THE MALLIAVIN DERIVATIVE FOR G ENERALIZATION  OF FRACTIONAL BESSEL TYPE PROCESS

Vũ Thị Hương

doi:10.34238/tnu-jst.4243

THE MALLIAVIN DERIVATIVE FOR G ENERALIZATION OF FRACTIONAL BESSEL TYPE PROCESS

About this article

Received: 01/04/21 Revised: 28/05/21 Published: 31/05/21

Authors

Vu Thi Huong , University of Transport and Communications

Abstract

This paper considersa generalization of fractional Bessel type process. It is also a type of singular stochastic differential equations driven by fractional Brownian motion which were studied by some authors. Undersome assumptions of coefficients, this equation has a unique positive solution. The main purpose of this paper is to show the formula of the Malliavin derivative for this process. The techniques of Malliavin calculus were applied for stochastic differential equations driven by a fractional Brownian motion. We obtain that the Malliavin derivative for this process is an exponent function of the drift coefficient's derivative. This result is useful to estimate inverse moments of the solution. From that, we can estimate the rate of convegence of the numerical approximation in the Lp- norm.

Keywords

Fractional Brownian motion; Fractional Bessel process; Fractional stochastic differential equation; Malliavin derivative; Malliavin calculus

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References

[1] Y. Hu, D. Nualart, and X. Song, "A singular stochastic differential equation driven by fractional Brownian motion," Statistics Probability Letters, vol. 78, no. 14, pp. 2075-2085, 2008.

[2] Y. Mishura, Anton Yurchenko-Tytarenko, "Fractional Cox-Ingersoll-Ross process with non-zero "mean"," Mod. Stoch. Theory Appl., vol. 5, no. 1, pp. 99 - 111, 2018.

[3] J. Hong, C. Huang, M. Kamrani, X.Wang, " Optimal strong convergence rate of a backward Euler type scheme for the Cox–Ingersoll–Ross model driven by fractional Brownian motion," Stochastic Processes and their Applications, vol. 130, no. 5, pp. 2675-2692, 2020.

[4] T. H. Vu, "Existence and uniqueness of solution for generalization of fractional Bessel type process," (in Vietnamese), TNU Journal of Science and Technology, vol. 225, no. 02: Natural Sciences - Engineering - Technology, pp. 39-44, 2020.

[5] F. Biagini, Y. Hu, B. Oksendal and T. Zhang, Stochastic Calculus for Fractional Brownian Motion and Applications. Springer, London, 2008.

[6] D. Nualart, The Malliavin Calculus and Related Topics , 2nd Edition, SpringerVerlag Berlin Heidelberg, 2006.

[7] D. Nualart and B. Saussereau, "Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion," Stochastic processes and their applications, vol. 119, no. 2, pp. 391-409, February 2009.

DOI: https://doi.org/10.34238/tnu-jst.4243

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