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Muckenhoupt's $(A_p)$ condition and the existence of the optimal martingale measure

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  • Dmitry Kramkov
  • Kim Weston

Abstract

In the problem of optimal investment with utility function defined on $(0,\infty)$, we formulate sufficient conditions for the dual optimizer to be a uniformly integrable martingale. Our key requirement consists of the existence of a martingale measure whose density process satisfies the probabilistic Muckenhoupt $(A_p)$ condition for the power $p=1/(1-a)$, where $a\in (0,1)$ is a lower bound on the relative risk-aversion of the utility function. We construct a counterexample showing that this $(A_p)$ condition is sharp.

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  • Dmitry Kramkov & Kim Weston, 2015. "Muckenhoupt's $(A_p)$ condition and the existence of the optimal martingale measure," Papers 1507.05865, arXiv.org.
    • Handle: RePEc:arx:papers:1507.05865
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    Cited by:

    1. Gu, Lingqi & Lin, Yiqing & Yang, Junjian, 2016. "On the dual problem of utility maximization in incomplete markets," Stochastic Processes and their Applications, Elsevier, vol. 126(4), pages 1019-1035.

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