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Department of Mathematics pres.

Financial/Actuarial Mathematics Seminar

Optimal consumption with reference to past spending maximum

We investigate an infinite-horizon optimal consumption problem under exponential utility, together with nonnegativity constraint on consumption and the behavioral reference point to the consumption peak. The performance is measured by the distance between the current consumption rate and a fraction of the historical consumption maximum, which renders the control problem path dependent. To apply dynamic programming arguments, the consumption running maximum process is chosen as an auxiliary controlled state process. The associate Hamilton-Jacobi-Bellman (HJB) equation can be expressed in a piecewise manner in three different regions. By employing the dual transform of the two dimensional value function, we obtain the fully explicit classical solution of the dual PDE using endogenous boundary conditions and smooth-fit principle. The feedback optimal investment and consumption strategies are provided via a rigorous verification theorem. Speaker(s): Shuoqing Deng (UM)