Occupation times and beyond

Ming Yang

Stochastic Processes and their Applications, 2002, vol. 97, issue 1, 77-93

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.

Keywords: Stopping; time; Local; time; Quasi-Gaussian; local; martingale; Moderate; function; Continuous; additive; functional; Exit; time; Decoupling; inequality; Ito's; formula (search for similar items in EconPapers)
Date: 2002
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Citations: View citations in EconPapers (1)

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