## Abstract

We investigate some portfolio problems that consist of maximizing expected terminal wealth under the constraint of an upper bound for the risk, where we measure risk by the variance, but also by the Capital-at-Risk (CaR). The solution of the mean-variance problem has the same structure for any price process which follows an exponential Lévy process. The CaR involves a quantile of the corresponding wealth process of the portfolio. We derive a weak limit law for its approximation by a simpler Levy process, often the sum of a drift term, a Brownian motion and a compound Poisson process. Certain relations between a Lev́y process and its stochastic exponential are investigated.

Original language | English |
---|---|

Pages (from-to) | 17-44 |

Number of pages | 28 |

Journal | Finance and Stochastics |

Volume | 8 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2004 |

## Keywords

- Capital-at-risk
- Downside risk measure
- Exponential Lévy process
- Portfolio optimization
- Stochastic exponential
- Value-at-risk
- Weak limit law for Lévy processes