DOI QR코드

DOI QR Code

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)
  • 투고 : 2018.09.09
  • 심사 : 2020.03.05
  • 발행 : 2020.05.31

초록

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.

키워드

과제정보

This research is supported by the Distinguished Young Scholars Foundation of Anhui Province (1608085J06), the National Natural Science Foundation of China (11271020). The authors are very grateful to the anonymous referee and the editor for their insightful and valuable comments, which have improved the presentation of the paper.

참고문헌

  1. 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
  2. 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
  3. 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
  4. 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
  5. 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
  6. N. Chakravarti and K. L. Sebastian, Fractional Brownian motion model for polymers, Chem. Phys. Lett. 267 (1997), no. 1-2, 9-13. https://doi.org/10.1016/S0009-2614(97)00075-4
  7. J. Cherayil and P. Biswas, Path integral description of polymers using fractional Brownian walks, J. Chem. Phys. 99 (1993), no. 11, 9230-9236. https://doi.org/10.1063/1.465539
  8. 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
  9. 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
  10. 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
  11. 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
  12. 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
  13. 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
  14. 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
  15. 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
  16. 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
  17. 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
  18. 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. https://doi.org/10.1007/s10959-012-0474-8
  19. 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
  20. 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
  21. 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.
  22. G. Shen and Q. Chen, Derivative for the intersection local time of weighted fractional Brownian motion, Submitted.
  23. 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
  24. 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
  25. 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
  26. 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
  27. L. Yan, Derivative for the intersection local time of fractional Brownian motions, Preprint, 2016.
  28. 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
  29. 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
  30. 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