×

On local linearization method for stochastic differential equations driven by fractional Brownian motion. (English) Zbl 1469.60175

Summary: We propose a local linearization scheme to approximate the solutions of non-autonomous stochastic differential equations driven by fractional Brownian motion with Hurst parameter \(1/2<H<1\) Toward this end, we approximate the drift and diffusion terms by means of a first-order Taylor expansion. This becomes the original equation into a local fractional linear stochastic differential equation, whose solution can be figured out explicitly. As in the Brownian motion case (i.e., \(H=1/2)\), the rate of convergence, in our case, is twice the one of the Euler scheme. Numerical examples are given to demonstrate the performance of the method.

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G22 Fractional processes, including fractional Brownian motion
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Alòs, E.; León, J. A.; Vives, J., On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility, Finance Stoch, 11, 4, 571-589 (2007) · Zbl 1145.91020 · doi:10.1007/s00780-007-0049-1
[2] Araya, H.; León, J. A.; Torres, S., Numerical scheme for stochastic differential equations driven by fractional Brownian motion with, J. Theor. Probab (2019) · Zbl 1464.60070 · doi:10.1007/s10959-019-00902-3
[3] Biscay, R.; Jimenez, J. C.; Riera, J.; Valdés, P., Local linearization method for the numerical solution of stochastic differential equations, Ann. Inst. Stat. Math, 48, 4, 631-644 (1996) · Zbl 1002.60545 · doi:10.1007/BF00052324
[4] Carbonell, F.; Jimenez, J. C.; Biscay, R.; De la Cruz, H., The local linearization method for numerical integration of random differential equations, Bit Numer. Math., 45, 1, 1-14 (2005) · Zbl 1081.65013 · doi:10.1007/s10543-005-2645-9
[5] Deya, A.; Neuenkirch, A.; Tindel, S., A Milstein-type scheme without Lévy area terms for sdes driven by fractional Brownian motion, Ann. Inst. H Poincaré Probab. Statist., 48, 2, 518-550 (2012) · Zbl 1260.60135 · doi:10.1214/10-AIHP392
[6] Doss, H., Liens entre èquations diffèrentielles stochastiques et ordinaires, Ann Inst. H. Poincarè Sect. B(N.S), 13, 99-125 (1977) · Zbl 0359.60087
[7] Dung, N. T., Fractional geometric mean-reversion processes, J. Math. Anal. Appl., 380, 1, 396-402 (2011) · Zbl 1215.60030 · doi:10.1016/j.jmaa.2011.03.016
[8] Fiel, A.; León, J. A.; Márquez-Carreras, D., Stability for some linear stochastic fractional systems, Commun. Stochastic Anal, 8, 2, 205-225 (2014) · doi:10.31390/cosa.8.2.05
[9] Garzón, J.; Gorostiza, L. G.; León, J. A., Approximations of fractional stochastic differential equations by means of transport processes, COSA, 5, 3, 433-456 (2011) · Zbl 1331.60064 · doi:10.31390/cosa.5.3.01
[10] Grecksch, W.; Anh, V., Approximation of stochastic differential equations with modified fractional Brownian motion, Z Anal. Anwend., 17, 3, 715-727 (1998) · Zbl 0922.60052 · doi:10.4171/ZAA/846
[11] Gubinelli, M., Controlling rough paths, J. Funct. Anal, 216, 1, 86-140 (2004) · Zbl 1058.60037 · doi:10.1016/j.jfa.2004.01.002
[12] Hu, Y.; Liu, Y.; Nualart, D., Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusion, Ann. Appl. Probab, 26, 2, 1147-1207 (2016) · Zbl 1339.60095 · doi:10.1214/15-AAP1114
[13] Jiménez, J.; Pedroso, L.; Carbonell, F.; Hernández, V., Local linearization method for numerical integration of delay differential equations, SIAM J. Numer. Anal, 44, 6, 2584-2609 (2006) · Zbl 1154.34040 · doi:10.1137/040607356
[14] Jimenez, J. C.; Biscay, R., Approximation of continuous time stochastic processes by the local linearization method revisited, Stochastic Anal. Appl, 20, 1, 105-121 (2002) · Zbl 1007.60058 · doi:10.1081/SAP-120002423
[15] Jimenez, J. C.; Mora, C.; Selva, M., A weak local linearization scheme for stochastic differential equations with multiplicative noise, J. Comput. Appl. Math., 313, 202-217 (2017) · Zbl 1353.65007 · doi:10.1016/j.cam.2016.09.013
[16] Kaj, I.; Taqqu, M. S.; V., Vares, In and Out of Equilibrium 2. Progress in Probability, Convergence to fractional Brownian Motion and to the Telecom Process: the Integral Representation Approach, 383-427 (2008), Germany: Birkhäuser, Germany · Zbl 1154.60020
[17] Kloeden, P.; Platen, E., Numerical Solution of Stochastic Differential Equations (1992), New York: Springer, New York · Zbl 0925.65261
[18] Klüppelberg, C.; Kühn, C., Fractional Brownian motion as a weak limit of Poisson shot noise processes with applications to finance, Stochastic Processes Appl., 113, 2, 333-351 (2004) · Zbl 1075.60020 · doi:10.1016/j.spa.2004.03.015
[19] Kubilius, K.; Mishura, Y.; Ralchenko, K., Estimation in Fractional Diffusion Models (2017), Switzerland: Bocconi and Springer Series, Switzerland · Zbl 1388.60006
[20] Kubilius, K.; Skorniakov, V.; Melichov, D., Estimation of parameters of SDE driven by fractional Brownian motion with polynomial drift, J. Stat. Comput. Simul, 86, 10, 1954-1969 (2016) · Zbl 1510.62341 · doi:10.1080/00949655.2015.1095301
[21] León, J. A.; Tindel, S., Malliavin calculus for fractional delay equations, J. Theor. Probab, 25, 3, 854-889 (2012) · Zbl 1259.60069 · doi:10.1007/s10959-011-0349-4
[22] Mishura, Y.; Shevchenko, G., The rate of convergence for Euler approximations of solutions of stochastic differential equations driven by fractional Brownian motion, Stochastics Int. J. Probab. Stochastic Processes, 80, 5, 489-511 (2008) · Zbl 1154.60046 · doi:10.1080/17442500802024892
[23] Nourdin, I.; Neuenkirch, A., Exact rate of convergence of some approximation schemes associated to sdes driven by a fractional Brownian motion, J. Theor. Probab, 20, 4, 871-899 (2007) · Zbl 1141.60043 · doi:10.1007/s10959-007-0083-0
[24] Nualart, D.; Răşcanu, A., Differential equations driven by fractional Brownian motion, Collect. Math, 53, 55-81 (2002) · Zbl 1018.60057
[25] Ozaki, T.; Hannan, E. J.; Krishnaiah, P. R.; Rao, M. M., Handbook of Statistics, 5, Non-linear time series models and dynamical systems, 25-83 (1985), North-Holland: Elsevier, North-Holland · Zbl 0578.62074
[26] Ozaki, T., Statistical identification of storage models with application to stochastic hydrology, J. Am. Water Resources Assoc, 21, 4, 663-675 (1985) · doi:10.1111/j.1752-1688.1985.tb05381.x
[27] Panloup, F.; Tindel, S.; Varvenne, M., A general drift estimation procedure for stochastic differential equations with additive fractional noise, Electron. J. Statist, 14, 1, 1075-1136 (2020) · Zbl 1439.62186 · doi:10.1214/20-EJS1685
[28] Pavlov, B. V.; Rodionova, O. E., The method of local linearization in the numerical solution of stiff systems of ordinary differential equations, USSR Comput. Math. Math. Phys, 27, 3, 30-38 (1987) · Zbl 0656.65076 · doi:10.1016/0041-5553(87)90076-0
[29] Kilbas, A. A.; Samko, S. G.; Marichev, O. I., Fractional Integrals and Derivatives (1987), Amsterdam: Gordon and Breach Science Publishers, Amsterdam · Zbl 0617.26004
[30] Sottinen, T., Fractional Brownian motion, random walks and binary market models, Finance Stochast, 5, 3, 343-355 (2001) · Zbl 0978.91037 · doi:10.1007/PL00013536
[31] Tvedt, J., Freight rates and assets in bulk shipping (1995), Norwegian School of Economics and Bussines Administration: Norwegian School of Economics and Bussines Administration, Bergen, Norway
[32] Zähle, M., Integration with respect to fractal functions and stochastic calculus I, Probab. Theory Relat. Fields, 111, 3, 333-374 (1998) · Zbl 0918.60037 · doi:10.1007/s004400050171
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.