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Concentration phenomena for critical fractional Schrödinger systems
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 |
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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Comparison theorem for stochastic differential delay equations with jumps, Acta Appl. Math., 116 (2011), 119-132.
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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.
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F.-Y. Wang,
The stochastic order and critical phenomena for superprocesses, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 9 (2006), 107-128.
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F.-Y. Wang,
Distribution-dependent SDEs for Landau type equations, Stoch. Proc. Appl., 128 (2018), 595-621.
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J.-M. Wang,
Stochastic comparison for Lévy-type processes, J. Theor. Probab., 26 (2013), 997-1019.
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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.
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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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