Bernoulli

  • Bernoulli
  • Volume 12, Number 6 (2006), 955-969.

A complete characterization of local martingales which are functions of Brownian motion and its maximum

Jan Obloj

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Abstract

We prove the max-martingale conjecture of Obłój and Yor. We show that for a continuous local martingale ( N t:t0) and a function H :R×R +R , H (N t,sup s tN s) is a local martingale if and only if there exists a locally integrable function f such that H (x,y)= 0 yf(s)ds-f(y)(x-y)+H(0,0) . This readily implies, via Lévy's equivalence theorem, an analogous result with the maximum process replaced by the local time at 0 .

Article information

Source
Bernoulli, Volume 12, Number 6 (2006), 955-969.

Dates
First available in Project Euclid: 4 December 2006

Permanent link to this document
https://projecteuclid.org/euclid.bj/1165269146

Digital Object Identifier
doi:10.3150/bj/1165269146

Mathematical Reviews number (MathSciNet)
MR2274851

Zentralblatt MATH identifier
1130.60050

Keywords
Azéma-Yor martingales continuous martingales maximum process max-martingales Motoo's theorem

Citation

Obloj, Jan. A complete characterization of local martingales which are functions of Brownian motion and its maximum. Bernoulli 12 (2006), no. 6, 955--969. doi:10.3150/bj/1165269146. https://projecteuclid.org/euclid.bj/1165269146


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References

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