Backwards SDE with random terminal time and applications to semilinear elliptic PDE. (English) Zbl 0895.60067

Summary: Suppose \(\{{\mathcal I}_t\}\) is the filtration induced by a Wiener process \(W\) in \(R^d\), \(\tau\) is a finite \(\{{\mathcal I}_t\}\) stopping time (terminal time), \(\xi\) is an \({\mathcal I}_\tau\)-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 \[ (y-\widetilde y) \bigl[f(s,y,z)-f(s, \widetilde y,z) \bigr]\leq-a | y-\widetilde y|^2 \] \((f\) need not be Lipschitz in \(y)\), and \[ \bigl| f(s,y,z)-f(s,y,\overline z) \bigr|\leq b\| z-\overline z\|, \] 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 \[ dY(s)=1_{\{s\leq\tau\}} \biggl[Z(s) dW(s)-f\bigl( s,Y(s),Z(s)\bigr) ds\biggr] \] 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.


60H20 Stochastic integral equations
35J60 Nonlinear elliptic equations
91B28 Finance etc. (MSC2000)
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