Acta mathematica scientia, Series B >
MULTI-DIMENSIONAL REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS AND THE COMPARISON THEOREM
Received date: 2006-10-22
Revised date: 2008-11-29
Online published: 2010-09-20
Supported by
Wu acknowledges the National Natural Science Foundation (10371067), the National Basic Research Program of China (973 Program, 2007CB814904), the Natural Science Foundation of Shandong Province (Z2006A01), the Doctoral Fund of Education Ministry of China, and Youth Growth Foundation of Shandong University at Weihai, P.R.China. Xiao acknowledges the Natural Science Foundation of Shandong Province (ZR2009AQ017), and Independent Innovation Foundation of Shandong University, IIFSDU.
In this article, we study the multi-dimensional reflected backward stochastic differential equations. The existence and uniqueness result
of the solution for this kind of equation is proved by the fixed point argument where every element of the solution is forced to stay above the given stochastic process, i.e., multi-dimensional obstacle, respectively. We also give a kind of multi-dimensional comparison theorem for the reflected BSDE and then use it as the tool to prove an existence result for the multi-dimensional reflected BSDE where the coefficient is continuous and has linear growth.
WU Zhen , XIAO Hua . MULTI-DIMENSIONAL REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS AND THE COMPARISON THEOREM[J]. Acta mathematica scientia, Series B, 2010 , 30(5) : 1819 -1836 . DOI: 10.1016/S0252-9602(10)60175-0
[1] Pardoux E, Peng S. Adapted solutions of a backward stochastic differential equation. Systems and Control Letters, 1990, 14: 55--61
[2] Duffie D, Epstein L. Stochastic differential utility. Econometrica, 1992, 60: 353--394
[3] El-Karoui N, Peng S, Quenez M C. Backward stochastic differential equation in finance. Math Finance, 1997, 7: 1--71
[4] Hamadene S, Lepeltier J P. Backward equations, stochastic control and zero-sum t stochastic differential games. Stochastic and Stochastic Reports, 1995, 54: 221--231
[5] Lepeltier J P, Martin J San. Backward stochastic differential equations with continuous coefficient. Statistics and Probability Letters, 1997, 32: 425--430
[6] El-Karoui N, et al. Reflected solutions of backward SDE's and related obstacle problems For PDE's. Ann Probab, 1997, 25: 702--737
[7] Cvitanic J, Karatzas I. Backward SDE's with reflection and Dynkin games. Ann Proba, 1996, 24: 2024--2056
[8] Hamadene S, Lepeltier J P. Reflected BSDEs and mixed games. Stochastic Process Appl, 2000, 85: 177--188
[9] Geiβ C, Manthey R. Comparison theorems for stochastic differential equations in finite and infinite dimensions. Stochastic Process Appl, 1994, 53: 23--35
[10] Zhou H. Comparison theorem for multi-dimensional backward stochastic differential equations and applications
[D]. Jinan: Shandong University, 1999 (in Chinese)
[11] Buckdahn R, Peng S. Stationary backward stochastic differential equations and associated partial differential equations. Probability Theory and Related Fields, 1999, 115: 383--399
[12] Wu Z. Forward-backward stochastic differential equations with stopping time. Acta Mathematica Scientia, 2004, 24B(1): 91--99
[13] Ding X, Wu R. A new proof for comparison theorems for stochastic differential inequalities with respect to semi-martingales. Stochastic Process Appl, 1998, 78: 155--171
/
| 〈 |
|
〉 |