Bounding the Maximal Height of a Diffusion by the Time Elapsed

Goran Peskir ()
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Goran Peskir: University of Aarhus

Journal of Theoretical Probability, 2001, vol. 14, issue 3, 845-855

Abstract: 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.

Keywords: diffusion process; stopping time; maximal inequality; Lenglart's domination principle; Brownian motion; stochastic differential equation; scale function; speed measure (search for similar items in EconPapers)
Date: 2001
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Citations: View citations in EconPapers (3)

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DOI: 10.1023/A:1017505509361

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