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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  October 2019 Early access  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 and 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.

[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.

[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.

[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.

[5]

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

[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.

[7]

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

[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.

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[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.

[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.

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

[19]

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

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

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

[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.

[23]

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

[24]

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

[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.

[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.

show all references

References:
[1]

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

[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.

[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.

[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.

[5]

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

[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.

[7]

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

[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.

[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.

[10]

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

[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.

[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.

[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.

[14]

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

[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.

[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.

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

[19]

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

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

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

[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.

[23]

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

[24]

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

[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.

[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.

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