MULTI-DIMENSIONAL REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS AND THE COMPARISON THEOREM

doi:10.1016/S0252-9602(10)60175-0

Acta mathematica scientia,Series B ›› 2010, Vol. 30 ›› Issue (5): 1819-1836.doi: 10.1016/S0252-9602(10)60175-0

• Articles • Previous Articles

MULTI-DIMENSIONAL REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS AND THE COMPARISON THEOREM

WU Zhen, XIAO Hua

1.School of Mathematics, Shandong University, Jinan 250100, China
2.School of Mathematics and Statistics, Shandong University at Weihai, Weihai 264209, China

Received:2006-10-22 Revised:2008-11-29 Online:2010-09-20 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.

Abstract

Abstract:

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.

Key words: backward stochastic differential equations, comparison theorem, local time

CLC Number:

60H10

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.

Trendmd

References

[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

MULTI-DIMENSIONAL REFLECTED BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS AND THE COMPARISON THEOREM

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 11

Metrics

Comments

Recommended 10

[1]	Yuecai HAN, Yifang SUN. SOLUTIONS TO BSDES DRIVEN BY BOTH FRACTIONAL BROWNIAN MOTIONS AND THE UNDERLYING STANDARD BROWNIAN MOTIONS [J]. Acta mathematica scientia,Series B, 2018, 38(2): 681-694.
[2]	Jingjun GUO, Chujin LI. ON COLLISION LOCAL TIME OF TWO INDEPENDENT FRACTIONAL ORNATEIN-UHLENBECK PROCESSES [J]. Acta mathematica scientia,Series B, 2017, 37(2): 316-328.
[3]	JIA Guang-Yan, ZHANG Na. INVARIANT REPRESENTATION FOR STOCHASTIC DIFFERENTIAL OPERATOR BY BSDES WITH UNIFORMLY CONTINUOUS COEFFICIENTS AND ITS APPLICATIONS [J]. Acta mathematica scientia,Series B, 2013, 33(5): 1407-1418.
[4]	ZHONG Yu-Quan. LOCAL TIME ANALYSIS OF ADDITIVE L\'EVY PROCESSES WITH DIFFERENT LÉVY EXPONENTS [J]. Acta mathematica scientia,Series B, 2009, 29(5): 1155-1164.
[5]	Li Juan. FULLY COUPLED FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH GENERAL MARTINGALE [J]. Acta mathematica scientia,Series B, 2006, 26(3): 443-450.
[6]	TUN Zhen. FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH STOPPING TIME [J]. Acta mathematica scientia,Series B, 2004, 24(1): 91-99.
[7]	JIANG Yi-Wen, LIU Lu-Qin. REPRESENTATION OF ADDITIVE FUNCTIONALS AND LOCAL TIMES FOR JUMP MARKOV PROCESSES AND THEIR FUNCTIONAL LIMIT THEOREM [J]. Acta mathematica scientia,Series B, 2003, 23(1): 117-123.
[8]	ZHONG Yu-Quan, HU Di-He. SELF-INTERSECTION LOCAL TIME OF ADDITIVE L´\|EVY PROCESS [J]. Acta mathematica scientia,Series B, 2002, 22(2): 261-268.
[9]	MO Jia-Qi, OU Yang-Cheng. A CLASS OF NONLOCAL BOUNDARY VALUE PROBLEMS OF NONLINEAR ELLIPTIC SYSTEMS IN UNBOUNDED DOMAINS [J]. Acta mathematica scientia,Series B, 2001, 21(1): 93-97.
[10]	Chen Songliu, Mo Jiaqi. THE NONLOCAL PROBLEMS FOR SINGULARLY PERTURBED REACTION DIFFUSION SYSTEMS [J]. Acta mathematica scientia,Series B, 1998, 18(S1): 78-82.
[11]	Li Cunhang. SOME PROPERTIES OF (α, d, β)-SUPERPROCESSES [J]. Acta mathematica scientia,Series B, 1996, 16(2): 234-240.