Abstract
The main theorem of the paper is that, for a large class of one-dimensional diffusions (i. e. strong Markov processes with continuous sample paths): if x(t) is a continuous stochastic process possessing the hitting probabilities and mean exit times of the given diffusion, then x(t) is Markovian, with the transition probabilities of the diffusion.
For a diffusion x(t) with natural boundaries at ± ∞, there is constructed a sequence π n (t, x) of functions with the property that the π n (t, x (t)) are martingales, reducing in the case of the Brownian motion to the familiar martingale polynomials.
It is finally shown that if a stochastic process x (t) is a martingale with continuous paths, with the additional property that
is a martingale, then x(t) is a diffusion with generator D mD+ and natural boundaries at ± ∞. This generalizes a martingale characterization given by Lévy for the Brownian motion.
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A revised version of the author's M. I. T. Ph. D. Thesis. I am very grateful to Professor Henry Mckean for all that he contributed to the thesis.
This work has been supported in part by the U.S.Army Signal Corps., the Air Force Office of Scientific Research, and the Office of Naval Research; in part by the National Institutes of Health (Grant NB-01865-05); and in part by the U.S. Air Force (ASD Contract AF 33(616)-7783).
Forwarding Address: 521 J, Engineering Mechanics, Stanford, California 94305 U.S.A.
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Arbib, M.A. Hitting and martingale characterizations of one-dimensional diffusions. Z. Wahrscheinlichkeitstheorie verw Gebiete 4, 232–247 (1965). https://doi.org/10.1007/BF00533754
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DOI: https://doi.org/10.1007/BF00533754