Large deviation principle for reflected diffusion process fractional Brownian motion

Yıl 2021, Cilt: 5 Sayı: 1, 127 - 137, 31.03.2021

Raphael Diatta Ibrahima Sane , Alassane Diédhiou

https://doi.org/10.31197/atnaa.767867

Öz

In this paper we establish a large deviation principle for solution of perturbed reflected stochastic
differential equations driven by a fractional Brownian motion B^H with Hurst index H ∈ (0;1).
The key is to prove a uniform Freidlin-Wentzell estimates of solution on the set of continuous
square integrable functions in the dual of Schwartz space . We have built in the whole interval of H ∈ (0;1) a new approch different from that of Y. Inahama [10] for LDP of εBH in [6].Thanks to this we establish the LDP for the process diffusion of reflected stochastic differential
equations via the principle of contraction on the set of continuous square integrable functions in the dual of
Schwartz space.The existence and uniqueness of the solutions of such equations (1) and (2) are obtained by [7].

Anahtar Kelimeler

Fractional Brownian motion, Large deviation principle, Contraction principle, Reflected stochastic differential equation, Skorohod problem

Destekleyen Kurum

Assane SECK University of Ziguinchor,

Proje Numarası

6

Teşekkür

We would like to thank the UASZ and the Laboratory of Mathematics and Application

Kaynakça

• [1] R.Becker, F. Mhlanga, Application of white noise calculus to the computation of greeks, Communication on Stochastic Analysis, vol 7 , 4 ,(2013), 493-510.
• [2] C. Bender, An Ito formula for generalized functionals of a fractional Brownian motion with arbitrary Hurst parameter, Stochastic Process. Appl, vol 104 , 1 ,(2003), 81-106.
• [3] L. Bo, T. Zhang, Large deviation for perturbed reflected diffusion processes, Stochastics, vol 81, 6 , (2009),531-543.
• [4] X. Chen, W. V. Li, J. Rosinski, Q. Shao ,Large deviations for local times and intersection local times of fractional Brownian motions and Riemann-Liouville processes, Ann. Probab,vol 39, 2 , (2011), 729-778.
• [5] A.Dembo, O.Zeitouni, Large deviation techniques and applications, second ed; Springer-verlage, New York, (1998).
• [6] R.Diatta, A.Diedhiou, Large Deviation Principle Applied for a Solution of Mixed Stochastic Differential Equation Involving Independent Standard Brownian Motion and Fractional Brownian Motion ,Applied Mathematical Sciences, Vol 14, 11, (2020), 511-530.
• [7] R. Doney, T. Zhang, Perturbed Skorohod equations and perturbed reflected diffusion processes, Ann. Poincarre, vol 41 ,(2005), 107-121.
• [8] H. Doss, P. Priouret, Petites perturbations de systemes dynamiques avec reflection, Lecteur Notes in Math, Springer, New York,(1983), 986.
• [9] M. I. Freidlin, A. D. Wentzell, Random perturbations of dynanmical systems, second ed., Springer-Verlag. New York (1998).
• [10] Y. Inahama, Laplace approximation for rough differential equation driven by fractional Brownian motion, The Annals of Probability,vol 41, 1, (2013),170-205.
• [11] M. M. Meerschaert, E. X. Y. Nane, Large deviations for local time fractional Brownian motion and applications,J. Math. Anal. Appl. vol 346, 2, (2008), 432?445.
• [12] N. Huy, V. Hoan, N, Thach, Regularized solution of a Cauchy problem for stochastic elliptic equation, Mathematical Methods in the Applied Sciences.
• [13] T.B. Ngoc ,D. O'Regan, N.H. Tuan, On inverse initial value problems for the stochastic strongly damped wave equation, Applicable Analysis,(2020), 1-18.
• [14] H.Holden, B. Oksendal, J. Ube, T. Zhang , Stochastic Partial Diferential Equations A Modelling, White Noise Functional Approach, Springer, second edition, (2010).
• [15] D. Siska , Stochastic differential equations driven by fractional Brownian motion a white noise distribution, theory approach (2004).
• [16] N.H. Tuan,N.T. Thach, L.H. Cam Vu, N.H.Can , On a final value problem for a biparabolic equation with statistical discrete data, Applicable Analysis,(2020), 1-24.
• [17] W. Wang, Z. Chen , Large deviations for subordinated fractional Brownian motion and applications, J. Math. Anal. Appl. vol 458, 2, (2018), 1678-1692.