The Annals of Applied Probability

Portfolio choice with jumps: A closed-form solution

Yacine Aït-Sahalia, Julio Cacho-Diaz, and T. R. Hurd

Full-text: Open access


We analyze the consumption-portfolio selection problem of an investor facing both Brownian and jump risks. We bring new tools, in the form of orthogonal decompositions, to bear on the problem in order to determine the optimal portfolio in closed form. We show that the optimal policy is for the investor to focus on controlling his exposure to the jump risk, while exploiting differences in the Brownian risk of the asset returns that lies in the orthogonal space.

Article information

Ann. Appl. Probab., Volume 19, Number 2 (2009), 556-584.

First available in Project Euclid: 7 May 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62P05: Applications to actuarial sciences and financial mathematics 60J75: Jump processes
Secondary: 93E20: Optimal stochastic control

Optimal portfolio jumps Merton problem factor models closed-form solution


Aït-Sahalia, Yacine; Cacho-Diaz, Julio; Hurd, T. R. Portfolio choice with jumps: A closed-form solution. Ann. Appl. Probab. 19 (2009), no. 2, 556--584. doi:10.1214/08-AAP552.

Export citation


  • Aase, K. K. (1984). Optimum portfolio diversification in a general continuous-time model. Stochastic Process. Appl. 18 81–98.
  • Ang, A. and Bekaert, G. (2002). International asset allocation with regime shifts. Review of Financial Studies 15 1137–1187.
  • Ang, A. and Chen, J. (2002). Asymmetric correlations of equity portfolios. Journal of Financial Economics 63 443–494.
  • Bae, K.-H., Karolyi, G. A. and Stulz, R. M. (2003). A new approach to measuring financial contagion. Review of Financial Studies 16 717–763.
  • Choulli, T. and Hurd, T. R. (2001). The role of Hellinger processes in mathematical finance. Entropy 3 150–161.
  • Cvitanić, J., Polimenis, V. and Zapatero, F. (2008). Optimal portfolio allocation with higher moments. Annals of Finance 4 1–28.
  • Das, S. and Uppal, R. (2004). Systemic risk and international portfolio choice. Journal of Finance 59 2809–2834.
  • Emmer, S. and Klüppelberg, C. (2004). Optimal portfolios when stock prices follow an exponential Lévy process. Finance Stoch. 8 17–44.
  • Grauer, R. and Hakansson, N. (1987). Gains from international diversification: 1968–1985 returns on portfolios of stocks and bonds. Journal of Finance 42 721–739.
  • Han, S. and Rachev, S. (2000). Portfolio management with stable distributions. Math. Methods Oper. Res. 51 341–352.
  • Hartmann, P., Straetmans, S. and de Vries, C. (2004). Asset market linkages in crisis periods. Review of Economics and Statistics 86 313–326.
  • Jeanblanc-Picqué, M. and Pontier, M. (1990). Optimal portfolio for a small investor in a market model with discontinuous prices. Appl. Math. Optim. 22 287–310.
  • Kallsen, J. (2000). Optimal portfolios for exponential Lévy processes. Math. Methods Oper. Res. 51 357–374.
  • Liu, J., Longstaff, F. and Pan, J. (2003). Dynamic asset allocation with event risk. J. Finance 58 231–259.
  • Longin, F. and Solnik, B. (2001). Extreme correlation of international equity markets. J. Finance 56 649–676.
  • Madan, D. (2004). Equilibrium asset pricing with non-Gaussian factors and exponential utilities. Technical report, Univ. Maryland.
  • Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: The continuous-time case. Review of Economics and Statistics 51 247–257.
  • Merton, R. C. (1971). Optimum consumption and portfolio rules in a continuous-time model. J. Econom. Theory 3 373–413.
  • Ortobelli, S., Huber, I., Rachev, S. T. and Schwartz, E. S. (2003). Portfolio choice theory with non-Gaussian distributed returns. In Handbook of Heavy Tailed Distributions in Finance (S. T. Rachev, ed.) 547–594. Elsevier, Amsterdam.
  • Shirakawa, H. (1990). Optimal dividend and portfolio decisions with Poisson and diffusion-type return process. Technical report, Tokyo Institute of Technology.
  • Solnik, B. (1974). Why not diversify internationally rather than domestically? Financial Analysts Journal 30 48–53.