Typically, situations in which investors receive random endowments in addition to their standard wealth process turn out to be extremely challenging in terms of the mathematical analysis. This can also be seen in the paper „Optimal Investment with Time-Varying Stochastic Endowments “, where the authors approach the merton problem (utility maximization problem) for an investor receiving an additional random endowment with stochastic controls and the HJB equation. Even though this is a classical approach for dealing with the merton problem, it can be seen that very complex and technical arguments are needed to ensure a rigorous argumentation to reach the desired results. The idea of this master thesis now was to solve this problem with the so-called martingale method, hoping to have a more straight forward way with less technically constructed arguments. With the martingale method one splits up a utility maximization problem into two seperate problems: In a first part one actually applies the martingale method to determine the in terms of utility optimal achievable terminal payoff, as well as the existence of an admissible replicating strategy. This simplifies the optimization problem enormously, as instead of dealing with a complex stochastic process, one just solves a static optimization problem. As the martingale method just gives the existence of an optimal strategy, in a second step a concrete formula for that strategy is determined. The master’s thesis shows that the martingale method turns out to also beautifully work for the merton problem with random endowment. In opposite to the stochastic control approach, the argumentation is very straight forward and breaks down the complex incorporation of the random endowment into rather simple and understandable mathematics.     «

Typically, situations in which investors receive random endowments in addition to their standard wealth process turn out to be extremely challenging in terms of the mathematical analysis. This can also be seen in the paper „Optimal Investment with Time-Varying Stochastic Endowments “, where the authors approach the merton problem (utility maximization problem) for an investor receiving an additional random endowment with stochastic controls and the HJB equation. Even though this is a classical a...     »