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Unbounded stochastic control problems may lead to Hamilton-Jacobi-Bellmanequations whose Hamiltonians are not always defined, especially when thediffusion term is unbounded with respect to the control. We obtain existenceand uniqueness of viscosity solutions growing at most like $o(1+|x|^p)$ atinfinity for such HJB equations and more generally for degenerate parabolicequations with a superlinear convex gradient nonlinearity. If the correspondingcontrol problem has a bounded diffusion with respect to the control, then ourresults apply to a larger class of solutions, namely those growing like$O(1+|x|^p)$ at infinity. This latter case encompasses some equations relatedto backward stochastic differential equations.

Uniqueness results for convex Hamilton-Jacobi equations under $p>1$ growth conditions on data