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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

$\begin{array}{cc}\hfill X\left(t\right)& =x+{\int }_{0}^{t}f\left(s,X\left(s\right),Y\left(s\right),Z\left(s\right)\right)ds+{\int }_{0}^{t}\sigma \left(s,X\left(s\right),Y\left(s\right),Z\left(s\right)\right)dB\left(s\right),\hfill \\ \hfill Y\left(t\right)& =h\left(X\left(T\right)\right)+{\int }_{t}^{T}g\left(s,X\left(s\right),Y\left(s\right),Z\left(s\right)\right)ds-{\int }_{0}^{t}Z\left(s\right)dB\left(s\right)\hfill \end{array}$

for $t\in \left[0,T\right]$. 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 $\left\{\left(X\left(s\right),Y\left(s\right),Z\left(s\right)\right):s\in \left[0,T\right]\right\}$. 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:
 60H30 Applications of stochastic analysis 93E03 General theory of stochastic systems 60G44 Martingales with continuous parameter 35K55 Nonlinear parabolic equations