Four step scheme for general Markovian forward-backward SDES |
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Authors: | Jin Ma Jiongmin Yong Yanhong Zhao |
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Institution: | 1.Department of Mathematics,University of Southern California,Los Angeles,USA;2.Department of Mathematics,University of Central Florida,Orlando,USA;3.Department of Mathematics,Purdue University,West Lafayette,USA |
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Abstract: | This paper studies a class of forward-backward stochastic differential equations (FBSDE) in a general Markovian framework.
The forward SDE represents a large class of strong Markov semimartingales, and the backward generator requires only mild regularity
assumptions. The authors show that the Four Step Scheme introduced by Ma, et al. (1994) is still effective in this case. Namely,
the authors show that the adapted solution of the FBSDE exists and is unique over any prescribed time duration; and the backward
components can be determined explicitly by the forward component via the classical solution to a system of parabolic integro-partial
differential equations. An important consequence the authors would like to draw from this fact is that, contrary to the general
belief, in a Markovian set-up the martingale representation theorem is no longer the reason for the well-posedness of the
FBSDE, but rather a consequence of the existence of the solution of the decoupling integralpartial differential equation.
Finally, the authors briefly discuss the possibility of reducing the regularity requirements of the coefficients by using
a scheme proposed by F. Delarue (2002) to the current case. |
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