Journal of Applied Mathematics and Stochastic Analysis
Volume 2009 (2009), Article ID 854768, 26 pages
Research Article

On Variant Reflected Backward SDEs, with Applications

1Department of Mathematics, University of Southern California, Los Angels, CA 90089, USA
2Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

Received 11 December 2008; Accepted 11 March 2009

Academic Editor: Vo Anh

Copyright © 2009 Jin Ma and Yusun Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We study a new type of reflected backward stochastic differential equations (RBSDEs), where the reflecting process enters the drift in a nonlinear manner. This type of the reflected BSDEs is based on a variance of the Skorohod problem studied recently by Bank and El Karoui (2004), and is hence named the “Variant Reflected BSDEs” (VRBSDE) in this paper. The special nature of the Variant Skorohod problem leads to a hidden forward-backward feature of the BSDE, and as a consequence this type of BSDE cannot be treated in a usual way. We shall prove that in a small-time duration most of the well-posedness, comparison, and stability results are still valid, although some extra conditions on the boundary process are needed. We will also provide some possible applications where the VRBSDE can be potentially useful. These applications show that the VRBSDE could become a novel tool for some problems in finance and optimal stopping problems where no existing methods can be easily applicable.