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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 $$\align X(t) &= x+\int_0^t f(s,X(s),Y(s),Z(s)) ds +\int_0^t \sigma(s,X(s),Y(s),Z(s)) dB(s),\\ Y(t)&= h(X(T))+\int_t^T g(s,X(s),Y(s),Z(s)) ds -\int_0^t Z(s) dB(s) \endalign$$ for $t\in[0,T]$. 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 $\{(X(s),Y(s),Z(s)):s\in[0,T]\}$. 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.

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