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http://dx.doi.org/10.4134/BKMS.b180852

THE LOCAL TIME OF THE LINEAR SELF-ATTRACTING DIFFUSION DRIVEN BY WEIGHTED FRACTIONAL BROWNIAN MOTION  

Chen, Qin (Department of Mathematics Fuyang Normal University)
Shen, Guangjun (School of Mathematics and Finance Chuzhou University)
Wang, Qingbo (Department of Mathematics Anhui Normal University)
Publication Information
Bulletin of the Korean Mathematical Society / v.57, no.3, 2020 , pp. 547-568 More about this Journal
Abstract
In this paper, we introduce the linear self-attracting diffusion driven by a weighted fractional Brownian motion with weighting exponent a > -1 and Hurst index |b| < a + 1, 0 < b < 1, which is analogous to the linear fractional self-attracting diffusion. For the 1-dimensional process we study its convergence and the corresponding weighted local time. As a related problem, we also obtain the renormalized intersection local time exists in L2 if max{a1 + b1, a2 + b2} < 0.
Keywords
Weighted fractional Brownian motion; self-attracting diffusion; intersection local time;
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1 T. Bojdecki, L. G. Gorostiza, and A. Talarczyk, Occupation time limits of inhomogeneous Poisson systems of independent particles, Stochastic Process. Appl. 118 (2008), no. 1, 28-52. https://doi.org/10.1016/j.spa.2007.03.008   DOI
2 T. Bojdecki, L. G. Gorostiza, and A. Talarczyk, Self-similar stable processes arising from high-density limits of occupation times of particle systems, Potential Anal. 28 (2008), no. 1, 71-103. https://doi.org/10.1007/s11118-007-9067-z   DOI
3 N. Chakravarti and K. L. Sebastian, Fractional Brownian motion model for polymers, Chem. Phys. Lett. 267 (1997), no. 1-2, 9-13.   DOI
4 J. Cherayil and P. Biswas, Path integral description of polymers using fractional Brownian walks, J. Chem. Phys. 99 (1993), no. 11, 9230-9236.   DOI
5 M. Cranston and Y. Le Jan, Self-attracting diffusions: two case studies, Math. Ann. 303 (1995), no. 1, 87-93. https://doi.org/10.1007/BF01460980   DOI
6 T. E. Duncan, Y. Hu, and B. Pasik-Duncan, Stochastic calculus for fractional Brownian motion. I. Theory, SIAM J. Control Optim. 38 (2000), no. 2, 582-612. https://doi.org/10.1137/S036301299834171X   DOI
7 R. T. Durrett and L. C. G. Rogers, Asymptotic behavior of Brownian polymers, Probab. Theory Related Fields 92 (1992), no. 3, 337-349. https://doi.org/10.1007/BF01300560   DOI
8 J. Garzon, Convergence to weighted fractional Brownian sheets, Commun. Stoch. Anal. 3 (2009), no. 1, 1-14. https://doi.org/10.31390/cosa.3.1.01
9 J. Guo, Y. Hu, and Y. Xiao, Higher-order derivative of intersection local time for two independent fractional Brownian motions, J. Theoret. Probab. 32 (2019), no. 3, 1190-1201. https://doi.org/10.1007/s10959-017-0800-2   DOI
10 S. Herrmann and B. Roynette, Boundedness and convergence of some self-attracting diffusions, Math. Ann. 325 (2003), no. 1, 81-96. https://doi.org/10.1007/s00208-002-0370-0   DOI
11 T. Mountford and P. Tarres, An asymptotic result for Brownian polymers, Ann. Inst. Henri Poincare Probab. Stat. 44 (2008), no. 1, 29-46. https://doi.org/10.1214/07-AIHP113   DOI
12 Y. Hu and B. Oksendal, Chaos expansion of local time of fractional Brownian motions, Stochastic Anal. Appl. 20 (2002), no. 4, 815-837. https://doi.org/10.1081/SAP-120006109   DOI
13 A. Jaramillo and D. Nualart, Asymptotic properties of the derivative of self-intersection local time of fractional Brownian motion, Stochastic Process. Appl. 127 (2017), no. 2, 669-700. https://doi.org/10.1016/j.spa.2016.06.023   DOI
14 P. Jung and G. Markowsky, On the Tanaka formula for the derivative of self-intersection local time of fractional Brownian motion, Stochastic Process. Appl. 124 (2014), no. 11, 3846-3868. https://doi.org/10.1016/j.spa.2014.07.001   DOI
15 P. Jung and G. Markowsky, Holder continuity and occupation-time formulas for fBm self-intersection local time and its derivative, J. Theoret. Probab. 28 (2015), no. 1, 299-312.   DOI
16 I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Graduate Texts in Mathematics, 113, Springer-Verlag, New York, 1988. https://doi.org/10.1007/978-1-4684-0302-2
17 L. C. G. Rogers and D. Williams, Diffusions, Markov Processes, and Martingales. Vol. 2, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1987.
18 G. Shen and Q. Chen, Derivative for the intersection local time of weighted fractional Brownian motion, Submitted.
19 S. Herrmann and M. Scheutzow, Rate of convergence of some self-attracting diffusions, Stochastic Process. Appl. 111 (2004), no. 1, 41-55. https://doi.org/10.1016/j.spa.2003.10.012   DOI
20 G. Shen, L. Yan, and J. Cui, Berry-Esseen bounds and almost sure CLT for quadratic variation of weighted fractional Brownian motion, J. Inequal. Appl. 2013 (2013), 275, 13 pp. https://doi.org/10.1186/1029-242X-2013-275   DOI
21 G. Shen, X. Yin, and L. Yan, Least squares estimation for Ornstein-Uhlenbeck processes driven by the weighted fractional Brownian motion, Acta Math. Sci. Ser. B (Engl. Ed.) 36 (2016), no. 2, 394-408. https://doi.org/10.1016/S0252-9602(16)30008-X
22 X. Sun and L. Yan, Central limit theorems and parameter estimation associated with a weighted-fractional Brownian motion, J. Statist. Plann. Inference 192 (2018), 45-64. https://doi.org/10.1016/j.jspi.2017.07.001   DOI
23 X. Sun, L. Yan, and Q. Zhang, The quadratic covariation for a weighted fractional Brownian motion, Stoch. Dyn. 17 (2017), no. 4, 1750029, 41 pp. https://doi.org/10.1142/S0219493717500290   DOI
24 L. Yan, Derivative for the intersection local time of fractional Brownian motions, Preprint, 2016.
25 L. Yan, Y. Sun, and Y. Lu, On the linear fractional self-attracting diffusion, J. Theoret. Probab. 21 (2008), no. 2, 502-516. https://doi.org/10.1007/s10959-007-0113-y   DOI
26 L. Yan, Z. Wang, and H. Jing, Some path properties of weighted-fractional Brownian motion, Stochastics 86 (2014), no. 5, 721-758. https://doi.org/10.1080/17442508.2013.878345   DOI
27 L. Yan and X. Yu, Derivative for self-intersection local time of multidimensional fractional Brownian motion, Stochastics 87 (2015), no. 6, 966-999. https://doi.org/10.1080/17442508.2015.1019883   DOI
28 T. Bojdecki, L. G. Gorostiza, and A. Talarczyk, Some extensions of fractional Brownian motion and sub-fractional Brownian motion related to particle systems, Electron. Comm. Probab. 12 (2007), 161-172. https://doi.org/10.1214/ECP.v12-1272   DOI
29 E. Alos, O. Mazet, and D. Nualart, Stochastic calculus with respect to Gaussian processes, Ann. Probab. 29 (2001), no. 2, 766-801. https://doi.org/10.1214/aop/1008956692   DOI
30 S. M. Berman, Local nondeterminism and local times of Gaussian processes, Indiana Univ. Math. J. 23 (1973/74), 69-94. https://doi.org/10.1512/iumj.1973.23.23006   DOI