September  2020, 40(9): 5173-5188. doi: 10.3934/dcds.2020224

Limit theorems for additive functionals of path-dependent SDEs

1. 

Center for Applied Mathematics, Tianjin University, Tianjin 300072, China

2. 

Department of Mathematics, Swansea University, Bay Campus, SA1 8EN, UK

* Corresponding author: Chenggui Yuan

Received  June 2019 Published  June 2020

Fund Project: This work is supported in part by NNSFC (11771326, 11431014, 11831014)

By using limit theorems of uniform mixing Markov processes and martingale difference sequences, the strong law of large numbers, central limit theorem, and the law of iterated logarithm are established for additive functionals of path-dependent stochastic differential equations.

Citation: Jianhai Bao, Feng-Yu Wang, Chenggui Yuan. Limit theorems for additive functionals of path-dependent SDEs. Discrete & Continuous Dynamical Systems - A, 2020, 40 (9) : 5173-5188. doi: 10.3934/dcds.2020224
References:
[1]

J. H. BaoF.-Y. Wang and C. G. Yuan, Hypercontractivity for functional stochastic differential equations, Stochastic Process. Appl., 125 (2015), 3636-3656.  doi: 10.1016/j.spa.2015.04.001.  Google Scholar

[2]

J. Bao, F.-Y. Wang and C. Yuan, Ergodicity for neutral type SDEs with infinite length of memory, Math. Nach., arXiv: 1805.03431. Google Scholar

[3]

W. BoltA. A. Majewski and T. Szarek, An invariance principle for the law of the iterated logarithm for some Markov chains, Studia Math., 212 (2012), 41-53.  doi: 10.4064/sm212-1-3.  Google Scholar

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O. Butkovsky and M. Scheutzow, Invariant measures for stochastic functional differential equations, Electron. J. Probab., 22 (2017), Paper No. 98, 23 pp. doi: 10.1214/17-EJP122.  Google Scholar

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O. Butkovsky, Subgeometric rates of convergence of Markov processes in the Wasserstein metric, Ann. Appl. Probab., 24 (2014), 526-552.  doi: 10.1214/13-AAP922.  Google Scholar

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P. CattiauxD. Chafai and A. Guillin, Central limit theorems for additive functionals of ergodic Markov diffusions processes, ALEA Lat. Am. J. Probab. Math. Stat., 9 (2012), 337-382.   Google Scholar

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X. Chen, The law of the iterated logarithm for functionals of Harris recurrent Markov chains: Self-normalization, J. Theoret. Probab., 12 (1999), 421-445.  doi: 10.1023/A:1021630228280.  Google Scholar

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Y. Derriennic and M. Lin, The central limit theorem for Markov chains started at a point, Probab. Theory Related Fields, 125 (2003), 73-76.  doi: 10.1007/s004400200215.  Google Scholar

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W. Doeblin, Sur deux problemes de M. Kolmogoroff concernant les chanes d énombrables, Bull. Soc. Math. France, 66 (1938), 210-220.   Google Scholar

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G. Dos Reis, W. Salkeld and J. Tugaut, Freidlin-Wentzell LDP in path space for McKean-Vlasov equations and the Functional Iterated Logarithm Law, arXiv: 1708.04961v3. Google Scholar

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S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2003.  Google Scholar

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F. Q. Gao, Long time asymptotics of unbounded additive functionals of Markov processes, Electron. J. Probab., 22 (2017), Paper No. 94, 21 pp. doi: 10.1214/17-EJP104.  Google Scholar

[13]

M. Hairer and J. C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. Math. (2), 164 (2006), 993-1032. doi: 10.4007/annals.2006.164.993.  Google Scholar

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M. HairerJ. C. Mattingly and M. Scheutzow, Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations, Probab. Theory Related Fields, 149 (2011), 223-259.  doi: 10.1007/s00440-009-0250-6.  Google Scholar

[15] P. Hall and C. C. Heyde, Martingale Limit Theory and its Applications, Academic Press, New York-London, 1980.   Google Scholar
[16]

C. C. Heyde and D. J. Scott, Invariance principles for the law of the iterated logarithm for martingales and processes with stationary increments, Ann. Probab., 1 (1973), 428-436.  doi: 10.1214/aop/1176996937.  Google Scholar

[17]

I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff Publishing, Groningen, 1971.  Google Scholar

[18]

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

[19]

J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Grundlehren der Mathematischen Wissenschaften, 288. Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-662-02514-7.  Google Scholar

[20]

C. Kipnis and S. R. S. Varadhan, Central limit theorem for additive functionals of reversible Markov process and applications to simple exclusions, Comm. Math. Phys., 104 (1986), 1-19.  doi: 10.1007/BF01210789.  Google Scholar

[21]

T. Komorowski and A. Walczuk, Central limit theorem for Markov processes with spectral gap in the Wasserstein metric, Stochastic Process. Appl., 122 (2012), 2155-2184.  doi: 10.1016/j.spa.2012.03.006.  Google Scholar

[22]

A. Kulik, Ergodic Behavior of Markov Processes. With Applications to Limit Theorems, De Gruyter Studies in Mathematics, 67, De Gruyter, Berlin, 2018.  Google Scholar

[23]

S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Springer-Verlag London, Ltd., London, 1993. doi: 10.1007/978-1-4471-3267-7.  Google Scholar

[24]

A. Shirikyan, Law of large numbers and central limit theorem for randomly forced PDE's, Probab. Theory Related Fields, 134 (2006), 215-247.  doi: 10.1007/s00440-005-0427-6.  Google Scholar

[25]

V. Strassen, An invariance principle for the law of the iterated logarithm, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 3 (1964), 211-226.  doi: 10.1007/BF00534910.  Google Scholar

[26]

M.-K. von Renesse and M. Scheutzow, Existence and uniqueness of solutions of stochastic functional differential equations, Random Oper. Stoch. Equ., 18 (2010), 267-284.  doi: 10.1515/ROSE.2010.015.  Google Scholar

[27]

A. Walczuk, Central limit theorem for an additive functional of a Markov process, stable in the Wesserstein metric, Ann. Univ. Mariae Curie-Sklodowska Sect. A, 62 (2008), 149-159.  doi: 10.2478/v10062-008-0016-0.  Google Scholar

[28]

L. M. Wu, Forward-backward martingale decomposition and compactness results for additive functionals of stationary ergodic Markov processes, Ann. Inst. H. Poincaré Probab. Statist., 35 (1999), 121-141.  doi: 10.1016/S0246-0203(99)80008-9.  Google Scholar

show all references

References:
[1]

J. H. BaoF.-Y. Wang and C. G. Yuan, Hypercontractivity for functional stochastic differential equations, Stochastic Process. Appl., 125 (2015), 3636-3656.  doi: 10.1016/j.spa.2015.04.001.  Google Scholar

[2]

J. Bao, F.-Y. Wang and C. Yuan, Ergodicity for neutral type SDEs with infinite length of memory, Math. Nach., arXiv: 1805.03431. Google Scholar

[3]

W. BoltA. A. Majewski and T. Szarek, An invariance principle for the law of the iterated logarithm for some Markov chains, Studia Math., 212 (2012), 41-53.  doi: 10.4064/sm212-1-3.  Google Scholar

[4]

O. Butkovsky and M. Scheutzow, Invariant measures for stochastic functional differential equations, Electron. J. Probab., 22 (2017), Paper No. 98, 23 pp. doi: 10.1214/17-EJP122.  Google Scholar

[5]

O. Butkovsky, Subgeometric rates of convergence of Markov processes in the Wasserstein metric, Ann. Appl. Probab., 24 (2014), 526-552.  doi: 10.1214/13-AAP922.  Google Scholar

[6]

P. CattiauxD. Chafai and A. Guillin, Central limit theorems for additive functionals of ergodic Markov diffusions processes, ALEA Lat. Am. J. Probab. Math. Stat., 9 (2012), 337-382.   Google Scholar

[7]

X. Chen, The law of the iterated logarithm for functionals of Harris recurrent Markov chains: Self-normalization, J. Theoret. Probab., 12 (1999), 421-445.  doi: 10.1023/A:1021630228280.  Google Scholar

[8]

Y. Derriennic and M. Lin, The central limit theorem for Markov chains started at a point, Probab. Theory Related Fields, 125 (2003), 73-76.  doi: 10.1007/s004400200215.  Google Scholar

[9]

W. Doeblin, Sur deux problemes de M. Kolmogoroff concernant les chanes d énombrables, Bull. Soc. Math. France, 66 (1938), 210-220.   Google Scholar

[10]

G. Dos Reis, W. Salkeld and J. Tugaut, Freidlin-Wentzell LDP in path space for McKean-Vlasov equations and the Functional Iterated Logarithm Law, arXiv: 1708.04961v3. Google Scholar

[11]

S. S. Dragomir, Some Gronwall Type Inequalities and Applications, Nova Science Publishers, Inc., Hauppauge, NY, 2003.  Google Scholar

[12]

F. Q. Gao, Long time asymptotics of unbounded additive functionals of Markov processes, Electron. J. Probab., 22 (2017), Paper No. 94, 21 pp. doi: 10.1214/17-EJP104.  Google Scholar

[13]

M. Hairer and J. C. Mattingly, Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. Math. (2), 164 (2006), 993-1032. doi: 10.4007/annals.2006.164.993.  Google Scholar

[14]

M. HairerJ. C. Mattingly and M. Scheutzow, Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations, Probab. Theory Related Fields, 149 (2011), 223-259.  doi: 10.1007/s00440-009-0250-6.  Google Scholar

[15] P. Hall and C. C. Heyde, Martingale Limit Theory and its Applications, Academic Press, New York-London, 1980.   Google Scholar
[16]

C. C. Heyde and D. J. Scott, Invariance principles for the law of the iterated logarithm for martingales and processes with stationary increments, Ann. Probab., 1 (1973), 428-436.  doi: 10.1214/aop/1176996937.  Google Scholar

[17]

I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff Publishing, Groningen, 1971.  Google Scholar

[18]

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

[19]

J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Grundlehren der Mathematischen Wissenschaften, 288. Springer-Verlag, Berlin, 1987. doi: 10.1007/978-3-662-02514-7.  Google Scholar

[20]

C. Kipnis and S. R. S. Varadhan, Central limit theorem for additive functionals of reversible Markov process and applications to simple exclusions, Comm. Math. Phys., 104 (1986), 1-19.  doi: 10.1007/BF01210789.  Google Scholar

[21]

T. Komorowski and A. Walczuk, Central limit theorem for Markov processes with spectral gap in the Wasserstein metric, Stochastic Process. Appl., 122 (2012), 2155-2184.  doi: 10.1016/j.spa.2012.03.006.  Google Scholar

[22]

A. Kulik, Ergodic Behavior of Markov Processes. With Applications to Limit Theorems, De Gruyter Studies in Mathematics, 67, De Gruyter, Berlin, 2018.  Google Scholar

[23]

S. P. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Springer-Verlag London, Ltd., London, 1993. doi: 10.1007/978-1-4471-3267-7.  Google Scholar

[24]

A. Shirikyan, Law of large numbers and central limit theorem for randomly forced PDE's, Probab. Theory Related Fields, 134 (2006), 215-247.  doi: 10.1007/s00440-005-0427-6.  Google Scholar

[25]

V. Strassen, An invariance principle for the law of the iterated logarithm, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 3 (1964), 211-226.  doi: 10.1007/BF00534910.  Google Scholar

[26]

M.-K. von Renesse and M. Scheutzow, Existence and uniqueness of solutions of stochastic functional differential equations, Random Oper. Stoch. Equ., 18 (2010), 267-284.  doi: 10.1515/ROSE.2010.015.  Google Scholar

[27]

A. Walczuk, Central limit theorem for an additive functional of a Markov process, stable in the Wesserstein metric, Ann. Univ. Mariae Curie-Sklodowska Sect. A, 62 (2008), 149-159.  doi: 10.2478/v10062-008-0016-0.  Google Scholar

[28]

L. M. Wu, Forward-backward martingale decomposition and compactness results for additive functionals of stationary ergodic Markov processes, Ann. Inst. H. Poincaré Probab. Statist., 35 (1999), 121-141.  doi: 10.1016/S0246-0203(99)80008-9.  Google Scholar

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