Adjoint Equations in Stochastic Optimal Control and Application to Portfolio Optimization with Borrowing Constraints

The paper derives the adjoint/Euler equation for the co-state process of optimal control of diffusions in both integral and differential forms, as a part of the Stochastic Maximum Principle. The result is applied to the consumption and portfolio selection problems with statevariable-dependen t utility functions, where tradi tional Con sum ptionbased Capital Asset Pricing Model techniques do not work. The method is illustrated on the example of portfolio selection with restrictions on the short position size, particularly the dynamics of exchange rates in a world of liquidity-valuing international investors.