EULER SCHEME FOR FRACTIONAL DELAY STOCHASTIC DIFFERENTIAL EQUATIONS BY ROUGH PATHS TECHNIQUES

doi:10.1007/s10473-019-0308-1

数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (3): 747-763.doi: 10.1007/s10473-019-0308-1

EULER SCHEME FOR FRACTIONAL DELAY STOCHASTIC DIFFERENTIAL EQUATIONS BY ROUGH PATHS TECHNIQUES

Johanna GARZÓN¹, Samy TINDEL², Soledad TORRES³

1. Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá D. C., Colombia;
2. Department of Mathematics, Purdue University, 150 N. University Street, W. Lafayette, IN 47907, USA;
3. Facultad de Ingeniería, CIMFAV Universidad de Valparaíso, Casilla 123-V, 4059 Valparaiso, Chile

收稿日期:2018-03-31 修回日期:2018-08-23 出版日期:2019-06-25 发布日期:2019-06-27
通讯作者: Samy TINDEL E-mail:stindel@purdue.edu
作者简介:Johanna GARZÓN,E-mail:mjgarzonm@unal.edu.co;Soledad TORRES,E-mail:soledad.torres@uv.cl
基金资助:
The first author is supported by MATH-AmSud 18-MATH-07 SaSMoTiDep Project and HERMES project 41305. S. Torres is partially supported by the Project ECOS-CONICYT C15E05, REDES 150038, MATH-AmSud 18-MATH-07 SaSMoTiDep Project and Fondecyt (1171335). S. Tindel is supported by NSF (Grant DMS-1613163).

EULER SCHEME FOR FRACTIONAL DELAY STOCHASTIC DIFFERENTIAL EQUATIONS BY ROUGH PATHS TECHNIQUES

Johanna GARZÓN¹, Samy TINDEL², Soledad TORRES³

1. Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá D. C., Colombia;
2. Department of Mathematics, Purdue University, 150 N. University Street, W. Lafayette, IN 47907, USA;
3. Facultad de Ingeniería, CIMFAV Universidad de Valparaíso, Casilla 123-V, 4059 Valparaiso, Chile

Received:2018-03-31 Revised:2018-08-23 Online:2019-06-25 Published:2019-06-27
Contact: Samy TINDEL E-mail:stindel@purdue.edu
Supported by:
The first author is supported by MATH-AmSud 18-MATH-07 SaSMoTiDep Project and HERMES project 41305. S. Torres is partially supported by the Project ECOS-CONICYT C15E05, REDES 150038, MATH-AmSud 18-MATH-07 SaSMoTiDep Project and Fondecyt (1171335). S. Tindel is supported by NSF (Grant DMS-1613163).

摘要/Abstract

摘要： In this note, we study a discrete time approximation for the solution of a class of delayed stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H∈ (1/2, 1). In order to prove convergence, we use rough paths techniques. Theoretical bounds are established and numerical simulations are displayed.

关键词: Fractional Brownian motion, stochastic differential equations, rough paths, discrete time approximation

Abstract: In this note, we study a discrete time approximation for the solution of a class of delayed stochastic differential equations driven by a fractional Brownian motion with Hurst parameter H∈ (1/2, 1). In order to prove convergence, we use rough paths techniques. Theoretical bounds are established and numerical simulations are displayed.

Key words: Fractional Brownian motion, stochastic differential equations, rough paths, discrete time approximation

中图分类号:

60H15

Johanna GARZÓN, Samy TINDEL, Soledad TORRES. EULER SCHEME FOR FRACTIONAL DELAY STOCHASTIC DIFFERENTIAL EQUATIONS BY ROUGH PATHS TECHNIQUES[J]. 数学物理学报(英文版), 2019, 39(3): 747-763.

Johanna GARZÓN, Samy TINDEL, Soledad TORRES. EULER SCHEME FOR FRACTIONAL DELAY STOCHASTIC DIFFERENTIAL EQUATIONS BY ROUGH PATHS TECHNIQUES[J]. Acta mathematica scientia,Series B, 2019, 39(3): 747-763.

参考文献

[1] Bardina X, Bascompte D, Rovira C, Tindel S. An analysis of a stochastic model for bacteriophage systems. Math Biosci, 2013, 241(1):99-108
[2] Calsina A, Palmada J-M, Ripoll J. Optimal latent period in a bacteriophage population model structured by infection-age. Math Models and Methods in Appl Sc, 2011, 21(4):1-26
[3] Carletti M. Mean-square stability of a stochastic model for bacteriophage infection with time delays. Math Biosci, 2007, 210(2):395-414
[4] Cheridito P. Arbitrage in fractional Brownian motion models. Finance Stoch, 2003, 7(4):533-553
[5] Dasgupta A, Kallianpur G. Arbitrage opportunities for a class of Gladyshev processes. Appl Math Optim, 200041(3):377-385
[6] Denk G, Meintrup D, Schäffler S. Transient noise simulation:Modeling and simulation of 1/fnoise//Antreich K, et al. Modeling, simulation, and optimization of integrated circuits. Birkhäuser. Int Ser Numer Math, 2003, 146:251-267
[7] Deya A, Neuenkirch A, Tindel, S. A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion. Ann Inst Henri Poincaré Probab Stat, 2012, 48(2):518-550
[8] Fernique X M. Regularité des trajectoires de fonctions aléatoires gaussiennes//Hennequin P L. École d'Été de Saint-Flour IV, Lecture Notes in Mathematics. Berlin:Springer, 1974, 480:2-95
[9] Ferrante M, Rovira C. Convergence of delay differential equations driven by fractional Brownian motion. Bernoulli, 2006, 12(1):85-100
[10] Ferrante M, Rovira C. Convergence of delay differential equations driven by fractional Brownian motion. J Evol Equ, 2010, 10(4):761-783
[11] Friz P, Victoir N. Multidimensional dimensional processes seen as rough paths. Cambridge:Cambridge University Press, 2010
[12] Friz P K, Riedel S. Convergence rates for the full Gaussian rough paths. Ann Inst Henri Poincaré Probab Stat, 2014, 50(1):154-194
[13] Guasoni P. No arbitrage under transaction costs, with fractional Brownian motion and beyond. Mathematical Finance, 2006, 16(3):569-582
[14] Gubinelli M. Controlling rough paths. J Funct Anal, 2004, 216(1):86-140
[15] Hu Y, Oksendal B. Fractional white noise calculus and applications to finance. Infinite dimensional analysis, quantum probability and related topics, 2003, 6(1):1-32
[16] Hu Y, Oksendal B, Sulem A. Optimal consumption and portfolio in a Black-Scholes market driven by fractional Brownian motion. Infinite dimensional analysis, quantum probability and related topics, 2003, 6(4):519-536
[17] Kou S, Sunney-Xie X. Generalized Langevin equation with fractional Gaussian noise:subdiffusion within a single protein molecule. Phys Rev Lett, 2004, 93(18)
[18] Küchle U, Platen E. Strong Discrete Time Approximation of Stochastic Differential Equations with Time Delay. Math Comput Simulation, 2000, 54(1/3):189-205
[19] Léon J, Tindel S. Malliavin calculus for fractional delay equations. J Theoret Probab, 2011, 25(3):854-889
[20] Liu Y, Tindel S. First-order Euler scheme for SDEs driven by fractional Brownian motions:the rough case. Arxiv, 2017
[21] Mishura Y, Shevchenko G. The rate of convergence for Euler approximations of solutions of stochastic differential equations driven by fractional Brownian motion. Stochastics, 2008, 80(5):489-511
[22] Mohammed S. Stochastic functional differential equations. Research Notes in Mathematics 99. Boston:Pitman Advanced Publishing Program, 1984
[23] Neuenkirch A, Nourdin I. Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion. J Theoret Probab, 2007, 20(4):871-899
[24] Neuenkirch A, Nourdin I, Tindel S. Delay equations driven by rough paths. Electron J Probab, 2008,13(67):2031-2068
[25] Nualart D, Răşcanu A. Differential equations driven by fractional Brownian motion. Collect Mat, 2002, 53(1):55-81
[26] Smith H. Models of virulent phage growth with application to phage therapy. SIAM J Appl Math, 2008, 68(6):1717-1737
[27] Szymanski J, Weiss M. Elucidating the origin of anomalous diffusion in crowded fluids. Phys Rev Lett, 2009, 103(3)
[28] Tejedor V, Bénichou O, Voituriez R, et al. Quantitative Analysis of Single Particle Trajectories:Mean Maximal Excursion Method. Biophysical J, 2010, 98(7):1364-1372
[29] Willinger W, Taqqu M S, Teverovsky V. Stock market prices and long-range dependence. Finance Stoch, 1999, 3(1):1-13
[30] Young L. An inequality of the Hölder type, connected with Stieltjes integration. Acta Math, 1936, 67(1):251-282
[31] Zähle M. Integration with respect to fractal functions and stochastic calculus I. Prob Theory Relat Fields, 1998, 111(3):333-374

EULER SCHEME FOR FRACTIONAL DELAY STOCHASTIC DIFFERENTIAL EQUATIONS BY ROUGH PATHS TECHNIQUES

EULER SCHEME FOR FRACTIONAL DELAY STOCHASTIC DIFFERENTIAL EQUATIONS BY ROUGH PATHS TECHNIQUES

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