# Characterization of the Levy Measures of Inverse Local Times of Gap Diffusion

• Frank B. Knight
Chapter
Part of the Progress in Probability and Statistics book series (PRPR, volume 1)

## Abstract

Let X(t) be a persistent nonsingular diffusion on an interval Q containing 0 (in the sense of [4]). If we assume a natural scale, then X is characterized by its speed measure m(dx) on Q, and finite endpoints are reflecting. There exists (P0-a.s.) the local time
$$l(t) = \mathop{{\lim }}\limits_{{{h_i} \to 0 + }} \frac{1}{{m[ - {h_1},{h_2})}}\int\limits_0^t {{I_{{[ - {h_1},{h_2})}}}(X(s))ds}$$
(1.1)
which is continuous in t ≥ 0. The right-continuous inverse ℓ(-1)(t) = inf{s: ℓ(s) > t} is an increasing finite process with homogeneous independent increments, and as such (see for example [4, §6.1]) it is characterized by its Lévy measure n(dy) on (0,∞) through the equation
$$E\exp [ - \lambda {l^{{( - 1)}}}(t)] = \exp [ - t{m_0}\lambda - t\int\limits_0^{\infty } {(1 - {e^{{ - \lambda y}}})n(dy)],\lambda > 0}$$
where m0 = m{0}. An interesting problem [4, p. 217] is to characterize the class of all n(dy) which can appear when Q and m(dx) vary. It is shown there that there is a unique measure µ on [0,∞) such that $$n(dy) = dy\int_{{0 - }}^{\infty } {{e^{{ - y\gamma }}}\mu (d\gamma )}$$, so the problem reduces to characterizing the class of µ.

## Keywords

Local Time Speed Measure Natural Scale Negative Excursion Hunt Process
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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