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Sample Path Behaviour of Wiener and Cauchy Processes

Profile image of Stanislav KeprtaStanislav Keprta

1998

Abstract

functions (cf. Révész (1990), pp. 33–34). Definition 1.1. The function ψ belongs to the upper-upper class of {V (t); t ≥ 0} (ψ(t) ∈ UUC(V (t))) if for almost all ω ∈ Ω there exists a t0 = t0(ω) > 0 such that V (t) < ψ(t) for all t > t0. Definition 1.2. The function ψ belongs to the upper-lower class of {V (t); t ≥ 0} (ψ(t) ∈ ULC(V (t))) if for almost all ω ∈ Ω there exists a sequence of positive numbers 0 < t1 = t1(ω) < t2 = t2(ω) < · · · with tn → ∞ such that V (ti) ≥ ψ(ti), i = 1, 2, . . . . Definition 1.3. The function ψ belongs to the lower-upper class of {V (t); t ≥ 0} (ψ(t) ∈ LUC(V (t))) if for almost all ω ∈ Ω there exists a sequence of positive numbers 0 < t1 = t1(ω) < t2 = t2(ω) < · · · with tn → ∞ such that V (ti) ≤ ψ(ti), i = 1, 2, . . . . Definition 1.4. The function ψ belongs to the lower-lower class of {V (t); t ≥ 0} (ψ(t) ∈ LLC(V (t))) if for almost all ω ∈ Ω there exists a t0 = t0(ω) > 0 such that V (t) > ψ(t) for all t > t0. Assum...

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References (13)

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  13. Spitzer, F. (1958) Some Theorems Concerning 2-dimensional Brownian Motion, Trans. Amer. Math. Soc., vol. 87, pp. 187-197.

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