Sojourns and extremes of a diffusion process on a fixed interval

Abstract

Let X(t), , be an Ito diffusion process on the real line. For u > 0 and t > 0, let Lt (u) be the Lebesgue measure of the set . Limit theorems are obtained for (i) the distribution of Lt (u) for u → ∞and fixed t, and (ii) the tail of the distribution of the random variable max[0, t] X(s). The conditions on the process are stated in terms of the drift and diffusion coefficients. These conditions imply the existence of a stationary distribution for the process.

Strong Asymptotic Equivalence and Inversion of Functions in the Class Kc

“…The last condition as a Tauberian condition played an important role in papers [8,18,19] by Stanojević and Grow, then in papers [15][16][17] by Stadtmüller and Trautner, and in many other papers related to the background and the applications of the theory of regularly varying functions (see, e.g., [5]). In particular, Berman [3,4] called the continuous functions satisfying condition (Sch) "regularly oscillating" functions and found some important applications of such functions. Cline [6] also considered the condition (Sch) and developed the entire theory of functions satisfying this condition.…”

Precise Large Deviations for Sums of Random Variables with Consistently Varying Tails in Multi-Risk Models

“…The class C was also thoroughly studied by Berman (1982), who called it 'regular oscillation' and by Cline (1994), who called it 'intermediate regular variation'.…”

On some extensions of Karamata’s theory and their applications

“…PRV functions and their various applications have been studied by Korenblyum [36], Matuszewska [38], Matuszewska and Orlicz [39], Stadtmüller and Trautner [48], [49], Berman [4,5], Yakymiv [53,54], Cline [19], Klesov et al [33], Djurčić and Torgašev [21], Buldygin et al [9], [11]- [13], [15]- [18]. Note that PRV functions are called regularly oscillating in Berman [4], weakly oscillating in Yakymiv [53] and intermediate regularly varying in Cline [19].…”

“…Note that PRV functions are called regularly oscillating in Berman [4], weakly oscillating in Yakymiv [53] and intermediate regularly varying in Cline [19].…”