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We show how a stochastic version of the Lagrange multiplier method can becombined with the stochastic maximum principle for jump diffusions to solvecertain constrained stochastic optimal control problems. Two different terminalconstraints are considered; one constraint holds in expectation and the otheralmost surely.  As an application of this method, we study the effects of inflation- and wagerisk on optimal consumption. To do this, we consider the optimal consumptionproblem for a budget constrained agent with a L\'evy income process andstochastic inflation. The agent must choose a consumption path such that hiswealth process satisfies the terminal constraint. We find expressions for theoptimal consumption of the agent in the case of CRRA utility, and give aneconomic interpretation of the adjoint processes.

Stochastic maximum principle with Lagrange multipliers and optimal consumption with L\'evy wage