Optimal strong convergence rate of a backward Euler type scheme for the Cox-Ingersoll-Ross model driven by fractional Brownian motion

被引：24

作者：

Hong, Jialin ^{[1
,2
]}

Huang, Chuying ^{[1
,2
]}

Kamrani, Minoo ^{[3
]}

Wang, Xu ^{[1
,2
]}

机构：

[1] Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China

[2] Univ Chinese Acad Sci, Sch Math Sci, Beijing 100049, Peoples R China

[3] Razi Univ, Fac Sci, Dept Math, Kermanshah, Iran

来源：

STOCHASTIC PROCESSES AND THEIR APPLICATIONS | 2020年 / 130卷 / 05期

基金：

中国国家自然科学基金;

关键词：

Cox-Ingersoll-Ross model; Fractional Brownian motion; Backward Euler scheme; Optimal strong convergence rate; Malliavin calculus; APPROXIMATION; SDES; DISCRETIZATION; CIR;

D O I：

10.1016/j.spa.2019.07.014

中图分类号：

O21 [概率论与数理统计]; C8 [统计学];

学科分类号：

020208 ; 070103 ; 0714 ;

摘要：

In this paper, we investigate the optimal strong convergence rate of numerical approximations for the Cox-Ingersoll-Ross model driven by fractional Brownian motion with Hurst parameter H is an element of (1/2, 1). To deal with the difficulties caused by the unbounded diffusion coefficient, we study an auxiliary equation based on Lamperti transformation. By means of Malliavin calculus, we prove that the backward Euler scheme applied to this auxiliary equation ensures the positivity of the numerical solution, and is of strong order one. Furthermore, a numerical approximation for the original model is obtained and converges with the same order. (C) 2019 Elsevier B.V. All rights reserved.

引用

页码：2675 / 2692

页数：18

共 26 条

[1] On the discretization schemes for the CIR (and Bessel squared) processes [J].

Alfonsi, Aurelien .

MONTE CARLO METHODS AND APPLICATIONS, 2005, 11 (04) :355-384

[2] Strong order one convergence of a drift implicit Euler scheme: Application to the CIR process [J].

Alfonsi, Aurelien .

STATISTICS & PROBABILITY LETTERS, 2013, 83 (02) :602-607

[3]

[Anonymous], 2014, Eur. Phys. J. B

[4]

Berkaoui A., 2008, ESAIM: Probability and Statistics, V12, P1, DOI DOI 10.1051/PS:2007030

[5] A THEORY OF THE TERM STRUCTURE OF INTEREST-RATES [J].

COX, JC ;

INGERSOLL, JE ;

ROSS, SA .

ECONOMETRICA, 1985, 53 (02) :385-407

[6] Stochastic analysis of the fractional Brownian motion [J].

Decreusefond, L ;

Üstünel, AS .

POTENTIAL ANALYSIS, 1999, 10 (02) :177-214

[7] An Euler-type method for the strong approximation of the Cox-Ingersoll-Ross process [J].

Dereich, Steffen ;

Neuenkirch, Andreas ;

Szpruch, Lukasz .

PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, 2012, 468 (2140) :1105-1115

[8] A Milstein-type scheme without Levy area terms for SDEs driven by fractional Brownian motion [J].

Deya, A. ;

Neuenkirch, A. ;

Tindel, S. .

ANNALES DE L INSTITUT HENRI POINCARE-PROBABILITES ET STATISTIQUES, 2012, 48 (02) :518-550

[9]

Euch O.E., 2018, MATH FINANCE

[10] Discretization of Stationary Solutions of Stochastic Systems Driven by Fractional Brownian Motion [J].

Garrido-Atienza, Maria J. ;

Kloeden, Peter E. ;

Neuenkirch, Andreas .

APPLIED MATHEMATICS AND OPTIMIZATION, 2009, 60 (02) :151-172

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