Optimal consumption with past maximum and habit formation
Abstract
In the first part of the talk, we consider the optimal consumption problem with reference to the past spending maximum. Under exponential utility function, by introducing the consumption running maximum as an auxiliary state process, we are able to solve explicitly the associated Hamilton-Jacobi-Bellman (HJB) equation in three different regions. The optimal investment and consumption strategies are provided in feedback form via a complete verification theorem. In the second part ofthe talk,we considerthe optimal consumption problemunderthe nonaddictive habitformation in incomplete semimartingale marketmodels. Additional difficulties arise due to the non-addictive nature of the habit and two Lagrange multipliers are introduced. The optimal stochastic Lagrange multiplier is constructed by the optimal solution froman auxiliary unconstrained dual problem, which generalizes the seminal work of Detemple and Karatzas from complete marketsto generalsemimartingale models.