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The Kella–Whitt Martingale and the Minimum

  • Andreas E. Kyprianou
Part of the EAA Series book series (EAAS)

Abstract

We move to the second of our two key martingales. In a similar spirit to Chapter  2, we shall use the martingale to study the law of the process \(\underline{X}= \{\underline{X}_{t}\,\colon t\geq 0\}\), where
$$\underline{X}_t : =\inf_{s\leq t}X_s, \quad t\geq 0. $$
As with the case of \(\overline{X}\), we characterise the law of \(\underline{X}\) when sampled at an independent and exponentially distributed time. Unlike the case of \(\overline{X}\) however, this will not turn out be exponentially distributed.

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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Andreas E. Kyprianou
    • 1
  1. 1.Department of Mathematical SciencesUniversity of BathBathUnited Kingdom

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