Loi normale multidimensionnelle

En théorie des probabilités, on appelle loi normale multidimensionnelle, ou normale multivariée ou loi multinormale ou loi de Gauss à plusieurs variables, la loi de probabilité qui est la généralisation multidimensionnelle de la loi normale. Idée générale gauche|vignette|Différentes densités de lois normales en un dimension. gauche|vignette|Densité d'une loi gaussienne en 2D. Une loi normale classique est une loi dite « en cloche » en une dimension. Comme le montre la figure, la densité en forme de cloche peut être translatée n'importe où ; l'abscisse où se trouve le pic est la moyenne, aussi appelée centre, ou espérance. Si on fait plusieurs tirages selon une loi normale, on obtient plusieurs nombres réels et la moyenne des valeurs obtenues se situent près du centre. La largeur de la cloche se mesure par la variance. Plus la variance est petite, plus la cloche est resserrée, plus les valeurs tirées auront tendance à être proche de la moyenne. À l'inverse, plus la variance

Robust Portfolio Optimization

Zheng Wei Yap

Since the 2008 Global Financial Crisis, the financial market has become more unpredictable than ever before, and it seems set to remain so in the forseeable future. This means an investor faces unprecedented risks, hence the increasing need for robust portfolio optimization to protect them against uncertainty, which is potentially devastating if unattended yet ignored in the classical Markowitz model, whose another deficiency is the absence of higher moments in its assumption of the distribution of asset returns. We establish an equivalence between the Markowitz model and the portfolio return value-at-risk optimization problem under multivariate normality of asset returns, so that we can add these excluded features into the former implicitly by incorporating them into the latter. We also provide a probabilistic smoothing spline approximation method and a deterministic model within the location-scale framework under elliptical distribution of the asset returns to solve the robust portfolio return value-at-risk optimization problem. In particular for the deterministic model, we introduce a novel eigendecomposition uncertainty set which lives in the positive definite space for the scale matrix without compromising on the computational complexity and conservativeness of the optimization problem, invent a method to determine the size of the involved uncertainty sets, test it out on real data, and explore its diversification properties. Although the value-at-risk has been the standard risk measure adopted by the banking and insurance industry since the early nineties, it has since attracted many criticisms, in particular from McNeil et al. (2005) and the Basel Committee on Banking Supervision in 2012, also known as Basel 3.5. Basel 4 even suggests a move away from the what" value-at-risk to the what-if" conditional value-at-risk' measure. We shall see that the former may be replaced with the latter or even other risk measures in our formulations easily.

EPFL2017

Efficient inference for spatial extreme value processes associated to log-Gaussian random functions

Jonathan A. Tawn, Jennifer Lynne Wadsworth

Max-stable processes arise as the only possible nontrivial limits for maxima of affinely normalized identically distributed stochastic processes, and thus form an important class of models for the extreme values of spatial processes. Until recently, inference for max-stable processes has been restricted to the use of pairwise composite likelihoods, due to intractability of higher-dimensional distributions. In this work we consider random fields that are in the domain of attraction of a widely used class of max-stable processes, namely those constructed via manipulation of log-Gaussian random functions. For this class, we exploit limiting d-dimensional multivariate Poisson process intensities of the underlying process for inference on all d-vectors exceeding a high marginal threshold in at least one component, employing a censoring scheme to incorporate information below the marginal threshold. We also consider the d-dimensional distributions for the equivalent max-stable process, and perform full likelihood inference by exploiting the methods of Stephenson & Tawn (2005), where information on the occurrence times of extreme events is shown to dramatically simplify the likelihood. The Stephenson-Tawn likelihood is in fact simply a special case of the censored Poisson process likelihood. We assess the improvements in inference from both methods over pairwise likelihood methodology by simulation.

Oxford University Press2014

A new representation for multivariate tail probabilities

Jonathan A. Tawn, Jennifer Lynne Wadsworth

Existing theory for multivariate extreme values focuses upon characterizations of the distributional tails when all components of a random vector, standardized to identical margins, grow at the same rate. In this paper, we consider the effect of allowing the components to grow at different rates, and characterize the link between these marginal growth rates and the multivariate tail probability decay rate. Our approach leads to a whole class of univariate regular variation conditions, in place of the single but multivariate regular variation conditions that underpin the current theories. These conditions are indexed by a homogeneous function and an angular dependence function, which, for asymptotically independent random vectors, mirror the role played by the exponent measure and Pickands' dependence function in classical multivariate extremes. We additionally offer an inferential approach to joint survivor probability estimation. The key feature of our methodology is that extreme set probabilities can be estimated by extrapolating upon rays emanating from the origin when the margins of the variables are exponential. This offers an appreciable improvement over existing techniques where extrapolation in exponential margins is upon lines parallel to the diagonal.

Bernoulli Society for Mathematical Statistics and Probability2013

Robust Portfolio Optimization

Zheng Wei Yap

EPFL2017