Optimal Investment Strategy under Lévy ambiguity

This paper examines an optimal investment problem of Abel and Eberly (1997) and Imai and Tsujimura (2016) under higher degree of ambiguity. To that end we introduces an exponential Lévy process as the underlying risk process of the project. The ambiguity indicates a manager’s disconfidence with respect to the underlying model. It can be formulated as allowing one to change the reference probability measure into a different equivalent probability measure. The difference between the reference measure and another equivalent measure indicates the manager’s misspecification of the underlying model. In the formulation, on the one hand, the firm’s manager chooses the investment level to maximize the firm’s value. On the other hand, Nature chooses the equivalent probability measure so that the firm’s value is minimized. Consequently, the optimal investment problem developed in this paper can be formulated as a maxmin expected utility problem. It is crucial to notice that an adoption of a Lévy process enables us express higher degree of ambiguity, that is, unlike the diffusion process, when the underlying asset follows an exponential Lévy process, a change of measure can affect not only its drift term of the diffusion but also its variance, skewness and kurtosis via changing the jump structure of the original Lévy process. Consequently, in this paper, we can express higher level of ambiguity, i.e, our model can describe potential misspecification with respect to its higher moments of the distribution.