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On Maximal Inequalities for some Jump Processes

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  • Pavel V. Gapeev

Abstract

We present a solution to the considered in [5] and [22] optimal stopping problem for some jump processes. The method of proof is based on reducing the initial problem to an integro-differential free-boundary problem where the normal reflection and smooth fit may break down and the latter then be replaced by the continuous fit. The derived result is applied for determining the best constants in maximal inequalities for a compound Poisson process with linear drift and exponential jumps.

Suggested Citation

  • Pavel V. Gapeev, 2006. "On Maximal Inequalities for some Jump Processes," SFB 649 Discussion Papers SFB649DP2006-060, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
  • Handle: RePEc:hum:wpaper:sfb649dp2006-060
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    File URL: http://sfb649.wiwi.hu-berlin.de/papers/pdf/SFB649DP2006-060.pdf
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    References listed on IDEAS

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    1. S. G. Kou & Hui Wang, 2004. "Option Pricing Under a Double Exponential Jump Diffusion Model," Management Science, INFORMS, vol. 50(9), pages 1178-1192, September.
    2. Gapeev Pavel V. & Kühn Christoph, 2005. "Perpetual convertible bonds in jump-diffusion models," Statistics & Risk Modeling, De Gruyter, vol. 23(1/2005), pages 15-31, January.
    3. Ernesto Mordecki, 1999. "Optimal stopping for a diffusion with jumps," Finance and Stochastics, Springer, vol. 3(2), pages 227-236.
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    More about this item

    Keywords

    Jump process; stochastic differential equation; maximum process; optimal stopping problem; compound Poisson process; Ito’s formula; integro-differential free-boundary problem; normal reflection; continuous and smooth fit; maximality principle; maximal inequalities;

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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