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Equivalent or absolutely continuous probability measures with given marginals

100%
Berti P., Pratelli L., Rigo P., Spizzichino F.
Dependence Modeling
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2015
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tom 3
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nr 1
EN
Let (X,A) and (Y,B) be measurable spaces. Supposewe are given a probability α on A, a probability β on B and a probability μ on the product σ-field A ⊗ B. Is there a probability ν on A⊗B, with marginals α and β, such that ν ≪ μ or ν ~ μ ? Such a ν, provided it exists, may be useful with regard to equivalent martingale measures and mass transportation. Various conditions for the existence of ν are provided, distinguishing ν ≪ μ from ν ~ μ.
2
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On the control of the difference between two Brownian motions: a dynamic copula approach

86%
Deschatre T.
Dependence Modeling
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2016
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tom 4
|
nr 1
EN
We propose new copulae to model the dependence between two Brownian motions and to control the distribution of their difference. Our approach is based on the copula between the Brownian motion and its reflection. We show that the class of admissible copulae for the Brownian motions are not limited to the class of Gaussian copulae and that it also contains asymmetric copulae. These copulae allow for the survival function of the difference between two Brownian motions to have higher value in the right tail than in the Gaussian copula case. Considering two Brownian motions B1t and B2t, the main result is that the range of possible values for [...] is the same for Markovian pairs and all pairs of Brownian motions, that is [...] with φ being the cumulative distribution function of a standard Gaussian random variable.
3
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On the control of the difference between two Brownian motions: an application to energy markets modeling

58%
Deschatre T.
Dependence Modeling
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2016
|
tom 4
|
nr 1
EN
We derive a model based on the structure of dependence between a Brownian motion and its reflection according to a barrier. The structure of dependence presents two states of correlation: one of comonotonicity with a positive correlation and one of countermonotonicity with a negative correlation. This model of dependence between two Brownian motions B1 and B2 allows for the value of [...] to be higher than 1/2 when x is close to 0, which is not the case when the dependence is modeled by a constant correlation. It can be used for risk management and option pricing in commodity energy markets. In particular, it allows to capture the asymmetry in the distribution of the difference between electricity prices and its combustible prices.
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