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Renewal theorems and stability for the reflected process

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  • Doney, Ron
  • Maller, Ross
  • Savov, Mladen

Abstract

Renewal-like results and stability theorems relating to the large-time behaviour of a random walk Sn reflected in its maximum, Rn=max0 [infinity]ER[tau](r)/r=1 is shown to hold when EX [infinity] and limr-->[infinity]R[tau](r)/r=1 almost surely (a.s.); alternatively expressed, the overshoot R[tau](r)-r is o(r) as r-->[infinity], in probability or a.s. Comparisons are also made with exit times of the random walk Sn across both two-sided and one-sided horizontal boundaries.

Suggested Citation

  • Doney, Ron & Maller, Ross & Savov, Mladen, 2009. "Renewal theorems and stability for the reflected process," Stochastic Processes and their Applications, Elsevier, vol. 119(4), pages 1270-1297, April.
    • Handle: RePEc:eee:spapps:v:119:y:2009:i:4:p:1270-1297
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    References listed on IDEAS

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    1. Johansen, Anders, 2003. "Characterization of large price variations in financial markets," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 324(1), pages 157-166.
    2. Asmussen, Søren & Avram, Florin & Pistorius, Martijn R., 2004. "Russian and American put options under exponential phase-type Lévy models," Stochastic Processes and their Applications, Elsevier, vol. 109(1), pages 79-111, January.
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