  We are concerned with multidimensional stochastic balance laws driven by
L\'{e}vy processes. Using bounded variation (BV) estimates for vanishing
viscosity approximations, we derive an explicit continuous dependence estimate
on the nonlinearities of the entropy solutions under the assumption that
L\'{e}vy noise only depends on the solution. This result is used to show the
error estimate for the stochastic vanishing viscosity method. In addition, we
establish fractional $BV$ estimate for vanishing viscosity approximations in
case the noise coefficient depends on both the solution and spatial variable.
