×

Convergence of a numerical scheme associated to stochastic differential equations with fractional Brownian motion. (English) Zbl 1467.65002

Summary: We are interested in finding an approximation for the solution of stochastic differential equations (SDEs) driven by fractional Brownian motion (fBm) with Hurst parameter \(H > \frac{ 1}{ 2} \). Based on Taylor expansion we derive a numerical scheme and investigate its convergence. Under some assumptions on drift and diffusion, we show that the introduced method is convergent with strong rate of convergence \(\Delta^H\), where \(\Delta\) is the diameter of partition used for discretization. In addition, we explain the simulation of the proposed method and show the accuracy of our results by presenting an example.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G22 Fractional processes, including fractional Brownian motion
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Afanasiev, V. N.; Kolmanosvkii, V. B.; Nosov, V. R., Mathematical Theory of Control Systems Design (1996), Springer · Zbl 0845.93001
[2] Buckwar, E.; Winkler, R., Improved linear multi-step methods for stochastic ordinary differential equations, Comput. Appl. Math., 205, 912-922 (2007) · Zbl 1128.65005
[3] Busenberg, S.; Cooke, K. L., The effect of integral conditions in certain equations modelling epidemics and population growth, J. Math. Biol., 10, 13-32 (1980) · Zbl 0464.92022
[4] Deya, A.; Neuenkirch, A.; Tindel, S., A Milstein-type scheme without Levy area terms for sdes driven by fractional Brownian motion, Ann. Inst. Henri Poincaré Probab. Stat., 48, 518-550 (2012) · Zbl 1260.60135
[5] Hu, Y.; Liu, Y.; Nualart, D., Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusion, Ann. Appl. Probab., 26, 1147-1207 (2016) · Zbl 1339.60095
[6] Hu, Y.; Nualart, D.; Song, X., A singular stochastic differential equation driven by fractional Brownian motion, Stat. Probab. Lett., 78, 2075-2085 (2008) · Zbl 1283.60089
[7] Huang, S.; Cambanis, S., Stochastic and multiple Wiener integrals for Gaussian processes, Ann. Probab., 6, 585-614 (1978) · Zbl 0387.60064
[8] Kamrani, M., Numerical solution of stochastic fractional differential equations, Numer. Algorithms, 68, 81-93 (2015) · Zbl 1386.65038
[9] Kamrani, M.; Jamshidi, N., Implicit Euler approximation of stochastic evolution equations with fractional Brownian motion, Commun. Nonlinear Sci. Numer. Simul., 44, 1-10 (2017) · Zbl 1462.60098
[10] Kloeden, P. E.; Platen, E., Numerical Solution of Stochastic Differential Equations (1992), Springer: Springer Berlin, Germany · Zbl 0925.65261
[11] Lakshmikantham, V.; Bainov, D.; Simeonov, P., Theory of Impulsive Differential Equations, vol. 6 (1989), World Scientific Press: World Scientific Press Singapore · Zbl 0719.34002
[12] Mishura, Y.; Shevchenko, G., The rate of convergence for Euler approximations of solutions of stochastic differential equations driven by fractional Brownian motion, Stochastics, 80, 489-511 (2008) · Zbl 1154.60046
[13] Neuenkirch, A., Optimal approximation of SDE’s with additive fractional noise, J. Complex., 22, 459-474 (2006) · Zbl 1106.65003
[14] Neuenkirch, A., Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion, Stoch. Process. Appl., 118, 2294-2333 (2008) · Zbl 1154.60338
[15] Nourdin, I.; Neunkirch, A., Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion, J. Theor. Probab., 20, 871-899 (2007) · Zbl 1141.60043
[16] Nualart, D.; Saussereau, B., Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion, Stoch. Process. Appl., 119, 391-409 (2009) · Zbl 1169.60013
[17] Zähle, M., Stochastic differential equations with fractal noise, Math. Nachr., 278, 1097-1106 (2005) · Zbl 1075.60075
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.