Optimal consumption problems in discontinuous markets

• Relatore: Giorgia Callegaro

Abstract: we study an extension of Merton’s classical portfolio optimization problem (1969-1970) to a particular case of (complete) discontinuous market, with a single jump. The market consists of a non-risky asset, a "standard risky" asset and a risky asset with discontinuous price dynamics (e.g., a defaultable bond or a mortality linked security).

We consider three different problems of maximization of the expected utility from consumption, in the cases when the investment horizon is fixed, when it is finite, but possibly uncertain and when it is infinite.
We solve the problems by means of the martingale approach in a general stochastic coefficients model, in which, in the logarithmic utility case, we characterize the optimal investment-consumption strategy. Furthermore, we compare the optimal consumption rates for the three different problems, finding quite intuitive results.
In the constant and deterministic coefficients’ case explicit solutions are also obtained in the power and exponential utility cases. As alternative technique, the Dynamic Programming approach is profitably used to obtain the optimal solutions.

Referente:
Luca Di Persio