On \(q\)-optimal martingale measures in exponential Lévy models
The authors introduce a sufficient condition to identify the \(q\)-optimal signed martingale measure or the \(q\)-optimal absolutely continuous martingale measure in exponential Lévy models. The proof is based on a verification procedure in terms of a hedging problem, which holds true in a general semimartingale setting and is based on duality for convex optimization. Some explicit results are obtained in the one-dimensional case for the stock models with \(p\)-th moment, where \(p\) is the conjugate exponent to \(q\). The convergence of the \(q\)-optimal signed, respectively absolutely continuous, martingale measures to the minimal entropy equivalent martingale measure for one-dimensional exponential Lévy processes is studied. One of the theorems entails, for instance, the convergence under the assumption of bounded upward jump heights without any assumptions on the activity of jumps. If the positivity condition for the measure fails and the \(q\)-optimal measures are absolutely continuous, resp., signed, this theorem provides the first convergence result to the minimal entropy measure for exponential Lévy processes. Finally, the convergence results for the measures are applied to establish the convergence of an approximating sequence of optimal investment problems (with and without consumption) to the exponential utility maximization problem.
scientific article; zbMATH DE number 5520650
scientific article