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Occupation times and beyond

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  • Yang, Ming

Abstract

Let Xt be a continuous local martingale satisfying X0=0 and K1q(t)[less-than-or-equals, slant] t[less-than-or-equals, slant]K2q(t) a.s. for a nondecreasing function q with constants K1 and K2. Define for a Borel function and Mt*=sup0[less-than-or-equals, slant]s[less-than-or-equals, slant]t Ms. If f is in L2 and f[not equal to]0 then for any slowly increasing function [phi] there exist two positive constants c and C such that for all stopping times TSuppose that f2 is even and is moderate. If [phi] satisfies one of the 3 conditions: (i) [phi] is slowly increasing, (ii) [phi] is concave if f[negated set membership]L2, and (iii) [phi] is moderate if is convex, then there exist two positive constants c and C such that for all stopping times TDefine Tr=inf {t>0; Mt=r}, r>0. The growth rate function of ETr[gamma] can be found for appropriate [gamma], as an application of the above inequalities. The method of proving the main result also yields a similar type of two-sided inequality for the integrable Brownian continuous additive functional over all stopping times.

Suggested Citation

  • Yang, Ming, 2002. "Occupation times and beyond," Stochastic Processes and their Applications, Elsevier, vol. 97(1), pages 77-93, January.
  • Handle: RePEc:eee:spapps:v:97:y:2002:i:1:p:77-93
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    References listed on IDEAS

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    1. Novikov, Alexander & Valkeila, Esko, 1999. "On some maximal inequalities for fractional Brownian motions," Statistics & Probability Letters, Elsevier, vol. 44(1), pages 47-54, August.
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    Cited by:

    1. Michael J. Klass & Ming Yang, 2012. "Maximal Inequalities for Additive Processes," Journal of Theoretical Probability, Springer, vol. 25(4), pages 981-1012, December.

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