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Levy Measure
Stochastic Process
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On the $\gamma$Variation of Processes with Stationary Independent Increments
On the $\gamma$Variation of Processes with Stationary Independent Increments,10.1214/aoms/1177692473,The Annals of Mathematical Statistics,Itrel Monro
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On the $\gamma$Variation of Processes with Stationary Independent Increments
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Citations: 15
)
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Itrel Monroe
Let $\{X_t; t \geqq 0\}$ be a
stochastic process
in $R^N$ defined on the probability space $(\Omega, \mathscr{F}, \mathbf{P})$ which has stationary independent increments. Let $\nu$ be the
Levy measure
for $X_t$ and let $\beta = \inf\{\alpha > 0: \int_{x < 1}x^\alpha\nu(dx) < \infty\}$. For each $\omega \in \Omega$, let $V_\gamma(\mathbf{X}(\bullet, \omega); a, b) = \sup \sum^m_{j=1} X(t_j, \omega)  X(_{t1}, \omega)^\gamma$ where the supremum is over all finite subdivisions $a = t_0 < t_1 < \cdots < t_m = b$. Then if $\gamma > \beta, \mathbf{P}\{\mathbf{V}_\gamma(\mathbf{X}(\bullet, \omega); a, b) < \infty\} = 1$.
Journal:
The Annals of Mathematical Statistics
, vol. 43, no. 1972, pp. 12131220, 1972
DOI:
10.1214/aoms/1177692473
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Citation Context
(5)
...The results of Blumenthal and Getoor [5, Theorems 4.1, 4.2] and Monroe [
23, Theorem 2
] combine to provide a similar result for normalized Lévy processes...
...Theorem 6.4 [5,
23
] Let X ={ Xt ,t ≥ 0} be a normalized Lévy process in R d . Then...
Jamison Wolf
.
Random Fractals Determined by Lévy Processes
...Theorem 1.5 [
15, Theorem 2
] Let (Xt)t≥0 be a L´evy process in Rn without a...
...[
15, Theorem 1
] shows that the index of the process τ(s) is half that of the...
David R. E. Williams
.
Pathwise solutions of SDE's driven by Levy processes
...Remark 4.1 In [
13
] the characterisation of the sample path pvariation of all L´evy processes is completed...
David R. E. Williams
.
Diffeomorphic flows driven by Levy processes
...ments of a stochastically continuous process with stationary independent increments, the behavior of V~(g, S) has been extensively studied in [3, 8, 9, 11, 12, 16] and [
17
]...
Patrick L. Brockett
.
Variational sums of infinitesimal systems
...Getoor [3] and Monroe [
19
] on "strong" variation generalize in an obvious way to ich...
Olav Kallenberg
.
Path properties of processes with independent and interchangeable incr...
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Citations
(15)
Random Fractals Determined by Lévy Processes
Jamison Wolf
Journal:
Journal of Theoretical Probability  J THEOR PROBABILITY
, vol. 23, no. 4, pp. 11821203, 2010
First order p variations and Besov spaces
(
Citations: 2
)
Mathieu Rosenbaum
Journal:
Statistics & Probability Letters  STAT PROBAB LETT
, vol. 79, no. 1, pp. 5562, 2009
Generalized fractional OrnsteinUhlenbeck processes
Kotaro Endo
,
Muneya Matsui
Published in 2008.
Rough functions: p Variation, calculus, and index estimation
(
Citations: 2
)
R. Norvaiša
Journal:
Lithuanian Mathematical Journal  LITH MATH J
, vol. 46, no. 1, pp. 102128, 2006
Quadratic variation, pvariation and integration with applications to stock price modelling
Rimas Norvaisa
Published in 2001.