Abstract.
The problem of finding adapted solutions to systems of coupled linear forward—backward stochastic differential equations (FBSDEs, for short) is investigated. A necessary condition of solvability leads to a reduction of general linear FBSDEs to a special one. By some ideas from controllability in control theory, using some functional analysis, we obtain a necessary and sufficient condition for the solvability of a class of linear FBSDEs. Then a Riccati-type equation for matrix-valued (not necessarily square) functions is derived using the idea of the Four-Step Scheme (introduced in [11] for general FBSDEs). The solvability of such a Riccati-type equation is studied which leads to a representation of adapted solutions to linear FBSDEs.
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Accepted 29 April 1997
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Jiongmin Yong, . Linear Forward—Backward Stochastic Differential Equations . Appl Math Optim 39, 93–119 (1999). https://doi.org/10.1007/s002459900100
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DOI: https://doi.org/10.1007/s002459900100