  In this paper, a probabilistic interpretation for the viscosity solution of a
parabolic partial differential equation is obtained by virtue of the solution
of a class of quadratic backward stochastic differential equations (BSDEs, for
short). Furthermore, we prove the convergence and the large deviation principle
for the solution of this class of quadratic BSDEs, which is associated with a
family of Markov processes with the diffusion coefficients that tend to be
zero.
