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State-dependent utility maximization in L\'evy markets

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  • Jose E. Figueroa-Lopez
  • Jin Ma
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    Abstract

    We revisit Merton's portfolio optimization problem under boun-ded state-dependent utility functions, in a market driven by a L\'evy process $Z$ extending results by Karatzas et. al. (1991) and Kunita (2003). The problem is solved using a dual variational problem as it is customarily done for non-Markovian models. One of the main features here is that the domain of the dual problem enjoys an explicit "parametrization", built on a multiplicative optional decomposition for nonnegative supermartingales due to F\"ollmer and Kramkov (1997). As a key step in obtaining the representation result we prove a closure property for integrals with respect to Poisson random measures, a result of interest on its own that extends the analog property for integrals with respect to a fixed semimartingale due to M\'emin (1980). In the case that (i) the L\'evy measure of $Z$ is atomic with a finite number of atoms or that (ii) $\Delta S_{t}/S_{t^{-}}=\zeta_{t} \vartheta(\Delta Z_{t})$ for a process $\zeta$ and a deterministic function $\vartheta$, we explicitly characterize the admissible trading strategies and show that the dual solution is a risk-neutral local martingale.

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    File URL: http://arxiv.org/pdf/0901.2070
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 0901.2070.

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    Date of creation: Jan 2009
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    Handle: RePEc:arx:papers:0901.2070

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