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Forward-backward stochastic differential equations and quasilinear parabolic PDEs. (English) Zbl 0943.60057

This paper considers a forward-backward stochastic differential equation (FBSDE) which is a system of the type

X(t)=x+ 0 t f(s,X(s),Y(s),Z(s))ds+ 0 t σ(s,X(s),Y(s),Z(s))dB(s),Y(t)=h(X(T))+ t T g(s,X(s),Y(s),Z(s))ds- 0 t Z(s)dB(s)

for t[0,T]. Under some technical conditions like Lipschitz, linear growth, or measurability and a simple and very natural monotonicity condition on f and g, the authors prove results on existence and uniqueness of a solution {(X(s),Y(s),Z(s)):s[0,T]}. Furthermore, they establish a priori estimates and continuous dependence upon a parameter. Finally, they connect FBSDEs to systems of quasilinear parabolic PDEs of second order. Using their purely probabilistic approach, they prove the existence of viscosity solutions of the PDE.


MSC:
60H30Applications of stochastic analysis
93E03General theory of stochastic systems
60G44Martingales with continuous parameter
35K55Nonlinear parabolic equations