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September  2018, 17(5): 2125-2133. doi: 10.3934/cpaa.2018100

Order preservation for path-distribution dependent SDEs

Xing Huang 1, , Chang Liu 2, and  Feng-Yu Wang 1,3,,

1. 

Center for Applied Mathematics, Tianjin University, Tianjin 300072, China
2. 

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
3. 

Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, United Kingdom

* Corresponding author

Received  October 2017 Revised  January 2018 Published  April 2018

Fund Project: The third author is supported by NNSFC (11771326, 11431014).

Sufficient and necessary conditions are presented for the order preservation of path-distribution dependent SDEs. Differently from the corresponding study of distribution independent SDEs, to investigate the necessity of order preservation for the present model we need to construct a family of probability spaces in terms of the ordered pair of initial distributions.

Keywords: Path-distribution dependent SDEs, Wasserstein distance, order preservation.
Mathematics Subject Classification: Primary: 60H1075, 60G44.
Citation: Xing Huang, Chang Liu, Feng-Yu Wang. Order preservation for path-distribution dependent SDEs. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2125-2133. doi: 10.3934/cpaa.2018100
References:
[1]

J. Bao and C. Yuan, Comparison theorem for stochastic differential delay equations with jumps, Acta Appl. Math., 116 (2011), 119-132.   Google Scholar

[2]

M.-F. Chen and F.-Y. Wang, On order-preservation and positive correlations for multidimensional diffusion processes, Prob. Theory. Relat. Fields, 95 (1993), 421-428.   Google Scholar

[3]

L. Gal'cuk and M. Davis, A note on a comparison theorem for equations with different diffusions, Stochastics, 6 (1982), 147-149.   Google Scholar

[4]

X. Huang, M. Röckner and F.-Y. Wang, Nonlinear Fokker–Planck equations for probability measures on path space and path-distribution dependent SDEs, preprint, arXiv: 1709.00556. Google Scholar

[5]

X. Huang and F.-Y. Wang, Order-preservation for multidimensional stochastic functional differential equations with jumps, J. Evol. Equat., 14 (2014), 445-460.   Google Scholar

[6]

N. Ikeda and S. Watanabe, A comparison theorem for solutions of stochastic differential equations and its applications, Osaka J. Math., 14 (1977), 619-633.   Google Scholar

[7]

T. Kamae, U. Krengel and G. L. O'Brien, Stochastic inequalities on partially ordered spaces, Ann. Probab., 5 (1977), 899-912.   Google Scholar

[8]

X. Mao, A note on comparison theorems for stochastic differential equations with respect to semimartingales, Stochastics, 37 (1991), 49-59.   Google Scholar

[9]

G. L. O'Brien, A new comparison theorem for solution of stochastic differential equations, Stochastics, 3 (1980), 245-249.   Google Scholar

[10]

S. Peng and Z. Yang, Anticipated backward stochastic differential equations, Ann. Probab., 37 (2009), 877-902.   Google Scholar

[11]

S. Peng and X. Zhu, Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations, Stochastic Process. Appl., 116 (2006), 370-380.   Google Scholar

[12]

F.-Y. Wang, The stochastic order and critical phenomena for superprocesses, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 9 (2006), 107-128.   Google Scholar

[13]

F.-Y. Wang, Distribution-dependent SDEs for Landau type equations, Stoch. Proc. Appl., 128 (2018), 595-621.   Google Scholar

[14]

J.-M. Wang, Stochastic comparison for Lévy-type processes, J. Theor. Probab., 26 (2013), 997-1019.   Google Scholar

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Z. Yang, X. Mao and C. Yuan, Comparison theorem of one-dimensional stochastic hybrid systems, Systems Control Lett., 57 (2008), 56-63.   Google Scholar

[16]

X. Zhu, On the comparison theorem for multi-dimensional stochastic differential equations with jumps (in Chinese), Sci. Sin. Math., 42 (2012), 303-311.   Google Scholar

show all references

References:
[1]

J. Bao and C. Yuan, Comparison theorem for stochastic differential delay equations with jumps, Acta Appl. Math., 116 (2011), 119-132.   Google Scholar

[2]

M.-F. Chen and F.-Y. Wang, On order-preservation and positive correlations for multidimensional diffusion processes, Prob. Theory. Relat. Fields, 95 (1993), 421-428.   Google Scholar

[3]

L. Gal'cuk and M. Davis, A note on a comparison theorem for equations with different diffusions, Stochastics, 6 (1982), 147-149.   Google Scholar

[4]

X. Huang, M. Röckner and F.-Y. Wang, Nonlinear Fokker–Planck equations for probability measures on path space and path-distribution dependent SDEs, preprint, arXiv: 1709.00556. Google Scholar

[5]

X. Huang and F.-Y. Wang, Order-preservation for multidimensional stochastic functional differential equations with jumps, J. Evol. Equat., 14 (2014), 445-460.   Google Scholar

[6]

N. Ikeda and S. Watanabe, A comparison theorem for solutions of stochastic differential equations and its applications, Osaka J. Math., 14 (1977), 619-633.   Google Scholar

[7]

T. Kamae, U. Krengel and G. L. O'Brien, Stochastic inequalities on partially ordered spaces, Ann. Probab., 5 (1977), 899-912.   Google Scholar

[8]

X. Mao, A note on comparison theorems for stochastic differential equations with respect to semimartingales, Stochastics, 37 (1991), 49-59.   Google Scholar

[9]

G. L. O'Brien, A new comparison theorem for solution of stochastic differential equations, Stochastics, 3 (1980), 245-249.   Google Scholar

[10]

S. Peng and Z. Yang, Anticipated backward stochastic differential equations, Ann. Probab., 37 (2009), 877-902.   Google Scholar

[11]

S. Peng and X. Zhu, Necessary and sufficient condition for comparison theorem of 1-dimensional stochastic differential equations, Stochastic Process. Appl., 116 (2006), 370-380.   Google Scholar

[12]

F.-Y. Wang, The stochastic order and critical phenomena for superprocesses, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 9 (2006), 107-128.   Google Scholar

[13]

F.-Y. Wang, Distribution-dependent SDEs for Landau type equations, Stoch. Proc. Appl., 128 (2018), 595-621.   Google Scholar

[14]

J.-M. Wang, Stochastic comparison for Lévy-type processes, J. Theor. Probab., 26 (2013), 997-1019.   Google Scholar

[15]

Z. Yang, X. Mao and C. Yuan, Comparison theorem of one-dimensional stochastic hybrid systems, Systems Control Lett., 57 (2008), 56-63.   Google Scholar

[16]

X. Zhu, On the comparison theorem for multi-dimensional stochastic differential equations with jumps (in Chinese), Sci. Sin. Math., 42 (2012), 303-311.   Google Scholar

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