Optimal portfolio selection in a Lévy market with uncontrolled cash flow and only risky assets

Abstract

This article considers an investor who has an exogenous cash flow evolving according to a Lévy process and invests in a financial market consisting of only risky assets, whose prices are governed by exponential Lévy processes. Two continuous-time portfolio selection problems are studied for the investor. One is a benchmark problem, and the other is a mean-variance problem. The first problem is solved by adopting the stochastic dynamic programming approach, and the obtained results are extended to the second problem by employing the duality theory. Closed-form solutions of these two problems are derived. Some existing results are found to be special cases of our results.

Keywords:

• benchmark and mean-variance criteria
• Lévy processes
• Hamilton–Jacobi–Bellman equation
• exogenous cash flow
• duality theory

Acknowledgements

This research is supported by grants of the National Science Foundation for Distinguished Young Scholars of China (No. 70825002), National Natural Science Foundation of China (Nos. 71231008 and 71201173), Humanity and Social Science Foundation of Ministry of Education of China (Nos. 12YJCZH267 and 12YJCZH219), Philosophy and Social Science Programming Foundation of Guangdong Province (No. GD11YYJ07), Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (2012WYM_001), Fundamental Research Funds for the Central Universities (13wkpy28), and ‘985 Project’ of Sun Yat-sen University.