E. Self-repelling diffusions via an infinite dimensional approach. Stoch PDE: Anal Comp, 2015, 3: 506-530
[3] Bertoin J. Sur une intégrale pour les processus á $\alpha$-variation borné. Ann Probab, 1989 17: 1521-1535
[4] Biagini F, Hu Y, Øksendal B, Zhang T.Stochastic Calculus for fBm and Applications. Berlin: Springer, 2008
[5] Cranston M, Le Jan Y. Self-attracting diffusions: Two case studies. Math Ann, 1995, 303: 87-93
[6] Cranston M, Mountford T S. The strong law of large numbers for a Brownian polymer. Ann Probab, 1996, 24: 1300-1323
[7] Durrett R, Rogers L C G. Asymptotic behavior of Brownian polymer. Prob Theory Rel Fields, 1991, 92: 337-349
[8] FöIllmer H. Calcul d'Itô sans probabilités//Azéme J, Yor M. Séinaire de Probabilités XV.1980, 50: 143-145
[9] Gan Y, Yan L.Least squares estimation for the linear self-repelling diffusion driven by fractional Brownian motion
(in Chinese). Sci China Math, 2018, 48: 1143-1158
[10] Gauthier C E. Self attracting diffusions on a sphere and application to a periodic case. Electron Commun Probab, 2016, 21: Art 53
[11] Herrmann S, Roynette B. Boundedness and convergence of some self-attracting diffusions. Math Ann, 2003, 325: 81-96
[12] Herrmann S, Scheutzow M. Rate of convergence of some self-attracting diffusions. Stochastic Process Appl, 2004, 111: 41-55
[13] Hu Y. Integral transformations and anticipative calculus for fBms. Memoirs Amer Math Soc, 2005, 175: Art 825
[14] Mishura Y.Stochastic Calculus for fBm and Related Processes. Berlin: Springer, 2008
[15] Mountford T, Tarrés P. An asymptotic result for Brownian polymers. Ann Inst H Poincare Probab Statist, 2008, 44: 29-46
[16] Nourdin I.Selected Aspects of Fractional Brownian Motion. Berlin: Springer, 2012
[17] Nualart D.Malliavin Calculus and Related Topics. Berlin: Springer-Verlag, 2006
[18] Pipiras V, Taqqu M. Integration questions related to the fractional Brownian motion. Probab Theory Related Fields, 2001, 118: 251-281
[19] Sun X, Yan L, Ge Y. The laws of large numbers associated with the linear self-attracting diffusion driven by fractional Brownian motion and applications. Journal of Theoretical Probability, 2022, 35: 1423-1478
[20] Tudor Ciprian A.Analysis of Variations for Self-Similar Processes. Berlin: Springer-Verlag, 2013
[21] Yan L, Sun Y, Lu Y. On the linear fractional self-attracting diffusion. J Theort Probab, 2008, 21: 502-516