  We show how a stochastic version of the Lagrange multiplier method can be
combined with the stochastic maximum principle for jump diffusions to solve
certain constrained stochastic optimal control problems. Two different terminal
constraints are considered; one constraint holds in expectation and the other
almost surely.
  As an application of this method, we study the effects of inflation- and wage
risk on optimal consumption. To do this, we consider the optimal consumption
problem for a budget constrained agent with a L\'evy income process and
stochastic inflation. The agent must choose a consumption path such that his
wealth process satisfies the terminal constraint. We find expressions for the
optimal consumption of the agent in the case of CRRA utility, and give an
economic interpretation of the adjoint processes.
