Order preservation for path-distribution dependent SDEs

Xing Huang; Chang Liu; Feng-Yu Wang

doi:10.3934/cpaa.2018100

Article Contents

2018, Volume 17, Issue 5: 2125-2133. Doi: 10.3934/cpaa.2018100

This issue Previous Article Concentration phenomena for critical fractional Schrödinger systems Next Article A note concerning a property of symplectic matrices

Order preservation for path-distribution dependent SDEs

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
^* Corresponding author

Received: September 30, 2017

Revised: December 31, 2017

Published: April 2018

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 60H1075, 60G44.

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References

[1]	J. Bao and C. Yuan, Comparison theorem for stochastic differential delay equations with jumps, Acta Appl. Math., 116 (2011), 119-132.
[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.
[3]	L. Gal'cuk and M. Davis, A note on a comparison theorem for equations with different diffusions, Stochastics, 6 (1982), 147-149.
[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.
[5]	X. Huang and F.-Y. Wang, Order-preservation for multidimensional stochastic functional differential equations with jumps, J. Evol. Equat., 14 (2014), 445-460.
[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.
[7]	T. Kamae, U. Krengel and G. L. O'Brien, Stochastic inequalities on partially ordered spaces, Ann. Probab., 5 (1977), 899-912.
[8]	X. Mao, A note on comparison theorems for stochastic differential equations with respect to semimartingales, Stochastics, 37 (1991), 49-59.
[9]	G. L. O'Brien, A new comparison theorem for solution of stochastic differential equations, Stochastics, 3 (1980), 245-249.
[10]	S. Peng and Z. Yang, Anticipated backward stochastic differential equations, Ann. Probab., 37 (2009), 877-902.
[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.
[12]	F.-Y. Wang, The stochastic order and critical phenomena for superprocesses, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 9 (2006), 107-128.
[13]	F.-Y. Wang, Distribution-dependent SDEs for Landau type equations, Stoch. Proc. Appl., 128 (2018), 595-621.
[14]	J.-M. Wang, Stochastic comparison for Lévy-type processes, J. Theor. Probab., 26 (2013), 997-1019.
[15]	Z. Yang, X. Mao and C. Yuan, Comparison theorem of one-dimensional stochastic hybrid systems, Systems Control Lett., 57 (2008), 56-63.
[16]	X. Zhu, On the comparison theorem for multi-dimensional stochastic differential equations with jumps (in Chinese), Sci. Sin. Math., 42 (2012), 303-311.