  We characterize the value of swing contracts in continuous time as the unique
viscosity solution of a Hamilton-Jacobi-Bellman equation with suitable boundary
conditions. The case of contracts with penalties is straightforward, and in
that case only a terminal condition is needed. Conversely, the case of
contracts with strict constraints gives rise to a stochastic control problem
with a nonstandard state constraint. We approach this problem by a penalty
method: we consider a general constrained problem and approximate the value
function with a sequence of value functions of appropriate unconstrained
problems with a penalization term in the objective functional. Coming back to
the case of swing contracts with strict constraints, we finally characterize
the value function as the unique viscosity solution with polynomial growth of
the Hamilton-Jacobi-Bellman equation subject to appropriate boundary
conditions.
