The arc-sine law and its analogs for processes governed by signed and complex measures

Hochberg, Kenneth J.; Orsingher, Enzo

doi:10.1016/0304-4149(94)90029-9

Cited by 35 publications

(40 citation statements)

References 7 publications

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“…For instance, let us quote the works of Beghin et al [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] and the references therein. We observe that (5) and (7) are closely related to the continuous -iterated Laplacian d 2 /d 2 .…”

Section: G ( ) =̃(mentioning

confidence: 99%

From Pseudorandom Walk to Pseudo-Brownian Motion: First Exit Time from a One-Sided or a Two-Sided Interval

Lachal

2014

International Journal of Stochastic Analysis

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Let be a positive integer, a positive constant and ( ) ≥1 be a sequence of independent identically distributed pseudorandom variables. We assume that the 's take their values in the discrete set {− , − + 1, . . . , − 1, } and that their common pseudodistribution is characterized by the (positive or negative) real numbers P{ = } = 0 + (−1) −1 ( 2 + ) for any ∈ {− , − + 1, . . . , − 1, }. Let us finally introduce ( ) ≥0 the associated pseudorandom walk defined on Z by 0 = 0 and = ∑ =1 for ≥ 1. In this paper, we exhibit some properties of ( ) ≥0 . In particular, we explicitly determine the pseudodistribution of the first overshooting time of a given threshold for ( ) ≥0 as well as that of the first exit time from a bounded interval. Next, with an appropriate normalization, we pass from the pseudorandom walk to the pseudo-Brownian motion driven by the high-order heat-type equation / = (−1) −1 2 / 2 . We retrieve the corresponding pseudodistribution of the first overshooting time of a threshold for the pseudo-Brownian motion (Lachal, 2007). In the same way, we get the pseudodistribution of the first exit time from a bounded interval for the pseudo-Brownian motion which is a new result for this pseudoprocess.

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Section: G ( ) =̃(mentioning

confidence: 99%

From Pseudorandom Walk to Pseudo-Brownian Motion: First Exit Time from a One-Sided or a Two-Sided Interval

Lachal

2014

International Journal of Stochastic Analysis

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“…It is worth mentioning the work by D. Levin and T. Lyons [45] on rough paths, conjecturing that the signed measure (with infinite total variation) associated to the pseudo-process could exist on the quotient space of equivalence classes of paths corresponding to different parametrizations of the same path. Properties of the pseudo-process X(t) associated with the signed measure P, corresponding to the fundamental solution of (8) via the identity P x (X(t) ∈ dy) = p(t, x, dy) were studied by several authors, in particular Hochberg [31], Orsingher [55,32,54], Lachal [43], Nishioka [56,57]. It shall be noticed that, in the case N = 4, paths of X(t) are not continuous.…”

Section: Introductionmentioning

confidence: 99%

Probabilistic representation formula for the solution of fractional high-order heat-type equations

2019

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We propose a probabilistic construction for the solution of a general class of fractional high order heat-type equations in the one-dimensional case, by using a sequence of random walks in the complex plane with a suitable scaling. A time change governed by a class of subordinated processes allows to handle the fractional part of the derivative in space. We first consider evolution equations with space fractional derivatives of any order, and later we show the extension to equations with time fractional derivative (in the sense of Caputo derivative) of order α ∈ (0, 1).

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“…Refs. [1,11,14]). On the other hand, we would mention here, with ordinary heat kernel, that there have been more investigations on (1) but with −∆ instead of ∆ 2 (see e.g.…”

Section: Introductionmentioning

confidence: 97%

Higher-order stochastic partial differential equations with branching noises

Wang

Yan

2007

Front. Math. China

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The arc-sine law and its analogs for processes governed by signed and complex measures

Cited by 35 publications

References 7 publications

From Pseudorandom Walk to Pseudo-Brownian Motion: First Exit Time from a One-Sided or a Two-Sided Interval

From Pseudorandom Walk to Pseudo-Brownian Motion: First Exit Time from a One-Sided or a Two-Sided Interval

Probabilistic representation formula for the solution of fractional high-order heat-type equations

Higher-order stochastic partial differential equations with branching noises

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