• Previous Article
    Interplay of time-delay and velocity alignment in the Cucker-Smale model on a general digraph
  • DCDS-B Home
  • This Issue
  • Next Article
    Maintaining gene expression levels by positive feedback in burst size in the presence of infinitesimal delay
October  2019, 24(10): 5553-5567. doi: 10.3934/dcdsb.2019071

Existence and approximation of strong solutions of SDEs with fractional diffusion coefficients

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

1 Corresponding author

Received  July 2018 Revised  October 2018 Published  April 2019

Fund Project: The research was supported in part by the National Natural Science Foundations of China (Grant Nos. 61473125 and 11761130072) and the Royal Society-Newton Advanced Fellowship (REF NA160317).

In stochastic financial and biological models, the diffusion coefficients often involve the terms $ \sqrt{|x|} $ and $ \sqrt{|x(1-x)|} $, or more general $ |x|^{r} $ and $ |x(1-x)|^r $ for $ r $ $ \in $ $ (0, 1) $. These coefficients do not satisfy the local Lipschitz condition, which implies that the existence and uniqueness of the solution cannot be obtained by the standard conditions. This paper establishes the existence and uniqueness of the strong solution and the strong convergence of the Euler-Maruyama approximations under certain conditions for systems of stochastic differential equations for which one component has such a diffusion coefficient with $ r $ $ \in $ $ [1/2, 1) $.

Citation: Hao Yang, Fuke Wu, Peter E. Kloeden. Existence and approximation of strong solutions of SDEs with fractional diffusion coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5553-5567. doi: 10.3934/dcdsb.2019071
References:
[1]

L. Arnold, Stochastic Differential Equations: Theory and Applications, New York-London-Sydney, 1974.  Google Scholar

[2]

J. C. Cox and J. E. Ingersoll Jr. and S. A. Ross, A theory of the term structure of interest rates, Econometrica. 53 (1985), 385–407. doi: 10.2307/1911242.  Google Scholar

[3]

S. DereichA. Neuenkirch and L. Szpruch., An Euler-type method for the strong approximation of the Cox-Ingersoll-Ross process, Proc. Royal Society, series A, 468 (2012), 1105-1115.  doi: 10.1098/rspa.2011.0505.  Google Scholar

[4]

S. Fang and T. Zhang, A study of a class of stochastic differential equations with non-Lipschitzian coefficients, Probab. Theory Related Fields, 132 (2005), 356-390.  doi: 10.1007/s00440-004-0398-z.  Google Scholar

[5]

J. FontbonaH. RaminezV. Riquelme and F. Silva, Stochastic modeling and control of bioreactors, IFACPapersOnLine, 50 (2017), 12611-12616.   Google Scholar

[6]

D. T. Gillespie, The chemical Langevin and Fokker-Planck Equations for the reversible isomerization reaction, J. Phys. Chem. A, 106 (2002), 5063-5071.  doi: 10.1021/jp0128832.  Google Scholar

[7]

I. Gyöngy, A note on Euler's approximation, Potential Anal., 8 (1998), 205-216.  doi: 10.1023/A:1008605221617.  Google Scholar

[8]

I. Gyöngy and M. Rásonyi, A note on Euler approximations for SDEs with Hölder continuous diffusion coefficients, Stochastic Processes Appl., 121 (2011), 2189-2200.  doi: 10.1016/j.spa.2011.06.008.  Google Scholar

[9]

S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Finan. Stud., 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327.  Google Scholar

[10]

D. J. Higham, Modeling and simulating chemical reactions, SIAM Rev., 50 (2008), 347-368.  doi: 10.1137/060666457.  Google Scholar

[11]

D. J. HighamX. Mao and A. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40 (2003), 1041-1063.  doi: 10.1137/S0036142901389530.  Google Scholar

[12]

M. Hutzenthaler and A. Jentzen, Convergence of the stochastic Euler scheme for locally Lipschitz coefficients, Found. Comput. Math., 11 (2011), 657-706.  doi: 10.1007/s10208-011-9101-9.  Google Scholar

[13]

M. HutzenthalerA. Jentzen and P. E. Kloeden, Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467 (2011), 1563-1576.  doi: 10.1098/rspa.2010.0348.  Google Scholar

[14]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Process, North-Holland Publishing Company, 1981.  Google Scholar

[15]

K. Itô and M. Nisio, On stationary solutions of a stochastic differential equation, J. Math., 4 (1964), 1-75.  doi: 10.1215/kjm/1250524705.  Google Scholar

[16]

P. E. Kloeden and A. Neuenkirch, Convergence of numerical methods for stochastic differential equations in mathematical finance, in, Recent Developments in Computational Finance: Foundations, Algorithms and Applications, T. Gerstner and P.E. Kloeden (Editors), Interdisciplinary Mathematical Sciences Series, Vol. 14, World Scientific Publishing Co. Inc,, Singapore, 2013, 49–80. doi: 10.1142/9789814436434_0002.  Google Scholar

[17] V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, New York, 1986.   Google Scholar
[18]

K. S. Kumar, A class of degenerate stochastic differential equations with non-Lipschitz coefficients, Proc. Indian Acad. Sci.(Math Sci.), 123 (2013), 443-454.  doi: 10.1007/s12044-013-0141-8.  Google Scholar

[19]

T. Kurtz, Approximation of Population Processes, volume 36, CBMS-NSF Regional Conf. Series, SIAM, Philadelphia, 1981.  Google Scholar

[20] H. J. Kushner, Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory, MIT Press, Cambridge, MA, 1984.   Google Scholar
[21]

A. L. Lewis, Option Valuation Under Stochastic Volatility II, With Mathematica code. Finance Press, Newport Beach, CA, 2016.  Google Scholar

[22]

X. Mao, A. Truman and C. Yuan, Euler–Maruyama approximations in mean-reverting stochastic volatility model under regime-switching, J. Appl. Math. Stoch., 2006 (2006), Art. ID 80967, 20 pp. doi: 10.1155/JAMSA/2006/80967.  Google Scholar

[23]

X. Mao, Stochastic Differential Equations and Applications, 2nd Ed., Horwood, Chichester, UK, 2008. doi: 10.1533/9780857099402.  Google Scholar

[24]

A. N. Shiryaev, Probability, Springer, New York, 1996. doi: 10.1007/978-1-4757-2539-1.  Google Scholar

[25]

T. Yamada and S. Watanabe, On the uniqueness of stochastic differential equations, J. Math Kyoto Univ., 11 (1971), 155-167.  doi: 10.1215/kjm/1250523691.  Google Scholar

[26]

J. Yong and X. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

show all references

References:
[1]

L. Arnold, Stochastic Differential Equations: Theory and Applications, New York-London-Sydney, 1974.  Google Scholar

[2]

J. C. Cox and J. E. Ingersoll Jr. and S. A. Ross, A theory of the term structure of interest rates, Econometrica. 53 (1985), 385–407. doi: 10.2307/1911242.  Google Scholar

[3]

S. DereichA. Neuenkirch and L. Szpruch., An Euler-type method for the strong approximation of the Cox-Ingersoll-Ross process, Proc. Royal Society, series A, 468 (2012), 1105-1115.  doi: 10.1098/rspa.2011.0505.  Google Scholar

[4]

S. Fang and T. Zhang, A study of a class of stochastic differential equations with non-Lipschitzian coefficients, Probab. Theory Related Fields, 132 (2005), 356-390.  doi: 10.1007/s00440-004-0398-z.  Google Scholar

[5]

J. FontbonaH. RaminezV. Riquelme and F. Silva, Stochastic modeling and control of bioreactors, IFACPapersOnLine, 50 (2017), 12611-12616.   Google Scholar

[6]

D. T. Gillespie, The chemical Langevin and Fokker-Planck Equations for the reversible isomerization reaction, J. Phys. Chem. A, 106 (2002), 5063-5071.  doi: 10.1021/jp0128832.  Google Scholar

[7]

I. Gyöngy, A note on Euler's approximation, Potential Anal., 8 (1998), 205-216.  doi: 10.1023/A:1008605221617.  Google Scholar

[8]

I. Gyöngy and M. Rásonyi, A note on Euler approximations for SDEs with Hölder continuous diffusion coefficients, Stochastic Processes Appl., 121 (2011), 2189-2200.  doi: 10.1016/j.spa.2011.06.008.  Google Scholar

[9]

S. L. Heston, A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Finan. Stud., 6 (1993), 327-343.  doi: 10.1093/rfs/6.2.327.  Google Scholar

[10]

D. J. Higham, Modeling and simulating chemical reactions, SIAM Rev., 50 (2008), 347-368.  doi: 10.1137/060666457.  Google Scholar

[11]

D. J. HighamX. Mao and A. Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM J. Numer. Anal., 40 (2003), 1041-1063.  doi: 10.1137/S0036142901389530.  Google Scholar

[12]

M. Hutzenthaler and A. Jentzen, Convergence of the stochastic Euler scheme for locally Lipschitz coefficients, Found. Comput. Math., 11 (2011), 657-706.  doi: 10.1007/s10208-011-9101-9.  Google Scholar

[13]

M. HutzenthalerA. Jentzen and P. E. Kloeden, Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 467 (2011), 1563-1576.  doi: 10.1098/rspa.2010.0348.  Google Scholar

[14]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Process, North-Holland Publishing Company, 1981.  Google Scholar

[15]

K. Itô and M. Nisio, On stationary solutions of a stochastic differential equation, J. Math., 4 (1964), 1-75.  doi: 10.1215/kjm/1250524705.  Google Scholar

[16]

P. E. Kloeden and A. Neuenkirch, Convergence of numerical methods for stochastic differential equations in mathematical finance, in, Recent Developments in Computational Finance: Foundations, Algorithms and Applications, T. Gerstner and P.E. Kloeden (Editors), Interdisciplinary Mathematical Sciences Series, Vol. 14, World Scientific Publishing Co. Inc,, Singapore, 2013, 49–80. doi: 10.1142/9789814436434_0002.  Google Scholar

[17] V. B. Kolmanovskii and V. R. Nosov, Stability of Functional Differential Equations, Academic Press, New York, 1986.   Google Scholar
[18]

K. S. Kumar, A class of degenerate stochastic differential equations with non-Lipschitz coefficients, Proc. Indian Acad. Sci.(Math Sci.), 123 (2013), 443-454.  doi: 10.1007/s12044-013-0141-8.  Google Scholar

[19]

T. Kurtz, Approximation of Population Processes, volume 36, CBMS-NSF Regional Conf. Series, SIAM, Philadelphia, 1981.  Google Scholar

[20] H. J. Kushner, Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory, MIT Press, Cambridge, MA, 1984.   Google Scholar
[21]

A. L. Lewis, Option Valuation Under Stochastic Volatility II, With Mathematica code. Finance Press, Newport Beach, CA, 2016.  Google Scholar

[22]

X. Mao, A. Truman and C. Yuan, Euler–Maruyama approximations in mean-reverting stochastic volatility model under regime-switching, J. Appl. Math. Stoch., 2006 (2006), Art. ID 80967, 20 pp. doi: 10.1155/JAMSA/2006/80967.  Google Scholar

[23]

X. Mao, Stochastic Differential Equations and Applications, 2nd Ed., Horwood, Chichester, UK, 2008. doi: 10.1533/9780857099402.  Google Scholar

[24]

A. N. Shiryaev, Probability, Springer, New York, 1996. doi: 10.1007/978-1-4757-2539-1.  Google Scholar

[25]

T. Yamada and S. Watanabe, On the uniqueness of stochastic differential equations, J. Math Kyoto Univ., 11 (1971), 155-167.  doi: 10.1215/kjm/1250523691.  Google Scholar

[26]

J. Yong and X. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer, New York, 1999. doi: 10.1007/978-1-4612-1466-3.  Google Scholar

[1]

Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192

[2]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203

[3]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

[4]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088

[5]

Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021023

[6]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[7]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521

[8]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175

[9]

Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223

[10]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

[11]

Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113

[12]

Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115

[13]

Nabahats Dib-Baghdadli, Rabah Labbas, Tewfik Mahdjoub, Ahmed Medeghri. On some reaction-diffusion equations generated by non-domiciliated triatominae, vectors of Chagas disease. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021004

[14]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027

[15]

Xiaohu Wang, Dingshi Li, Jun Shen. Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2829-2855. doi: 10.3934/dcdsb.2020207

[16]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035

[17]

Zhihua Zhang, Naoki Saito. PHLST with adaptive tiling and its application to antarctic remote sensing image approximation. Inverse Problems & Imaging, 2014, 8 (1) : 321-337. doi: 10.3934/ipi.2014.8.321

[18]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[19]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810

[20]

Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (368)
  • HTML views (366)
  • Cited by (1)

Other articles
by authors

[Back to Top]