Backwards SDE with random terminal time and applications to semilinear elliptic PDE

Suppose ${\Im_t}$ is the filtration induced by a Wiener process $W$ in $R^d$, $\tau$ is a finite ${\Im_t}$ stopping time (terminal time), $\xi$ is an ${\Im_{\tau}}$-measurable random variable in $R^k$ (terminal value) and $f(\cdot, y, z)$ is a coefficient process, depending on $y \in R^k$ and $z \in L(R^d, R^k)$, satisfying $(y - \tilde{y})[f(s, y, z) - f(s, \tilde{y}, z)] \leq - a|y - \tilde{y}|^2$ ($f$ need not be Lipschitz in $y$), and $|f(s, y, z) - f(s, y, \tilde{z})| \leq b||z - \tilde{z}||$, for some real $a$ and $b$, plus other mild conditions. We identify a Hilbert space, depending on $\tau$ and on the number $\gamma \equiv b^2 - 2a$, in which there exists a unique pair of adapted processes $(Y, Z)$ satisfying the stochastic differential equation $$dY(s) = 1_{{s \leq \tau}} [Z(s) dW(s) - f(s, Y(s), Z(s)) ds]$$ with the given terminal condition $Y(\tau) - \xi$, provided a certain integrability condition holds. This result is applied to construct a continuous viscosity solution to the Dirichlet problem for a class of semilinear elliptic PDE’s.