Jacques Lévy Véhel

This article focuses on the estimation of the stability index and scale parameter of stable random variables. We study an estimator based on log-moments,. The main advantage of this estimator is that it has a simple closed-form expression. This allows us to prove an almost sure convergence result as well as a central limit theorem. We show how to improve the accuracy of this estimator by combining it with previously defined ones. The closed-form also enables us to consider the case of non-identically distributed data, and we show that our results still hold provided deviations from stationarity are ‘small’. Using a centro-symmetrization, we expand the previous estimators to skewed stable variables and we construct a test to check the skewness of the data. As applications, we show numerically that the stability index of multistable Lévy motion may be estimated accurately and consider a financial log&.

Market risk regulations adopted in response to recent crises aim to reduce financial risks. Nevertheless, a large number of practitioners feel that even, if these rules seem to succeed in lowering volatility, they appear to rigidify the financial structure of the economic system and tend to increase the probability of large jumps: prudential rules seem to produce an unexpected effect, the swap between volatility risk and jump risk. The attempt at reduction in volatility is accompanied by an increase in the intensity of jumps. The new regulations seem create a new risk. This article discusses this idea in three ways. First, we introduce a conventionalist framework to shed some light on this unexpected effect. Second, we precisely define volatility risk and the intensity of jumps to document the risk swap effect by analyzing a daily time series of the S&P 500. Third, we propose a model that allows one to appreciate a practical consequence of this swap on the risk measures. We conclude by challenging the main objective of regulation: we argue that concentrating on reducing volatility can create a new type of risk that increases the potential losses, which we term regulation risk.

We propose a Piecewise-Deterministic Markov Process (PDMP) to model the drug concentration in the case of multiple intravenous-bolus (multi-IV) doses and poor patient adherence situation: the scheduled time and doses of drug administration are not respected by the patient, the drug administration considers switching regime with random drug intake times. We study the randomness of drug concentration and derive probability results on the stochastic dynamics using the PDMP theory, focusing on two aspects of practical relevance: the variability of the concentration and the regularity of its stationary probability distribution. The main result show as the regularity of the concentration is governed by a parameter, which quantifies in a precise way the situations where drug intake times are too scarce concerning the elimination rate. Our approach is novel for the study of the regularity of the stationary distribution in PDMP models. This article extends the results given in [J. Lévy-Véhel and P.E. Lévy-Véhel, Variability and singularity arising from poor compliance in a pharmacodynamical model I: The multi-IV case, J. Pharmacokinet. Pharmacodyn. 40 (2013), pp. 15–39], by considering more realistic irregular dosing schedules. The computations permit precise assessment of the effect of various significant parameters such as the mean rate of intake, the elimination rate, and the mean dose. They quantify how much poor adherence will affect the regimen. Our results help to understand the consequences of poor adherence.

We define and study in this work a simple model designed for managing long-term market risk of financial institutions with long-term commitments. It allows the assessment of solvency capital requirements and the allocation of risk budgets. This model allows one to avoid over-assessment of solvency capital requirements specifically after market disruptions. It relies on a dampener component in charge of refining risk assessment after market failures. Rather than aiming at a realistic and thus complex description of equity prices movements, this model concentrates on minimal features enabling accurate computation of capital requirements. It is defined both in a discrete and continuous fashion. In the latter case, we prove the existence, uniqueness and stability of the solution of the stochastic functional differential equation that specifies the model. One difficulty is that the proposed underlying stochastic process has neither stationary nor independent increments. We are however able to perform statistical analyses in view of its validation. Numerical experiments show that our model outperforms more elaborate ones of common use as far as medium-term (between 6 months and 5 years) risk assessment is concerned.

This work defines two classes of processes, that we term {\it tempered fractional multistable motion} and {\it tempered multifractional stable motion}. They are extensions of fractional multistable motion and multifractional stable motion, respectively, obtained by adding an exponential tempering to the integrands. We investigate certain basic features of these processes, including scaling property, tail probabilities, absolute moment, sample path properties, pointwise H\"{o}lder exponent, H\"{o}lder continuity of quasi norm, (strong) localisability and semi-long-range dependence structure. These processes may provide useful models for data that exhibit both dependence and varying local regularity/intensity of jumps.

Stochastic volatility models have proven relevant for various purposes, e.g. explaining the term structure of implied volatility or correlations in absolute returns. When used in conjunction with jump models, they provide powerful tools that have bee

In this note, we provide a simple example of regulation risk. The idea is that, in certain situations, the very prudential rules (or, rather, some of them) imposed by the regulator in the framework of the Basel II/III Accords or Solvency II directive are themselves the source of a systemic risk. The instance of regulation risk that we bring to light in this work can be summarised as follows: wrongly assuming that prices evolve in a continuous fashion when they may in fact display large negative jumps, and trying to minimise Value at Risk (VaR) under a constraint of minimal volume of activity leads in effect to behaviours that will maximise VaR. Although much stylised, our analysis highlights some pitfalls of model-based regulation.

We propose a stochastic model for the drug concentration in the case of multiple oral doses and in a situation of poor patient adherence. Our model is able to take into account an irregular drug intake schedule. This article is the second in a series of three. It presents a multi-oral version of the results given in Lévy-Véhel and Lévy-Véhel (J Pharmacokinet Pharmacodyn 40(1):15–39, 2013), that dealt with the multi-IV bolus case. Under the assumption that the irregular dosing schedule follows a Poisson law, we study features of the drug concentration that have practical implications, such as its variability and the regularity of its cumulative probability distribution, which describes its predictive power with respect to the mean behaviour. We consider four variants: continuous-time, with either deterministic or random doses, and discrete-time, also with either deterministic or random doses. Our computations allow one to assess in a precise way the effect of various significant parameters such as the mean rate of intake, the elimination rate, the absorption rate and the mean dose. They quantify how much poor adherence will affect the efficacy of therapy. To appreciate this impact, we provide detailed comparisons with the variability of concentration in two reference situations: a fully adherent patient and a population of fully adherent patients with log-normally distributed pharmacokinetic parameters. Besides, the discrete-time versions of our models reveal unexpected links with objects which have been studied in the mathematical literature under the name of infinite Bernoulli convolutions (Erdós, Am J Math 61:974-975, 1939). This allows us to quantify the fact that, when the random dosing schedule is too sparse, the concentration behaves in a very erratic way. Our results complement the ones in Lévy-Véhel and Lévy-Véhel (J Pharmacokinet Pharmacodyn 40(1):15–39, 2013) and help understanding the consequences of poor adherence. They may have practical outcomes in terms of drug dosing and scheduling.

Multistable L\'evy motions are extensions of L\'evy motions where the stability index is allowed to vary in time. Several constructions of these processes have been introduced recently, based on Poisson and Ferguson-Klass-LePage series representations and on multistable measures. In this work, we prove a functional central limit theorem for the independent-increments multistable L\'evy motion, as well as of integrals with respect to these processes, using weighted sums of independent random variables. This allows us to construct continuous approximations of multistable L\'evy motions. In particular, we prove that multistable L\'evy motions are stochastic H\"older continuous and strongly localisable.