La modélisation stochastique des étiages: une revue bibliographique

La croissance continue de la population mondiale et l'augmentation du niveau de vie dans certaines parties de la planète exercent une pression de plus en plus forte sur la demande quantitative et qualitative de la ressource hydrique, nécessitant ainsi une gestion plus adéquate. Afin d'évaluer la fiabilité d'un système de ressources en eau et de déterminer son mode de gestion durant un étiage, il est utile d'avoir un outil de modélisation. Nous présentons ici une synthèse des travaux de modélisation réalisés dans le cadre de l'approche stochastique. Nous faisons d'abord le point sur la différence entre une sécheresse et un étiage, termes qui sont souvent confondus dans les publications, pour ensuite en présenter quelques indicateurs. L'approche stochastique peut être subdivisée en deux catégories: l'étude fréquentielle et les processus stochastiques. La plupart des études d'analyse de fréquence ont pour objet de calculer des débits d'étiage critiques xT correspondant à une certaine période de retour T, tel que P(X<xT)=1/T. L'approche par les processus stochastiques consiste à modéliser les événements de déficit ou les variables d'intérêt sans utiliser directement des modèles de débit. L'analyse de fréquence des débits ne tient pas compte des durées et émet des hypothèses trop simplistes de stationnarité. L'analyse des séquences permet l'obtention des lois de durées uniquement pour des processus de débits très simples. L'avantage de l'approche des processus ponctuels par rapport à l'analyse des séquences est qu'elle permet d'étudier des processus complexes, dépendants et non stationnaires. De plus, les processus ponctuels alternés permettent la modélisation des durée et la génération synthétique des temps d'occurrence des séries de surplus et de déficit. Nous présentons dans cet article les travaux de modélisation des étiages basés sur l'analyse fréquentielle, la théorie des séquences et sur les processus ponctuels. Nous n'avons pas inclus les études qui développent des distributions des faibles débits à partir de modèles physiques, ni les études de type régional.The increasing pressure on the water resources requires better management of the water deficit situations may it be unusual droughts or yearly recurring low-flows. It is therefore important to model the occurrence of these deficit events in order to quantify the related risks. Many approaches exist for the modeling of low-flow/drought events. We present here a literature review of the stochastic methods. We start by clarifying the difference between low-flows and droughts, two terms which are often used interchangeably. We then present some low-flow and drought indicators. The stochastic approach may be divided into two categories: Frequency analysis and stochastic processes. Most frequency analysis studies aim to assign to a flow value X a cumulative frequency, either directly using empirical distribution functions, or by fitting a theoretical distribution. This allows the computation of a critical flow xT corresponding to a return period T, such that P(X<xT)=1/T. These studies use mostly the annual minima of daily flows where the hydrological data is assumed independent and identically distributed. It is also common to analyze Qm, the annual minimum of the m-consecutive days average flow, m being generally 7, 10, 30, 60, 90, or 120 days, and to adopt as critical flow the m-day average having a return period of T years. The distributions which are used include the Normal, Weibull, Gumbel, Gamma, Log-Normal (2), Log-Pearson (3), Generalized Extreme Value, Pearson type 3, and Pearson type 5 distributions (GUMBEL, 1954; MATALAS, 1963; BERNIER, 1964; JOSEPH, 1970; CONDIE and NIX, 1975; HOANG, 1978; TASKER, 1987; RAYNAL-VILLASENOR and DOURIET, 1987; NATHAN and MCMAHON, 1990; ADAMCZYK, 1992).The approach using stochastic processes for low-flows may be direct (analytical) or indirect (experimental) (YEVJEVICH et al., 1983). The indirect approach (not described in this literature review) consists of obtaining flow models, generating synthetic flows and then empirically studying certain drought variables obtained from the synthetic data. The direct approach models deficit events and related variables without explicitely modeling flows. The stochastic processes are of two types and differ in the way that randomness is introduced in the model: ·- State modeling: The process may be modeled as a probabilistic transition between various states (Markov processes for example). The states of the process {Xt } are obtained from the hydrological observations {Yt } using thresholds. The number of states of {Xt } is finite and run series analysis may be used to study the properties of the drought parameters; or- Event modeling: The concept of random occurrence of an event is introduced, where an event is a transition between surplus and deficit and vice-versa. In this approach, stochastic point processes are appropriate. A deficit event is then considered a rare event and is characterized by its occurrence time.We review the low-flows studies based on frequency analysis, run series analysis and on point processes. However, we do not include the physically-based models nor the regional analysis studies.Run series analysis is applied to processes derived from flows and thresholds. A two-state process is obtained and Markov processes are often applied. The variables of interest are the duration of a deficit defined by the run length of series below the threshold (RL), the severity corresponding to the deficit volume over a negative run of length n (RSn), and the intensity In defined by the ratio RSn /RL (SALDARRIGA and YEVJEVICH, 1970; SEN, 1977; MILLAN and YEVJEVICH, 1971; MILLAN, 1972; SEN, 1980A; SEN, 1980B; SEN, 1980C; GÜVEN, 1983; MOYÉ et al., 1988; SEN, 1990). It is often assumed that the flow process is either independent or autoregressive of order 1 and that it is stationary except for SEN, 1980B.Point processes are based on the notion of the occurrence of an event. They are defined by the occurrence time tj of an event ej. We present a classification of some of the pertinent processes and their relation to each other. These include the Poisson process, both homongeneous and non-homogeneous, the renewal process, the doubly stochastic process and the self-exciting process. These processes are well suited for obtaining models of deficit durations (NORTH, 1981; LEE et al., 1986; ZELENHASIC et SALVAI, 1987; CHANG, 1989; MADSEN and ROSBJERG, 1995; ABI-ZEID, 1997). The advantage of this approach is its ability to take into account nonstationarity where alternating surplus-deficit point processes are defined from daily flow data. ABI-ZEID (1997) proposed a physically-based alternating non-homogeneous Poisson process that takes into account precipitation and temperature, and defined low-flow risk indices computed from these developed models.In conclusion, we remark that frequency analysis does not take into account well the duration aspcets and uses simplifying stationnarity hypothesis. Series analysis provides duration distributions for simple flow processes. The advantage of point processes is that they can model complex, dependent and non-stationary processes. Furthermore, alternating point processes can be used to model deficit durations and generate synthetic data such as occurrences of deficit and surplus events. We argue that the duration of low-flows is an important issue which has not received a lot of attention

Calibrage économétrique de processus stochastiques avec applications aux données boursières, bancaires et cambiales canadiennes

In this paper, we show how to calibrate the most usual stochastic processes: arithmetic and geometric Brownian motions,, mean-reverting processes and jump processes. This paper contains also many applications to Canadian financial data. We observe, among other phenomena, that a mean-reverting process is very appropriate to estimate the return on assets of the six biggest Canadian banks. Finally, we estimate a monofactorial model of interest rate.Stochastic processes, financial econometrics, banks, derivatives, financial engineering

Bornes quantitatives pour la convergence en temps long de processus de Markov

National audienceSi l'on sait assez bien décrire qualitativement le comportement de nombreux processus de Markov (existence et unicité d'une mesure invariante, convergence exponentielle à l'équilibre, etc), il est en général beaucoup plus difficile d'obtenir des bornes explicites pour la vitesse de convergence à l'équilibre

Approximation of quasi-stationary distributions for absorbed diffusions

National audienceThe theory of Markov processes with an absorbing state is commonly used in stochastic models of biological population, epidemics, chemical reactions and market dynamics

Probability laws related to the Jacobi theta and Riemann zeta function and Brownian excursions

This paper reviews known results which connect Riemann's integral representations of his zeta function, involving Jacobi's theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to one-dimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemann's zeta function which are related to these laws.Comment: LaTeX; 40 pages; review pape

Brownian Motion in wedges, last passage time and the second arc-sine law

We consider a planar Brownian motion starting from $O$ at time $t=0$ and stopped at $t=1$ and a set $F= \{OI_i ; i=1,2,..., n\}$ of $n$ semi-infinite straight lines emanating from $O$. Denoting by $g$ the last time when $F$ is reached by the Brownian motion, we compute the probability law of $g$. In particular, we show that, for a symmetric $F$ and even $n$ values, this law can be expressed as a sum of $\arcsin$ or $(\arcsin)^2$ functions. The original result of Levy is recovered as the special case $n=2$. A relation with the problem of reaction-diffusion of a set of three particles in one dimension is discussed