The Kella–Whitt Martingale and the Minimum

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


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.


  1. Azéma, J., Yor, M.: Une solution simple au problème de Skorokhod. In: Séminaire de Probabilités XIII. Lecture Notes in Mathematics, vol. 721, pp. 91–115. Springer, Berlin (1979) Google Scholar
  2. Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge (1996) zbMATHGoogle Scholar
  3. Bingham, N.H.: Fluctuation theory in continuous time. Adv. Appl. Probab. 7, 705–766 (1975) MathSciNetCrossRefzbMATHGoogle Scholar
  4. Feller, W.: An Introduction to Probability Theory and Its Applications, vol II, 2nd edn. Wiley, New York (1971) zbMATHGoogle Scholar
  5. Kella, O., Whitt, W.: Useful martingales for stochastic storage processes with Lévy input. J. Appl. Probab. 29, 396–403 (1992) MathSciNetCrossRefzbMATHGoogle Scholar
  6. Kennedy, D.: Some martingales related to cumulative sum tests and single-server queues. Stoch. Proc. Appl. 4, 261–269 (1976) CrossRefzbMATHGoogle Scholar
  7. Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion, 3rd edn. Springer, Berlin (2004) Google Scholar

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© 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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