no. G2
Probabilités
Density estimates for the exponential functionals of fractional Brownian motion
Tien Dung, Nguyen1  ; Thu Hang, Nguyen2 ; Phuong Thuy, Pham Thi3
1 Department of Mathematics, VNU University of Science, Vietnam, National University, Hanoi, 334 Nguyen Trai, Thanh Xuan, Hanoi, 084 Vietnam
2 Department of Mathematics, Hanoi University of Mining and Geology, 18 Pho Vien, Bac Tu Liem, Hanoi, 084 Vietnam
3 The faculty of Basic Sciences, Vietnam Air Defence and Air Force Academy, Son Tay, Ha Noi, 084 Vietnam
Comptes Rendus. Mathématique, Tome 360 (2022) no. G2, pp. 151-159.

In this note, we investigate the density of the exponential functional of the fractional Brownian motion. Based on the techniques of Malliavin’s calculus, we provide a log-normal upper bound for the density.

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Révisé le :
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DOI : 10.5802/crmath.274
Classification : 60G22, 60H07
Tien Dung, Nguyen 1 ; Thu Hang, Nguyen 2 ; Phuong Thuy, Pham Thi 3

1 Department of Mathematics, VNU University of Science, Vietnam, National University, Hanoi, 334 Nguyen Trai, Thanh Xuan, Hanoi, 084 Vietnam
2 Department of Mathematics, Hanoi University of Mining and Geology, 18 Pho Vien, Bac Tu Liem, Hanoi, 084 Vietnam
3 The faculty of Basic Sciences, Vietnam Air Defence and Air Force Academy, Son Tay, Ha Noi, 084 Vietnam 
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     title = {Density estimates for the exponential functionals of fractional {Brownian} motion},
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     pages = {151--159},
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Tien Dung, Nguyen; Thu Hang, Nguyen; Phuong Thuy, Pham Thi. Density estimates for the exponential functionals of fractional Brownian motion. Comptes Rendus. Mathématique, Tome 360 (2022) no. G2, pp. 151-159. doi : 10.5802/crmath.274. http://archive.numdam.org/articles/10.5802/crmath.274/

[1] Besalú, Mireia; Kohatsu-Higa, Arturo; Tindel, Samy Gaussian-type lower bounds for the density of solutions of SDEs driven by fractional Brownian motions, Ann. Probab., Volume 44 (2016) no. 1, pp. 399-443 | MR | Zbl

[2] Liu, Junfeng; Tudor, Ciprian A. Stochastic heat equation with fractional Laplacian and fractional noise: existence of the solution and analysis of its density, Acta Math. Sci., Volume 37 (2017) no. 6, pp. 1545-1566 | MR | Zbl

[3] Matsumoto, Hiroyuki; Yor, Marc Exponential functionals of Brownian motion. I. Probability laws at fixed time, Probab. Surv., Volume 2 (2005), pp. 312-347 | MR | Zbl

[4] Matsumoto, Hiroyuki; Yor, Marc Exponential functionals of Brownian motion. II. Some related diffusion processes, Probab. Surv., Volume 2 (2005), pp. 348-384 | MR | Zbl

[5] Nualart, David The Malliavin calculus and related topics, Probability and Its Applications, Springer, 2006 | Zbl

[6] Tan, Nguyen Van Smoothness and Gaussian density estimates for stochastic functional differential equations with fractional noise, Stat. Optim. Inf. Comput., Volume 8 (2020) no. 4, pp. 822-833 | DOI | MR

[7] Tien Dung, Nguyen The density of solutions to multifractional stochastic Volterra integro-differential equations, Nonlinear Anal., Theory Methods Appl., Volume 130 (2016), pp. 176-189 | DOI | MR | Zbl

[8] Tien Dung, Nguyen Kolmogorov distance between the exponential functionals of fractional Brownian motion, C. R. Math. Acad. Sci. Paris, Volume 357 (2019) no. 7, pp. 629-635 | MR | Zbl

[9] Tien Dung, Nguyen Gaussian lower bounds for the density via Malliavin calculus, C. R. Math. Acad. Sci. Paris, Volume 358 (2020) no. 1, pp. 79-88 | MR | Zbl

[10] Yor, Marc Exponential functionals of Brownian motion and related processes, Springer Finance, Springer, 2001 | Zbl

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