July  2019, 24(7): 3299-3318. doi: 10.3934/dcdsb.2018321

Stochastic invariance for neutral functional differential equation with non-lipschitz coefficients

Department of Probability and Statistics, School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, MO 510006, China

* Corresponding author: Jiaowan Luo

Received  February 2018 Revised  July 2018 Published  July 2019 Early access  January 2019

Fund Project: The second author is supported by NNSF grant 11271093.

In this paper, by the use of martingale property and spectral decomposition theory, we investigate the stochastic invariance for neutral stochastic functional differential equations (NSFDEs) and provide necessary and sufficient conditions for the invariance of closed sets of $ R^{d} $ with non-Lipschitz coefficients. A pathwise asymptotic estimate example is given to illustrate the feasibility and effectiveness of obtained result.

Citation: Chunhong Li, Jiaowan Luo. Stochastic invariance for neutral functional differential equation with non-lipschitz coefficients. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3299-3318. doi: 10.3934/dcdsb.2018321
References:
[1]

H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces, in MPS-SIAM Series on Optimization: SIAM and MPS, Philadelphia, 1nd edition, the Society for Industrial and Applied Mathematics and the Mathematical Programming Society, 2006.

[2]

J. C. A. Barata and M. S. Hussein, The Moore-Penrose Pseudoinverse: a tutorial review of the theory, Brazilian Journal of Physics, 42 (2012), 146-165. 

[3] F. Baudouin, An Introduction to the Geometry of Stochastic Flows, 1 edition, Imperial College Press, London, 2004.  doi: 10.1142/9781860947261.
[4] D. P. Bertsekas and S. E. Shreve, Stochastic Optimal Control: The Discrete Time Case, Academic Press, New York, 1978. 
[5]

R. BuckdahnM. Quincampoix and C. Rainer, Another proof for the equivalence between invariance of closed sets with respect to stochastic and deterministic systems, Bulletin Des Sciences Mathmatiques, 134 (2010), 207-214.  doi: 10.1016/j.bulsci.2007.11.003.

[6]

H. Cartan, Calcul différentiel, Hermann Paris, 15 (1967), 287-290. 

[7]

B. P. CheriditoH. M. Soner and N. Touzi, Small time path behavior of double stochastic integrals and applications to stochastic contral, The Annals of Applied Probability, 15 (2005), 2475-2495.  doi: 10.1214/105051605000000557.

[8]

I. Chueshov and M. Scheutzow, Invariance and monotonicity for stochastic delay differential equations, Discrete and Continuous Dynamical Systems - Series B, 18 (2012), 1533-1554.  doi: 10.3934/dcdsb.2013.18.1533.

[9]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, 1st edition, John Wiley and Sons, New York, 1986. doi: 10.1002/9780470316658.

[10]

E. Faou and T. Lelièvre, Conservative stochastic differential equations: Mathematical and numerical analysis, Mathematics of Computation, 78 (2009), 2047-2074.  doi: 10.1090/S0025-5718-09-02220-0.

[11]

E. Forgoston, L. Billings and P. Yecko, Set-based corral control in stochastic dynamical systems: making almost invariant sets more invariant, Chaos, 21 (2011), 013116, 11pp. doi: 10.1063/1.3539836.

[12]

A. Friedman, Stochastic Differential Equations and Applications, Dover Publications, Inc., Mineola, NY, 2006.

[13]

E. A. Jaber, B. Bouchard and C. Illand, Stochastic invariance of closed sets with non-Lipschitz coefficients, Electron. Commun. Probab., 22 (2017), Paper No. 53, 15 pp, arXiv: 1607.08717, 2016. doi: 10.1214/17-ECP88.

[14] F. C. Klebaner, Introduction to Stochastic Calculus With Applications, 2 edition, Imperial College Press, London, 2005.  doi: 10.1142/p386.
[15]

Y. Li and B. Liu, Existence of solution of nonlinear neutral stochastic diferential inclusions with infinite delay, Stochastic Analysis and Applications, 25 (2007), 397-415.  doi: 10.1080/07362990601139610.

[16]

D. S. Li and D. Y. Xu, Attracting and quasi-invariant sets of stochastic neutral partial functional differential equations, Acta Mathematica Scientia, 33 (2013), 578-588.  doi: 10.1016/S0252-9602(13)60021-1.

[17]

Z. Li, Global attractiveness and quasi-invariant sets of impulsive neutral stochastic functional differential equations driven by FBM, Neurocomputing, 177 (2016), 620-627. 

[18]

R. Lorenz, Weak approximation of stochastic delay differential equations with bounded memory by discrete time series, Solid State Communications, 65 (1988), 55-58. 

[19]

J. R. Magnus and H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics, John Wiley & Sons, Ltd., Chichester, 1988.

[20]

X. R. Mao, Stochastic Differential Equations and Application, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.

[21]

S. E. A. Mohammed, Stochastic Functional Differential Equations, 1st edition, Pitman Advanced Publishing Program, 1984.

[22]

G. D. Prato and H. Frankowska, Invariance of stochastic control systems with deterministic arguments, Journal of Differential Equations, 200 (2004), 18-52.  doi: 10.1016/j.jde.2004.01.007.

[23]

R. T. Rockafellar and R. J. B. Wets, Variational Analysis, Springer-Verlag, Berlin, 1998.

[24]

T. Stefan, Invariance of closed convex cones for stochastic partial differential equations, Journal of Mathematical Analysis and Applications, 451 (2017), 1077-1122.  doi: 10.1016/j.jmaa.2017.02.044.

[25]

G. Thoma Kurtz, Lectures on Stochastic Analysis, University of Wisconsin - Madison Madison, 2007.

[26]

L. WangZ. Wang and J. Wu, Positively invariant sets, monotone solutions, and contracting rectangles in neutral functional-differential equations, Functional Differential Equations, 7 (2000), 385-397. 

[27]

J. Zabczyk, Stochastic invariance and consistency of financial models, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 11 (2000), 67-80. 

show all references

References:
[1]

H. Attouch, G. Buttazzo and G. Michaille, Variational Analysis in Sobolev and BV Spaces, in MPS-SIAM Series on Optimization: SIAM and MPS, Philadelphia, 1nd edition, the Society for Industrial and Applied Mathematics and the Mathematical Programming Society, 2006.

[2]

J. C. A. Barata and M. S. Hussein, The Moore-Penrose Pseudoinverse: a tutorial review of the theory, Brazilian Journal of Physics, 42 (2012), 146-165. 

[3] F. Baudouin, An Introduction to the Geometry of Stochastic Flows, 1 edition, Imperial College Press, London, 2004.  doi: 10.1142/9781860947261.
[4] D. P. Bertsekas and S. E. Shreve, Stochastic Optimal Control: The Discrete Time Case, Academic Press, New York, 1978. 
[5]

R. BuckdahnM. Quincampoix and C. Rainer, Another proof for the equivalence between invariance of closed sets with respect to stochastic and deterministic systems, Bulletin Des Sciences Mathmatiques, 134 (2010), 207-214.  doi: 10.1016/j.bulsci.2007.11.003.

[6]

H. Cartan, Calcul différentiel, Hermann Paris, 15 (1967), 287-290. 

[7]

B. P. CheriditoH. M. Soner and N. Touzi, Small time path behavior of double stochastic integrals and applications to stochastic contral, The Annals of Applied Probability, 15 (2005), 2475-2495.  doi: 10.1214/105051605000000557.

[8]

I. Chueshov and M. Scheutzow, Invariance and monotonicity for stochastic delay differential equations, Discrete and Continuous Dynamical Systems - Series B, 18 (2012), 1533-1554.  doi: 10.3934/dcdsb.2013.18.1533.

[9]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, 1st edition, John Wiley and Sons, New York, 1986. doi: 10.1002/9780470316658.

[10]

E. Faou and T. Lelièvre, Conservative stochastic differential equations: Mathematical and numerical analysis, Mathematics of Computation, 78 (2009), 2047-2074.  doi: 10.1090/S0025-5718-09-02220-0.

[11]

E. Forgoston, L. Billings and P. Yecko, Set-based corral control in stochastic dynamical systems: making almost invariant sets more invariant, Chaos, 21 (2011), 013116, 11pp. doi: 10.1063/1.3539836.

[12]

A. Friedman, Stochastic Differential Equations and Applications, Dover Publications, Inc., Mineola, NY, 2006.

[13]

E. A. Jaber, B. Bouchard and C. Illand, Stochastic invariance of closed sets with non-Lipschitz coefficients, Electron. Commun. Probab., 22 (2017), Paper No. 53, 15 pp, arXiv: 1607.08717, 2016. doi: 10.1214/17-ECP88.

[14] F. C. Klebaner, Introduction to Stochastic Calculus With Applications, 2 edition, Imperial College Press, London, 2005.  doi: 10.1142/p386.
[15]

Y. Li and B. Liu, Existence of solution of nonlinear neutral stochastic diferential inclusions with infinite delay, Stochastic Analysis and Applications, 25 (2007), 397-415.  doi: 10.1080/07362990601139610.

[16]

D. S. Li and D. Y. Xu, Attracting and quasi-invariant sets of stochastic neutral partial functional differential equations, Acta Mathematica Scientia, 33 (2013), 578-588.  doi: 10.1016/S0252-9602(13)60021-1.

[17]

Z. Li, Global attractiveness and quasi-invariant sets of impulsive neutral stochastic functional differential equations driven by FBM, Neurocomputing, 177 (2016), 620-627. 

[18]

R. Lorenz, Weak approximation of stochastic delay differential equations with bounded memory by discrete time series, Solid State Communications, 65 (1988), 55-58. 

[19]

J. R. Magnus and H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics, John Wiley & Sons, Ltd., Chichester, 1988.

[20]

X. R. Mao, Stochastic Differential Equations and Application, Second edition. Horwood Publishing Limited, Chichester, 2008. doi: 10.1533/9780857099402.

[21]

S. E. A. Mohammed, Stochastic Functional Differential Equations, 1st edition, Pitman Advanced Publishing Program, 1984.

[22]

G. D. Prato and H. Frankowska, Invariance of stochastic control systems with deterministic arguments, Journal of Differential Equations, 200 (2004), 18-52.  doi: 10.1016/j.jde.2004.01.007.

[23]

R. T. Rockafellar and R. J. B. Wets, Variational Analysis, Springer-Verlag, Berlin, 1998.

[24]

T. Stefan, Invariance of closed convex cones for stochastic partial differential equations, Journal of Mathematical Analysis and Applications, 451 (2017), 1077-1122.  doi: 10.1016/j.jmaa.2017.02.044.

[25]

G. Thoma Kurtz, Lectures on Stochastic Analysis, University of Wisconsin - Madison Madison, 2007.

[26]

L. WangZ. Wang and J. Wu, Positively invariant sets, monotone solutions, and contracting rectangles in neutral functional-differential equations, Functional Differential Equations, 7 (2000), 385-397. 

[27]

J. Zabczyk, Stochastic invariance and consistency of financial models, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 11 (2000), 67-80. 

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