Abstract
In this article, we obtain sharp conditions for the existence of the high-order derivatives (k-th order) of intersection local time \( \widehat{\alpha }^{(k)}(0)\) of two independent d-dimensional fractional Brownian motions \(B^{H_1}_t\) and \(\widetilde{B}^{H_2}_s\) of Hurst parameters \(H_1\) and \(H_2\), respectively. We also study their exponential integrability.
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References
Berman, S.M.: Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J. 23(1), 69–94 (1973)
Hu, Y.: Analysis on Gaussian Spaces. World Scientific, Singapore (2017)
Hu, Y., Nualart, D.: Renormalized self-intersection local time for fractional Brownian motion. Ann. Probab. 33(3), 948–983 (2005)
Hu, Y., Nualart, D., Song, J.: Integral representation of renormalized self-intersection local times. J. Func. Anal. 255(9), 2507–2532 (2008)
Hu, Y., Huang, J., Nualart, D., Tindel, S.: Stochastic heat equations with general multiplicative Gaussian noises: Hölder continuity and intermittency. Electron. J. Probab. 20(55), 1–50 (2015)
Jung, P., Markowsky, G.: On the Tanaka formula for the derivative of self-intersection local time of fractional Brownian motion. Stoch. Process. Their Appl. 124, 3846–3868 (2014)
Jung, P., Markowsky, G.: H\(\ddot{o}\)lder continuity and occupation-time formulas for fBm self-intersection local time and its derivative. J. Theor. Probab. 28, 299–312 (2015)
Nualart, D., Ortiz-Latorre, S.: Intersection local time for two independent fractional Brownian motions. J. Theor. Probab. 20, 759–767 (2007)
Oliveira, M., Silva, J., Streit, L.: Intersection local times of independent fractional Brownian motions as generalized white noise functionals. Acta Appl. Math. 113(1), 17–39 (2011)
Pitt, L.D.: Local times for Gaussian vector fields. Indiana Univ. Math. J. 27(2), 309–330 (1978)
Wu, D., Xiao, Y.: Regularity of intersection local times of fractional Brownian motions. J. Theor. Probab. 23(4), 972–1001 (2010)
Yan, L.: Derivative for the intersection local time of fractional Brownian motions (2014). arXiv:1403.4102v3
Yan, L., Yu, X.: Derivative for self-intersection local time of multidimensional fractional Brownian motion. Stochastics 87(6), 966–999 (2015)
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Jingjun Guo acknowledges the support of National Natural Science Foundation of China #71561017, the Youth Academic Talent Plan of Lanzhou University of Finance and Economics. Yaozhong Hu is partially supported by a Grant from the Simons Foundation #209206 and by a General Research Fund of University of Kansas. Yanping Xiao acknowledges the support of Basic Charge of Research for Northwest Minzu University #31920170035.
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Guo, J., Hu, Y. & Xiao, Y. Higher-Order Derivative of Intersection Local Time for Two Independent Fractional Brownian Motions. J Theor Probab 32, 1190–1201 (2019). https://doi.org/10.1007/s10959-017-0800-2
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DOI: https://doi.org/10.1007/s10959-017-0800-2
Keywords
- Fractional Brownian motion
- Intersection local time
- k-th derivative of intersection local time
- Exponential integrability