# American Institute of Mathematical Sciences

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

## 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

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.

Citation: Xing Huang, Chang Liu, Feng-Yu Wang. Order preservation for path-distribution dependent SDEs. Communications on Pure and 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. [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.

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. [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.
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