Abstract: The authors investigated the weak viability property of backward stochastic differential equations(shortened by BSWVP). By means of sto chastic Lyapunov’s method and Chebychev’s inequality, the authors give a sufficient condition for BSWVP holds on a nonempty closed set. As application, the authors obtaineda decision condition for the weak viability for the viscosity solution of quasilinear parabolic partial differential equations in a closed set.

Key words: backward stochastic differential equation, weak viability property, quasilinear parabolic PDE, viscosity solution, stochastic Lyapunov functional