  We provide a stochastic representation for a general class of viscous
Hamilton-Jacobi (HJ) equations, which has convexity and superlinear
nonlinearity in its gradient term, via a type of backward stochastic
differential equation (BSDE) with constraint in the martingale part. We compare
our result with the classical representation in terms of (super)quadratic BSDE,
and show in particular that existence of a solution to the viscous HJ equation
can be obtained under more general growth assumptions on the coefficients,
including both unbounded diffusion coefficient and terminal data.
