Hitting and martingale characterizations of one-dimensional diffusions |
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Authors: | Michael A Arbib |
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Institution: | 1. Department of Mathematics and Research Laboratory of Electronics, Massachusetts Institute of Technology, USA
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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 $$\mathop \smallint \limits_0^{x\left( t \right)} m\left( {0,y} \right]dy - t$$ |
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