2004
DOI: 10.1214/009117904000000423
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p-variation of strong Markov processes

Abstract: Let ξt, t ∈ [0, T ], be a strong Markov process with values in a complete separable metric space (X, ρ) and with transition probability functionIt is shown that a certain growth condition on α(h, a), as a ↓ 0 and h stays fixed, implies the almost sure boundedness of the p-variation of ξt, where p depends on the rate of growth.

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Cited by 20 publications
(33 citation statements)
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“…This result is a generalisation of the main result of Manstavičius [32], which provides a criterion for a strong Markov process to have sample paths of a.s. finite p-variation. Our proof of Theorem 4.8 is a simplification of the stopping times technique adopted in [32].…”
Section: Introductionsupporting
confidence: 63%
See 1 more Smart Citation
“…This result is a generalisation of the main result of Manstavičius [32], which provides a criterion for a strong Markov process to have sample paths of a.s. finite p-variation. Our proof of Theorem 4.8 is a simplification of the stopping times technique adopted in [32].…”
Section: Introductionsupporting
confidence: 63%
“…In this section we give a criterion for p-variation tightness of strong Markov processes in a Polish space (Theorem 4.8), which is inspired by the work of Manstavičius [32].…”
Section: P-variation Tightness Of Strong Markov Processesmentioning
confidence: 99%
“…As Manstavičius [155] pointed out, the conclusion is sharp: it can fail when p = γ/β, for symmetric stable Lévy processes as will be shown in Remark 12.39. Theorem 12.27 (M. Manstavičius).…”
Section: Markov Processesmentioning
confidence: 88%
“…Since the process for given α does not have bounded p-variation for p = α, this shows that the assumption p > γ/β in Theorem 12.27 is sharp. Moreover, for this, according to Manstavičius [155], one does not need the full strength of Bretagnolle's theorem as quoted above; the earlier results of Blumenthal and Getoor [19] suffice.…”
Section: α-Stable Lévy Motionmentioning
confidence: 99%
“…Proof. Directly from Theorem 2 together with Theorem 1.3 in Manstavicius (2004). (2004) that for symmetric stable Lévy processes with index α ∈ (0, 2], the condition p > γ /β applies with γ /β = α.…”
Section: Markov Processesmentioning
confidence: 99%