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Bounding the Maximal Height of a Diffusion by the Time Elapsed

Author

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  • Goran Peskir

    (University of Aarhus
    University of Zagreb)

Abstract

Let X=(X t ) t≥0 be a one-dimensional time-homogeneous diffusion process associated with the infinitesimal generator $$\mathbb{L}_X = \mu (x)\frac{\partial }{{\partial x}} + \frac{{\sigma ^2 (x)}}{2}\frac{{\partial ^2 }}{{\partial x^2 }}$$ where x↦μ(x) and x↦σ(x)>0 are continuous. We show how the question of finding a function x↦H(x) such that $$c_1 E(H(\tau )) \leqslant E(\mathop {\max }\limits_{0 \leqslant {\text{t}} \leqslant \tau } \left| {X_t } \right|) \leqslant c_2 E(H(\tau ))$$ holds for all stopping times τ of X relates to solutions of the equation: $$\mathbb{L}_X (F) = 1$$ Explicit expressions for H are derived in terms of μ and σ. The method of proof relies upon a domination principle established by Lenglart and Itô calculus.

Suggested Citation

  • Goran Peskir, 2001. "Bounding the Maximal Height of a Diffusion by the Time Elapsed," Journal of Theoretical Probability, Springer, vol. 14(3), pages 845-855, July.
  • Handle: RePEc:spr:jotpro:v:14:y:2001:i:3:d:10.1023_a:1017505509361
    DOI: 10.1023/A:1017505509361
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    Cited by:

    1. Michael J. Klass & Ming Yang, 2012. "Maximal Inequalities for Additive Processes," Journal of Theoretical Probability, Springer, vol. 25(4), pages 981-1012, December.
    2. Chen Jia, 2019. "Sharp Moderate Maximal Inequalities for Upward Skip-Free Markov Chains," Journal of Theoretical Probability, Springer, vol. 32(3), pages 1382-1398, September.

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