Euler Approximation for A Class of Singular Multi-Dimensional SDEs Driven by an Additive Fractional Noise
Main Article Content
Abstract
We consider a class of multi-dimensional stochastic differential equations driven by fractional Brownian motion with Hurst index . In particular, the drift coefficient blows up at 0. We first prove that this equation has a unique positive solution, and then propose a semi-implicit Euler approximation scheme for the equation, and finally show that it is also positive, and study its rate of convergence.
Keywords:
Euler approximation, Fractional Brownian motion, Fractional stochastic differential equation.
References
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Y. Mishura, Stochastic Calculus for Fractional Brownian Motion and Related Processes, Springer, Berlin, 2008.
J. C. Cox, J. E. Ingersoll, S. A. Ross, A Theory of The Term Structure of Interest Rates, Journal of Finance,
Vol. 53, No. 2, 1985, pp. 385-408, https://doi.org/10.2307/1911242.
J. Gatheral, T. Jaisson, M. Rosenbaum, Volatility is Rough, Quantitative Finance,Vol. 18, No. 6, 2018, pp. 933-949, https://doi.org/10.1080/14697688.2017.1393551.
O. Euch, M. Rosenbaum, The Characteristic Function of Rough Heston Models, Math. Finance, Vol. 29, No. 1, 2019, pp. 3-38, http://dx.doi.org/10.1111/mafi.12173.
Y. Mishura, A. Y. Tytarenko, Fractional Cox-Ingersoll-Ross Process with Nonzero "Mean", Mod. Stoch. Theory Appl., Vol. 5, No. 1, 2018, pp. 99-111, https://doi.org/10.15559/18-vmsta97.
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, 2020, pp. 2675-2692, https://doi.org/10.1016/j.spa.2019.07.014.
A. Deya, A. Neuenkirch, S. Tindel, A Milstein-Type Scheme without Lévy Area Terms for SDEs Driven by Fractional Brownian Motion, Ann. Inst. Henri Poincaré Probab. Stat, Vol.48, No. 2, 2012, pp. 518-550, https://doi.org/10.1214/10-AIHP392.
Y. Hu, Y. Liu, D. Nualart, Rate of Convergence and Asymptotic Error Distribution of Euler pproximation Schemes for Fractional Diffusions, Ann. Appl. Probab., Vol. 26, No. 2, 2016, pp. 1147-1207, https://doi.org/10.1214/15-AAP1114.
D. Nualart, A. Rascanu, Differential Equations Driven by Fractional Brownian Motion, Collect. Math., Vol. 53, No. 1, 2002, pp. 55-81.
Y. Hu, D. Nualart, X. Song, A Singular Stochastic Differential Equation Driven by Fractional Brownian Motion, Statist. Probab. Lett., Vol. 78, No. 14, 2008, pp. 2075-2085, http://dx.doi.org/10.1016/j.spl.2008.01.080.
H. L. Ngo, D. Taguchi, Semi-implicit Euler- Maruyama Approximation for Noncolliding Particle Systems, Annals of Applied Probability, Vol. 30, No. 2, 2020, pp. 673-705, https://doi.org/10.1214/19-AAP1512.
S. Dereich, A. Neuenkirch, L. Szpruch, An Euler-Type Method for The Strong Approximation for The Cox-Ingersoll-Ross Process, Proc. R. Soc. A, Vol. 468, No. 2140, 2012, pp. 1105-1115, https://doi.org/10.1098/rspa.2011.0505.
G. H. Golub, C. F. Van Loan, Matrix Computations, The Johns Hopkins University Press, London, 2013.
J. F. Coeurjolly, Simulation and Identification of The Fractional Brownian Motion: A Bibliographical and Comparative Study, Journal of Statistical Software, Vol. 5, No. 7, 2000, pp. 1-53, https://doi.org/10.18637/jss.v005.i07.