Zero-One Laws and the Minimum of a Markov Process

Millar, P. W.

doi:10.2307/1997959

Cited by 32 publications

(48 citation statements)

References 11 publications

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“…For this to occur X i must be able to achieve its infimum at an exponential time or X j must be able to achieve its infimum at 0. According to [22,Prop. 2.1] there are three (non-exclusive) cases: Figure 2.…”

Section: Splitting and Conditioningmentioning

confidence: 99%

“…Proof of Proposition 3.1. Splitting at the infimum for MAPs can be proven along the lines of [22] or [11], and so we keep the proof rather brief.…”

Section: Figure 2 Scenarios Of Phase Switch At the Infimummentioning

confidence: 99%

“…It provides a valuable insight in the behaviour of the process and proves to be useful in various computations. In particular, the celebrated Wiener-Hopf factorization is just an application of splitting at the infimum, see [22,11] or [4,Lem. VI.6].…”

Section: Introductionmentioning

confidence: 99%

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Splitting and time reversal for Markov additive processes

Ivanovs

2017

Stochastic Processes and their Applications

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Abstract. We consider a Markov additive process with a finite phase space and study its path decompositions at the times of extrema, first passage and last exit. For these three families of times we establish splitting conditional on the phase, and provide various relations between the laws of post-and pre-splitting processes using time reversal. These results offer valuable insight into behaviour of the process, and while being structurally similar to the Lévy process case, they demonstrate various new features. As an application we formulate the Wiener-Hopf factorization, where time is counted in each phase separately and killing of the process is phase dependent. Assuming no positive jumps, we find concise formulas for these factors, and also characterize the time of last exit from the negative half-line using three different approaches, which further demonstrates applicability of path decomposition theory.

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Section: Splitting and Conditioningmentioning

confidence: 99%

“…Proof of Proposition 3.1. Splitting at the infimum for MAPs can be proven along the lines of [22] or [11], and so we keep the proof rather brief.…”

Section: Figure 2 Scenarios Of Phase Switch At the Infimummentioning

confidence: 99%

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Splitting and time reversal for Markov additive processes

Ivanovs

2017

Stochastic Processes and their Applications

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“…In this part, we provide two proofs for (1) of Theorem 1.6, though similar, from different viewpoints. The first proof is based on a fluctuation version of Williams' path decomposition of Brownian motion, originally due to Williams [84], and later extended in various ways by Millar [59,60], and Greenwood and Pitman [33]. We also refer readers to Pitman and Winkel [71] for a combinatorial explanation and various applications.…”

Section: 1mentioning

confidence: 99%

Patterns in Random Walks and Brownian Motion

Pitman¹,

Tang²

2015

Lecture Notes in Mathematics

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We ask if it is possible to find some particular continuous paths of unit length in linear Brownian motion. Beginning with a discrete version of the problem, we derive the asymptotics of the expected waiting time for several interesting patterns. These suggest corresponding results on the existence/non-existence of continuous paths embedded in Brownian motion. With further effort we are able to prove some of these existence and non-existence results by various stochastic analysis arguments. A list of open problems is presented.

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“…As we assume that θ has a finite mean, the post-maximum process has the same law as −X conditioned to stay positive (see for example Proposition 4.4 in [27] or Theorem 3.1 in [6]). Therefore, ζ(x) is the last passage time over the level x of −X conditioned to stay positive.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Predicting the Time at Which a Lévy Process Attains Its Ultimate Supremum

Baurdoux

Schaik

2014

Acta Appl Math

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We consider the problem of finding a stopping time that minimises the L 1 -distance to θ, the time at which a Lévy process attains its ultimate supremum. This problem was studied in [13] for a Brownian motion with drift and a finite time horizon. We consider a general Lévy process and an infinite time horizon (only compound Poisson processes are excluded. Furthermore due to the infinite horizon the problem is interesting only when the Lévy process drifts to −∞). Existing results allow us to rewrite the problem as a classic optimal stopping problem, i.e. with an adapted payoff process. We show the following. If θ has infinite mean there exists no stopping time with a finite L 1 -distance to θ, whereas if θ has finite mean it is either optimal to stop immediately or to stop when the process reflected in its supremum exceeds a positive level, depending on whether the median of the law of the ultimate supremum equals zero or is positive. Furthermore, pasting properties are derived. Finally, the result is made more explicit in terms of scale functions in the case when the Lévy process has no positive jumps.

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Zero-One Laws and the Minimum of a Markov Process

Cited by 32 publications

References 11 publications

Splitting and time reversal for Markov additive processes

Splitting and time reversal for Markov additive processes

Patterns in Random Walks and Brownian Motion

Predicting the Time at Which a Lévy Process Attains Its Ultimate Supremum

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