A Kneser-type theorem for backward doubly stochastic differential equations

Yufeng Shi; Qingfeng Zhu

doi:10.3934/dcdsb.2010.14.1565

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2010, Volume 14, Issue 4: 1565-1579. Doi: 10.3934/dcdsb.2010.14.1565

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A Kneser-type theorem for backward doubly stochastic differential equations

Yufeng Shi^1, and
Qingfeng Zhu^2,

1.
Institute for Financial Studies and School of Mathematics, Shandong University, Jinan, Shandong 250100
2.
School of Statistics and Mathematics, Shandong University of Finance, Jinan, Shandong 250014

Received: July 2009

Revised: February 2010

Published: June 2010

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Abstract

A class of backward doubly stochastic differential equations (BDSDEs in short) with continuous coefficients is studied. We give the comparison theorems, the existence of the maximal solution and the structure of solutions for BDSDEs with continuous coefficients. A Kneser-type theorem for BDSDEs is obtained. We show that there is either unique or uncountable solutions for this kind of BDSDEs.

Keywords:

Backward doubly stochastic differential equations,
comparison theorem,
maximal solution,
Kneser-type theorem.

Mathematics Subject Classification: 60H10.

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