MRL order, log-concavity and an application to peacocks

Bogso, Antoine-Marie

doi:10.1016/j.spa.2014.10.015

Cited by 6 publications

(7 citation statements)

References 20 publications

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“…In this paper, we provide a log-concavity characterization of the WDS ordering. This characterization is the same as that obtained in Bogso [4,Theorem 3.3] for the MRL ordering.…”

Section: Introductionsupporting

confidence: 80%

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Weak decreasing stochastic order

Bogso

Soh

2017

Statistics & Probability Letters

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We introduce the notion of weak decreasing stochastic (WDS) ordering for real-valued processes with negative means, which, to our knowledge, has not been studied before. Thanks to Madan-Yor's argument, it follows that the WDS ordering is a necessary and sufficient condition for a process with negative mean to be embeddable in a standard Brownian motion by the Cox and Hobson extension of the Azéma-Yor algorithm. Since the decreasing stochastic order is stronger than the WDS order, then, for every stochastically non-decreasing family of probability measures with densities, the Cox-Hobson stopping times provide an associated Markov process. The quantile process associated to a stochastically non-decreasing process is not necessarily Markovian.

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“…In this paper, we provide a log-concavity characterization of the WDS ordering. This characterization is the same as that obtained in Bogso [4,Theorem 3.3] for the MRL ordering.…”

Section: Introductionsupporting

confidence: 80%

“…Here is a characterization of WDS ordering in terms of log-concavity for probability measures with negative mean. This characterization is the same as that of the MRL ordering obtained in Bogso [4,Theorem 3.3].…”

Section: )supporting

confidence: 74%

Weak decreasing stochastic order

Bogso

Soh

2017

Statistics & Probability Letters

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“…There are many results in the literature related to the construction of self-similar processes. In Madan-Yor [23], Fan-Hamza-Klebaner [10], Hamza-Klebaner [13], Hirsch-Profeta-Roynette-Yor [15], Bogso [6] and Henry-Labordère-Tan-Touzi [14], the authors provided many constructions of self-similar martingales with given marginal distributions. In particular, Madan and Yor [23], Hamza and Klebaner [13], and Henry-Labordère, Tan and Touzi [14] exhibited several examples of discontinuous fake Brownian motions.…”

Section: Introductionmentioning

confidence: 99%

Self-similar martingales derived from Root embedding

Bogso¹,

Mbehou²

2019

Preprint

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Given a family (µ λ , λ ≥ 0) of integrable mean-zero probability measures such that, for every λ ≥ 0, µ λ is the image of µ1 under the homothety y −→ √ λy, we provide a necessary and sufficient condition on µ1 under which the Root embedding algorithm yields a self-similar martingale with one-dimensional marginals (µ λ , λ ≥ 0). Precisely, if τ λ and R λ denote the Root solution to the Skorokhod embedding problem (SEP) and the Root regular barrier for µ λ respectively, then this condition is equivalent to the property that (R λ , λ ≥ 0) is non-increasing in the sense of inclusion, which in turn is equivalent to the assertion that (τ λ , λ ≥ 0) is non-decreasing a.s. We show that there are many examples for which this result applies and we provide some numerical simulations to illustrate the monotonicity property of regular barriers (R λ , λ ≥ 0) in this case.

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“…Kellerer (1972) then showed that for ℝ-valued processes the martingale can in fact be chosen to be a Markov process. Recent contributions to the peacock literature include, next to the extensive work of Hirsch et al (2011), the articles by Bogso (2015Bogso ( , 2016, as well as Ewald and Yor (2015) who discuss applications in the economics of inequality. Further recent applications of peacocks involve Skorohod embeddings (see Källblad, Tan, & Touzi, 2017) and optimal transport under marginal constraints (see Beiglböck & Juillet, 2016).…”

Section: Introductionmentioning

confidence: 99%

On peacocks and lyrebirds: Australian options, Brownian bridges, and the average of submartingales

Ewald

Yor

2017

Mathematical Finance

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We introduce a class of stochastic processes, which we refer to as lyrebirds. These extend a class of stochastic processes, which have recently been coined peacocks, but are more commonly known as processes that are increasing in the convex order. We show how these processes arise naturally in the context of Asian and Australian options and consider further applications, such as the arithmetic average of a Brownian bridge and the average of submartingales, including the case of Asian and Australian options where the underlying features constant elasticity of variance or is of Merton jump diffusion type.

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MRL order, log-concavity and an application to peacocks

Cited by 6 publications

References 20 publications

Weak decreasing stochastic order

Weak decreasing stochastic order

Self-similar martingales derived from Root embedding

On peacocks and lyrebirds: Australian options, Brownian bridges, and the average of submartingales

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