The Wald Martingale and the Maximum

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


We introduce the first of our two key martingales and consider two immediate applications. In the first application, we will use the martingale to construct a change of measure and thereby consider the dynamics of X under the new law. In the second application, we shall use the martingale to study the law of the process \(\overline{X}= \{\overline{X}_{t}\,\colon t\geq 0\}\), where
$$\overline{X}_t =\sup_{s\leq t}X_s, \quad t\geq 0. $$
In particular, we shall discover that the position of the trajectory of \(\overline{X}\), when sampled at an independent and exponentially distributed time, is again exponentially distributed.


  1. Bingham, N.H.: Fluctuation theory in continuous time. Adv. Appl. Probab. 7, 705–766 (1975) MathSciNetCrossRefzbMATHGoogle Scholar
  2. Esscher, F.: On the probability function in the collective theory of risk. Skandinavisk Aktuarietidskrift 15, 175–195 (1932) Google Scholar
  3. Gerber, H.U.: When does the surplus reach a given target? Insurance Math. Econom. 9, 115–119 (1990) MathSciNetCrossRefzbMATHGoogle Scholar
  4. Gerber, H.U., Shiu, E.S.W.: Option pricing by Esscher transforms. Trans. - Soc. Actuar. 46, 99–191 (1994) Google Scholar
  5. Kyprianou, A.E.: Fluctuations of Lévy Processes with Applications, Introductory Lectures, 2nd edn. Springer, Heidelberg (2013) zbMATHGoogle Scholar
  6. Wald, A.: On cumulative sums of random variables. Ann. Math. Stat. 15, 283–296 (1944) MathSciNetCrossRefzbMATHGoogle Scholar
  7. Wald, A.: Some generalizations of the theory of cumulative sums of random variables. Ann. Math. Stat. 16, 287–293 (1945) MathSciNetCrossRefzbMATHGoogle Scholar

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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