A type of time-symmetric forward–backward stochastic differential equations

Shige Peng; Yufeng Shi

doi:10.1016/S1631-073X(03)00183-3

Probability Theory

A type of time-symmetric forward–backward stochastic differential equations
[Un type d'équations différentielles stochastiques progressives–rétrogrades symétriques par rapport au temps]

Présenté par : Paul Malliavin

Shige Peng ¹ ; Yufeng Shi ¹

¹ School of Mathematics and System Sciences, Shandong University, Jinan 250100, China

Comptes Rendus. Mathématique, Volume 336 (2003) no. 9, pp. 773-778.

Résumé
Abstract

Nous étudions dans cette Note un type d'équations différentielles stochastiques progressives–rétrogrades symétriques par rapport au temps. Sous certaines conditions de monotonie, nous donnons un théorème d'existence et unicité des solutions des équations par une méthode de continuation. Ensuite nous présentons une application.

In this Note, we study a type of time-symmetric forward–backward stochastic differential equations. Under some monotonicity assumptions, we establish the existence and uniqueness theorem by means of a method of continuation. We also give an application.

Reçu le : 2002-06-04
Accepté le : 2003-04-01
Publié le : 2003-04-30

DOI : 10.1016/S1631-073X(03)00183-3

Affiliations des auteurs :

Shige Peng ¹ ; Yufeng Shi ¹

¹ School of Mathematics and System Sciences, Shandong University, Jinan 250100, China

@article{CRMATH_2003__336_9_773_0,
     author = {Shige Peng and Yufeng Shi},
     title = {A type of time-symmetric forward{\textendash}backward stochastic differential equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {773--778},
     publisher = {Elsevier},
     volume = {336},
     number = {9},
     year = {2003},
     doi = {10.1016/S1631-073X(03)00183-3},
     language = {en},
}

TY  - JOUR
AU  - Shige Peng
AU  - Yufeng Shi
TI  - A type of time-symmetric forward–backward stochastic differential equations
JO  - Comptes Rendus. Mathématique
PY  - 2003
SP  - 773
EP  - 778
VL  - 336
IS  - 9
PB  - Elsevier
DO  - 10.1016/S1631-073X(03)00183-3
LA  - en
ID  - CRMATH_2003__336_9_773_0
ER  -

%0 Journal Article
%A Shige Peng
%A Yufeng Shi
%T A type of time-symmetric forward–backward stochastic differential equations
%J Comptes Rendus. Mathématique
%D 2003
%P 773-778
%V 336
%N 9
%I Elsevier
%R 10.1016/S1631-073X(03)00183-3
%G en
%F CRMATH_2003__336_9_773_0

Shige Peng; Yufeng Shi. A type of time-symmetric forward–backward stochastic differential equations. Comptes Rendus. Mathématique, Volume 336 (2003) no. 9, pp. 773-778. doi : 10.1016/S1631-073X(03)00183-3. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00183-3/

Bibliographie
Cité par

[1] F. Antonelli Backward–forward stochastic differential equations, Ann. Appl. Probab., Volume 3 (1993), pp. 777-793

[2] A. Bensoussan Stochastic Control by Functional Analysis Methods, North-Holland, Amsterdam, 1982

[3] J.-M. Bismut Conjugate convex functions in optimal stochastic control, J. Math. Anal. Appl., Volume 44 (1973), pp. 384-404

[4] N. El Karoui; S. Peng; M.-C. Quenez Backward stochastic differential equations in finance, Math. Finance, Volume 7 (1997), pp. 1-71

[5] Y. Hu; S. Peng Solution of forward–backward stochastic differential equations, Probab. Theory Related Fields, Volume 103 (1995), pp. 273-283

[6] J. Ma; P. Protter; J. Yong Solving forward–backward stochastic differential equations explicitly – a four step scheme, Probab. Theory Related Fields, Volume 98 (1994), pp. 339-359

[7] E. Pardoux; S. Peng Adapted solution of a backward stochastic differential equation, Systems Control Lett., Volume 14 (1990), pp. 55-61

[8] E. Pardoux; S. Peng Backward doubly stochastic differential equations and systems of quasilinear parabolic SPDE's, Probab. Theory Related Fields, Volume 98 (1994), pp. 209-227

[9] S. Peng Probabilistic interpretation for systems of quasilinear parabolic partial differential equations, Stochastics, Volume 37 (1991), pp. 61-74

[10] S. Peng Problem of eigenvalues of stochastic Hamiltonian systems with boundary conditions, Stochastic Process. Appl., Volume 88 (2000), pp. 259-290

[11] S. Peng; Y. Shi Infinite horizon forward–backward stochastic differential equations, Stochastic Process. Appl., Volume 85 (2000), pp. 75-92

[12] S. Peng; Z. Wu Fully coupled forward–backward stochastic differential equations and applications to optimal control, SIAM J. Control Optim., Volume 37 (1999), pp. 825-843

[13] J. Yong Finding adapted solutions of forward–backward stochastic differential equations – method of continuation, Probab. Theory Related Fields, Volume 107 (1997), pp. 537-572

Sicong Wang; Bin Teng; Yufeng Shi; Qingfeng Zhu A deep learning method for solving multi-dimensional coupled forward-backward doubly SDEs, Computers Mathematics with Applications, Volume 169 (2024), pp. 260-272 | DOI:10.1016/j.camwa.2024.07.015 | Zbl:7894803
Jian Song; Meng Wang On mean-field control problems for backward doubly stochastic systems, European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations, Volume 30 (2024), p. 27 (Id/No 20) | DOI:10.1051/cocv/2024012 | Zbl:1533.93851
Nana Zhao; Jinghan Wang; Yufeng Shi; Qingfeng Zhu General Time-Symmetric Mean-Field Forward-Backward Doubly Stochastic Differential Equations, Symmetry, Volume 15 (2023) no. 6, p. 1143 | DOI:10.3390/sym15061143
Bin Teng; Yufeng Shi; Qingfeng Zhu Solving high-dimensional forward-backward doubly SDEs and their related SPDEs through deep learning, Personal and Ubiquitous Computing, Volume 26 (2022) no. 4, p. 925 | DOI:10.1007/s00779-020-01500-5
Qing-feng Zhu; Liang-quan Zhang; Yu-feng Shi Infinite horizon forward-backward doubly stochastic differential equations and related SPDEs, Acta Mathematicae Applicatae Sinica. English Series, Volume 37 (2021) no. 2, pp. 319-336 | DOI:10.1007/s10255-021-1009-9 | Zbl:1469.60201
Qingfeng Zhu; Yufeng Shi; Bin Teng Forward-backward doubly stochastic differential equations with random jumps and related games, Asian Journal of Control, Volume 23 (2021) no. 2, pp. 962-978 | DOI:10.1002/asjc.2344 | Zbl:7878863
Shige Peng Backward Stochastic Differential Equations and Related Control Problems, Encyclopedia of Systems and Control (2021), p. 132 | DOI:10.1007/978-3-030-44184-5_234
Qingfeng Zhu; Yufeng Shi Nonzero-sum differential game of backward doubly stochastic systems with delay and applications, Mathematical Control and Related Fields, Volume 11 (2021) no. 1, pp. 73-94 | DOI:10.3934/mcrf.2020028 | Zbl:1471.91025
Qingfeng Zhu; Yufeng Shi; Jiaqiang Wen; Hui Zhang A Type of Time-Symmetric Stochastic System and Related Games, Symmetry, Volume 13 (2021) no. 1, p. 118 | DOI:10.3390/sym13010118
Jiaqiang Wen; Yufeng Shi Symmetrical martingale solutions of backward doubly stochastic Volterra integral equations, Computers Mathematics with Applications, Volume 79 (2020) no. 5, pp. 1435-1446 | DOI:10.1016/j.camwa.2019.09.006 | Zbl:1460.60058
Qingfeng Zhu; Lijiao Su; Fuguo Liu; Yufeng Shi; Yong'ao Shen; Shuyang Wang Mean-field type forward-backward doubly stochastic differential equations and related stochastic differential games, Frontiers of Mathematics in China, Volume 15 (2020) no. 6, pp. 1307-1326 | DOI:10.1007/s11464-020-0889-y | Zbl:1470.60174
Jinbiao Wu; Zaiming Liu Optimal control of mean-field backward doubly stochastic systems driven by Itô-Lévy processes, International Journal of Control, Volume 93 (2020) no. 4, pp. 953-970 | DOI:10.1080/00207179.2018.1502473 | Zbl:1436.93144
Jie Xu Stochastic maximum principle for delayed doubly stochastic control systems and their applications, International Journal of Control, Volume 93 (2020) no. 6, pp. 1371-1380 | DOI:10.1080/00207179.2018.1508850 | Zbl:1443.93141
Yufeng Shi; Jiaqiang Wen; Jie Xiong Backward doubly stochastic Volterra integral equations and their applications, Journal of Differential Equations, Volume 269 (2020) no. 9, pp. 6492-6528 | DOI:10.1016/j.jde.2020.05.006 | Zbl:1443.60056
AbdulRahman Al-Hussein; Boulakhras Gherbal Necessary and sufficient optimality conditions for relaxed and strict control of forward-backward doubly SDEs with jumps under full and partial information, Journal of Systems Science and Complexity, Volume 33 (2020) no. 6, pp. 1804-1846 | DOI:10.1007/s11424-020-9013-3 | Zbl:1460.93095
Dahbia Hafayed; Adel Chala An optimal control of a risk-sensitive problem for backward doubly stochastic differential equations with applications, Random Operators and Stochastic Equations, Volume 28 (2020) no. 1, pp. 1-18 | DOI:10.1515/rose-2020-2024 | Zbl:1433.93155
AbdulRahman Al-Hussein; Boulakhras Gherbal Existence and uniqueness of the solutions of forward-backward doubly stochastic differential equations with Poisson jumps, Random Operators and Stochastic Equations, Volume 28 (2020) no. 4, pp. 253-268 | DOI:10.1515/rose-2020-2044 | Zbl:1457.60085
Wencan Wang Optimal control of backward doubly stochastic system, IET Control Theory Applications, Volume 13 (2019) no. 12, pp. 1844-1854 | DOI:10.1049/iet-cta.2018.6249 | Zbl:1432.93386
Jiaqiang Wen; Yufeng Shi Backward doubly stochastic differential equations with random coefficients and quasilinear stochastic PDEs, Journal of Mathematical Analysis and Applications, Volume 476 (2019) no. 1, pp. 86-100 | DOI:10.1016/j.jmaa.2018.10.038 | Zbl:1478.60190
Wencan Wang; Jinbiao Wu; Zaiming Liu The optimal control of fully-coupled forward-backward doubly stochastic systems driven by Itô-Lévy processes, Journal of Systems Science and Complexity, Volume 32 (2019) no. 4, pp. 997-1018 | DOI:10.1007/s11424-018-7300-z | Zbl:1419.93068
Anis Matoussi; Dylan Possamaï; Wissal Sabbagh Probabilistic interpretation for solutions of fully nonlinear stochastic pdes, Probability Theory and Related Fields, Volume 174 (2019) no. 1-2, pp. 177-233 | DOI:10.1007/s00440-018-0859-4 | Zbl:1447.60116
Dahbia Hafayed; Adel Chala A general maximum principle for mean-field forward-backward doubly stochastic differential equations with jumps processes, Random Operators and Stochastic Equations, Volume 27 (2019) no. 1, pp. 9-25 | DOI:10.1515/rose-2019-2002 | Zbl:1414.93202
Chunrong Feng; Xince Wang; Huaizhong Zhao Quasi-linear PDEs and forward-backward stochastic differential equations: weak solutions, Journal of Differential Equations, Volume 264 (2018) no. 2, pp. 959-1018 | DOI:10.1016/j.jde.2017.09.030 | Zbl:1516.35184
Zhaojun Zong; Feng Hu $L^{p}$ solutions of infinite time interval backward doubly stochastic differential equations under monotonicity and general increasing conditions, Journal of Mathematical Analysis and Applications, Volume 458 (2018) no. 2, pp. 1486-1511 | DOI:10.1016/j.jmaa.2017.10.041 | Zbl:1388.60102
Zhaojun Zong; Feng Hu $L^{p}$ solutions of infinite time interval backward doubly stochastic differential equations, Filomat, Volume 31 (2017) no. 7, pp. 1857-1868 | DOI:10.2298/fil1707857z | Zbl:1488.60162
Jie Xu; Yuecai Han Stochastic maximum principle for delayed backward doubly stochastic control systems, Journal of Nonlinear Science and Applications, Volume 10 (2017) no. 1, pp. 215-226 | DOI:10.22436/jnsa.010.01.21 | Zbl:1415.49021
Adel Chala The general relaxed control problem of fully coupled forward-backward doubly system, S $\vec{e}$ MA Journal, Volume 74 (2017) no. 1, pp. 1-19 | DOI:10.1007/s40324-016-0076-y | Zbl:1380.93283
Hui-nan Leng Infinite horizon backward doubly stochastic differential equations with non-degenerate terminal functions and their stationary property, Acta Mathematicae Applicatae Sinica. English Series, Volume 32 (2016) no. 2, pp. 407-422 | DOI:10.1007/s10255-016-0567-8 | Zbl:1338.60147
Abdulrahman Al-Hussein; Boulakhras Gherbal Sufficient conditions of optimality for forward-backward doubly SDEs with jumps, Statistical methods and applications in insurance and finance. CIMPA school, Marrakech and Kelaat M'gouna, Morocco, April 8–20, 2013, Cham: Springer, 2016, pp. 173-191 | DOI:10.1007/978-3-319-30417-5_7 | Zbl:1403.93193
Chala Adel Necessary condition for optimality of forward-backward doubly system, Afrika Matematika, Volume 26 (2015) no. 3-4, pp. 575-584 | DOI:10.1007/s13370-014-0227-1 | Zbl:1328.49025
Adel Chala Near-relaxed control problem of fully coupled forward-backward doubly system, Communications in Mathematics and Statistics, Volume 3 (2015) no. 4, pp. 459-476 | DOI:10.1007/s40304-015-0068-8 | Zbl:1327.93408
Qingfeng Zhu; Yufeng Shi Optimal Control of Backward Doubly Stochastic Systems With Partial Information, IEEE Transactions on Automatic Control, Volume 60 (2015) no. 1, p. 173 | DOI:10.1109/tac.2014.2322212
Qingfeng Zhu; Yufeng Shi Mean-field forward-backward doubly stochastic differential equations and related nonlocal stochastic partial differential equations, Abstract and Applied Analysis, Volume 2014 (2014), p. 10 (Id/No 194341) | DOI:10.1155/2014/194341 | Zbl:1469.60200
Eddie C.M. Hui; Hua Xiao Differential games of partial information forward-backward doubly SDE and applications, ESAIM: Control, Optimisation and Calculus of Variations, Volume 20 (2014) no. 1, p. 78 | DOI:10.1051/cocv/2013055
Dr. Shige Peng Backward Stochastic Differential Equations and Related Control Problems, Encyclopedia of Systems and Control (2014), p. 1 | DOI:10.1007/978-1-4471-5102-9_234-1
Adel Chala On optimal control problem for backward stochastic doubly systems, ISRN Applied Mathematics, Volume 2014 (2014), p. 10 (Id/No 903912) | DOI:10.1155/2014/903912 | Zbl:1298.93355
AbdulRahman Al-Hussein; Boulakhras Gherbal Stochastic Maximum Principle for Hilbert Space Valued Forward-Backward Doubly SDEs with Poisson Jumps, System Modeling and Optimization, Volume 443 (2014), p. 1 | DOI:10.1007/978-3-662-45504-3_1
Yufeng Shi; Qingfeng Zhu Partially observed optimal controls of forward-backward doubly stochastic systems, ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 3, p. 828 | DOI:10.1051/cocv/2012035
Shaolin Ji; Qingmeng Wei; Xiumin Zhang A maximum principle for controlled time-symmetric forward-backward doubly stochastic differential equation with initial-terminal state constraints, Abstract and Applied Analysis, Volume 2012 (2012), p. 29 (Id/No 537376) | DOI:10.1155/2012/537376 | Zbl:1256.49007
Bo Zhu; Baoyan Han Backward doubly stochastic differential equations with infinite time horizon., Applications of Mathematics, Volume 57 (2012) no. 6, pp. 641-653 | DOI:10.1007/s10492-012-0039-2 | Zbl:1274.60193
Bao Yan Han Comparison Theorems for the Multi-Dimensional Backward Doubly Stochastic Differential Equations, Applied Mechanics and Materials, Volume 166-169 (2012), p. 3210 | DOI:10.4028/www.scientific.net/amm.166-169.3210
Bo Zhu; Baoyan Han Comparison theorems for the multidimensional BDSDEs and applications, Journal of Applied Mathematics, Volume 2012 (2012), p. 14 (Id/No 304781) | DOI:10.1155/2012/304781 | Zbl:1244.60064
Bo Zhu; Baoyan Han Stochastic PDEs and infinite horizon backward doubly stochastic differential equations, Journal of Applied Mathematics, Volume 2012 (2012), p. 17 (Id/No 582645) | DOI:10.1155/2012/582645 | Zbl:1267.35266
QingFeng Zhu; YuFeng Shi Forward-backward doubly stochastic differential equations and related stochastic partial differential equations, Science China. Mathematics, Volume 55 (2012) no. 12, pp. 2517-2534 | DOI:10.1007/s11425-012-4411-1 | Zbl:1305.60050
Auguste Aman $L^{p}$ -solutions of backward doubly stochastic differential equations, Stochastics and Dynamics, Volume 12 (2012) no. 3, p. 1150025 | DOI:10.1142/s0219493711500250 | Zbl:1247.60098
Liangquan Zhang; Yufeng Shi Maximum principle for forward-backward doubly stochastic control systems and applications, European Series in Applied and Industrial Mathematics (ESAIM): Control, Optimization and Calculus of Variations, Volume 17 (2011) no. 4, pp. 1174-1197 | DOI:10.1051/cocv/2010042 | Zbl:1236.93155
Yuecai Han; Shige Peng; Zhen Wu Maximum principle for backward doubly stochastic control systems with applications, SIAM Journal on Control and Optimization, Volume 48 (2010) no. 7, pp. 4224-4241 | DOI:10.1137/080743561 | Zbl:1222.49040
Qing-feng Zhu; Yu-feng Shi; Xian-jun Gong Solutions to general forward-backward doubly stochastic differential equations, Applied Mathematics and Mechanics. (English Edition), Volume 30 (2009) no. 4, pp. 517-526 | DOI:10.1007/s10483-009-0412-x | Zbl:1166.60318
Modeste N'zi; Jean-Marc Owo Backward doubly stochastic differential equations with discontinuous coefficients, Statistics Probability Letters, Volume 79 (2009) no. 7, pp. 920-926 | DOI:10.1016/j.spl.2008.11.011 | Zbl:1168.60353

Cité par 49 documents. Sources : Crossref, zbMATH

Commentaires - Politique