Optimal Portfolio for CRRA Utility Functions when Risky Assets are Exponential Additive Processes

Abstract

In this paper, we analyse a market where the risky assets follow exponential additive processes, which can be viewed as time‐inhomogeneous generalizations of geometric Levy processes. In this market we show that, when an investor wants to maximize a CRRA utility function of his/her terminal wealth, his/her optimal strategy consists in keeping proportions of wealth in the risky assets which depend only on time but not on the current wealth level or on the prices of the risky assets. In the time‐homogeneous case, the optimal strategy is to keep constant proportions of wealth, a result already found by Kallsen which extends the classical Merton’s result to this market. While the one‐dimensional case has been extensively treated and the multidimensional case has been treated only in the time‐homogeneous case Callegaro and Vargiolu (2009), Kallsen (2000), and Korn et al. (2003) to the authors’ knowledge this is the first time that such results are obtained for exponential additive processes in the multidimensional case. We use these results to show that the optimal solution in the presence of jumps has the form of the analogous one without jumps but with the asset yields vector reduced by suitable quantities: in the one‐dimensional case, we extend a result by Benth et al. (2001). We conclude with four examples.