T-Stability of the Euler-Maruyama Algorithm for the Generalized Black-Scholes Model with Fractional Brownian Motion

doi:10.23940/ijpe.18.04.p23.815820

Abstract

Abstract:

On account of the fact that a fractional Brownian motion (fBm) with the Hurst parameter H ∈(0,1/2)∪(1/2,1) cannot follow the laws of the semimartingale and the Markov process, little work is presented about the T-stability for stochastic differential equations (SDEs) with fBm. Here, three results are obtained for the generalized Black-Scholes model (SDE) with H∈(1/3,1/2). Firstly, the sufficient conditions of the stochastical and asymptotical stability in the large for such equation are presented by the aid of the Lyapunov exponent. Secondly, the Euler-Maruyama (EM) numerical algorithm with a given step-size for such model is constructed. Lastly, by taking advantage of the stable average function, the sufficient conditions of the T-stability that originated from the EM algorithm are presented. All the results show that on the basis of the stability of such equation, the T-stable region produced by the EM algorithm can be found. Moreover, one numerical example is afforded to the main conclusions.

Submitted on December 21, 2017; Revised on January 29, 2018; Accepted on March 5, 2018
References: 32

Add to citation manager EndNote|Reference Manager|ProCite|BibTeX|RefWorks

References 0

	K. Al, “Stochastic Integration with Respect to Fractional Brownian Motion”, Annales De Linstitut Henri Poincare, vol.39, no.1, pp.27-68, 2003
	S. Anna, “Approximation of the Solution of Stochastic Differential Equations Driven by Multifractional Brownian Motion”, Studia Universitatis Babes-Bolyai, Mathematica, vol.56, no.2, pp. 587-598, 2011
	A. Boudaoui, T. Caraballo, and A. Ouahab, “Impulsive Stochastic Functional Differential Inclusions Driven by a Fractional Brownian Motion with Infinite Delay”, Mathematical Methods in the Applied Sciences, vol. 39, no.6, pp. 1435-1451, 2016
	F. Biagini, B. ?ksendal, A. Sulem, and N. Wallner, “An Introduction to White Noise Theory and Malliavin Calculus for Fractional Brownian Motion”, Proceedings Mathematical Physical and Engineering Sciences, vol. 460, no.2041, pp. 347-372, 2004
	F. Black, and M. Scholes, “The Pricing of Options and Corporate Liabilities”, Journal of Political Economy, vol. 81, no.3, pp. 637-654, 1973
	L. Bai, and J. Ma, “Stochastic Differential Equations Driven by Fractional Brownian Motion and Poisson Point Process”, Bernoulli, vol. 21, no.1, pp. 303-334, 2015
	K. Burrage, P. Burrage, and T. Mitsui, “Numerical Solutions of Stochastic Differential Equations Implementation and Stability Issues”, Journal of Computational and Applied Mathematics, vol. 125, no.1, pp. 171-182, 2000
	W. R. Cao, “T-stability of the Semi-implicit Euler Method for Delay Differential Equations with Multiplicative Noise”, Applied Mathematics and Computation, vol. 216, no.3, pp. 999–1006, 2010
	L. Decreusefond, and A. Ustunel, “Stochastic Analysis of the Fractional Brownian Motion”, Potential Analysis, vol. 10, no.2, pp. 177–214, 1999
	T. E. Duncan, B. Maslowski, and B. Pasik-Duncan, “Stochastic Equations in Hilbert Space with a Multiplicative Fractional Gaussian Noise”, Stochastic Processes and Their Applications, vol.115, no.8, pp.1357-1383, 2005
	N. Dung, “Neutral Stochastic Differential Equations Driven by a Fractional Brownian Motion with Impulsive Effects and Varying-time Delays”, Journal of the Korean Statistical Society, vol.43, no.4, pp.599-608, 2014
	R. Elliott, and J. Van Der Hoek, “A General Fractional White Noise Theory and Applications to Finance”, Mathematical Finance, vol.13, no.2, pp.301-330, 2003
	R. Ezzati, M. Khodabin, and Z. Sadati, “Numerical Implementation of Stochastic Operational Matrix Driven by a Fractional Brownian Motion for Solving a Stochastic Differential Equation”, Abstract and Applied Analysis, vol.2014, no.1, pp.1-11, 2014
	M. J. Garrido-Atienza, P. E. Kloeden,and A. Neuenkirch, “Discretization of Stationary Solutions of Stochastic Systems Driven by Fractional Brownian Motion”, Applied Mathematics and Optimization, vol.60, no.2, pp.151-172, 2009
	J. Guerra, and D. Nualart, “Stochastic Differential Equations Driven by Fractional Brownian Motion and Standard Brownian Motion”, Stochastic Analysis and Applications, vol.26, no.5, pp.1053-1075, 2008
	Z. Guo, and H. Yuan, “Pricing European Option under the Time-changed Mixed Brownian Fractional Brownian Model”, Physica A Statistical Mechanics and Its Applications, vol.406, no.406, pp.73-79, 2014
	P. Guo, C. Zeng, C. Li, and Y. Chen, “Numerics for the Fractional Langevin Equation Driven by the Fractional Brownian Motion”, Fractional Calculus and Applied Analysis, vol.16, no.1, pp.123-141, 2013
	Y. Hu, and B. ?ksendal, “Fractional White Noise Calculus and Applications to Finance”, Infinite Dimensional Analysis Quantum Probability and Related Topics, vol.6, no.1, pp.1-32, 2003
	A. L. Jorge, and S. M. Jaime, “Linear Stochastic Differential Equations Driven by a Fractional Brownian Motion with Hurst Parameter less than 1/2”, Stochastic Analysis and Applications, vol.25, no.1, pp.105-126, 2006
	M. Jolis, “On the Wiener Integral with Respect to the Fractional Brownian Motion on an Interval”, Journal of Mathematical Analysis and Applications, vol.330, no.2, pp.1115-1127, 2007
	M. Kamrani, “Numerical Solution of Stochastic Fractional Differential Equations”, Numerical Algorithms, vol.68, no.1, pp.81-93, 2015
	J. Liao and X. Wang, “Stability of Stochastic Differential Equation with Linear Fractal noise”, Frontiers of Mathematics in China, 9, vol.9, no.3, pp.495-507, 2014
	B. B .Mandelbrot, and H. W. Van Ness, “Fractional Brownian Motions, Fractional Noises and Applications”, Siam Review, vol.10, no.4, pp.422-437, 1968
	B. Maslowski, and B. Schmalfuss, “Random Dynamical Systems and Stationary Solutions of Differential Equations Driven by the Fractional Brownian Motion”, Stochastic Analysis and Applications, vol.22, no.6, pp.1577-1607, 2004
	R. C. Merton, “Theory of Rational Option Pricing”, Bell Journal of Economics, vol.4, no.1, pp.141-183, 1973
	A. Neuenkirch, and I. Nourdin, “Exact Rate of Convergence of Some Approximation Schemes Associated to SDEs Driven by a FBM”, Journal of Theoretical Probability, vol.20, no.4, pp.871-899, 2007
	A. Neuenkirch, “Optimal Pointwise Approximation of Stochastic Differential Equations Driven by Fractional Brownian Motion”, Stochastic Processes and Their Applications, vol.118, no.12, pp.2294-2333, 2008
	H. S. Shu, C. L. Chen, and G. L. Wei, “Stability of Linear Stochastic Differential Equations with Respect to Fractional Brownian Motion”, Journal of Donghua University, vol.26, no.2, pp.531-536, 2009
	D. N. Tien, “Neutral Stochastic Differential Equations Driven by a Fractional Brownian Motion with Impulsive Effects and Varying-time Delays”, Journal of the Korean Statistical Society, vol.43, no.4, pp.599-608, 2014
	C. Zeng, Q. Yang, and Y. Q. Chen, “Solving Nonlinear Stochastic Differential Equations with Fractional Brownian Motion Using Reducibility Approach”, Nonlinear Dynamics, vol.67, no.4, pp.2719-2726, 2012
	C. Zeng, Y. Q. Chen, and Q. Yang, “Almost Sure and Moment Stability Properties of Fractional Order Black-Scholes Model”, Fractional Calculus Applied Analysis, vol.16, no.2, pp.317-331, 2013
	C. Zeng, Q. Yang, and Y. Q. Chen, “Lyapunov Techniques for Stochastic Differential Equations Driven by Fractional Brownian Motion”, Abstract and Applied Analysis, vol.2014, no.2, pp.1-9, 2014