  The paper deals with the existence and uniqueness of the solution of the
backward stochastic variational inequality: \begin{equation}
\left\{\begin{array} {l}-dY_{t}+\partial \varphi(Y_{t})dt \ni
F(t,Y_{t},Z_{t})dt-Z_{t}dB_{t},\;0\leq t<T \\ Y_{T}=\eta, \end{array}
\right.\end{equation} where $F$ satisfies a local boundedness condition.
