DESCHATRE Thomas

Based on empirical evidence of fast mean-reverting spikes, we model electricity price processes X + Z β as the sum of a continuous Itô semimartingale X and a a mean-reverting compound Poisson process Z β t = t 0 R xe −β(t−s) p(ds, dt) where p(ds, dt) is Poisson random measure with intensity λds ⊗ dt. In a first part, we investigate the estimation of (λ, β) from discrete observations and establish asymptotic efficiency in various asymptotic settings. In a second part, we discuss the use of our inference results for correcting the value of forward contracts on electricity markets in presence of spikes. We implement our method on real data in the French, Greman and Australian market over 2015 and 2016 and show in particular the effect of spike modelling on the valuation of certain strip options. In particular, we show that some out-of-the-money options have a significant value if we incorporate spikes in our modelling, while having a value close to 0 otherwise. Mathematics Subject Classification (2010): 62M86, 60J75, 60G35, 60F05.

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This thesis deals with dependence problems between stochastic processes in continuous time. In a first part, new copulas are established to model the dependence between two Brownian movements and to control the distribution of their difference. It is shown that the class of admissible copulas for Brownians contains asymmetric copulas. With these copulas, the survival function of the difference of the two Brownians is higher in its positive part than with a Gaussian dependence. The results are applied to the joint modeling of electricity prices and other energy commodities. In a second part, we consider a discretely observed stochastic process defined by the sum of a continuous semi-martingale and a compound Poisson process with mean reversion. An estimation procedure for the mean-reverting parameter is proposed when the mean-reverting parameter is large in a high frequency finite horizon statistical framework. In a third part, we consider a doubly stochastic Poisson process whose stochastic intensity is a function of a continuous semi-martingale. To estimate this function, a local polynomial estimator is used and a window selection method is proposed leading to an oracle inequality. A test is proposed to determine if the intensity function belongs to a certain parametric family. With these results, the dependence between the intensity of electricity price peaks and exogenous factors such as wind generation is modeled.

No summary available.

In this thesis, we study some dependence modeling problems between continuous time stochastic processes. These results are applied to the modeling and risk management of electricity markets. In a first part, we propose new copulae to model the dependence between two Brownian motions and to control the distribution of their difference. We show that the class of admissible copulae for the Brownian motions 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. Results are applied to the joint modeling of electricity and other energy commodity prices. In a second part, we consider a stochastic process which is a sum of a continuous semimartingale and a mean reverting compound Poisson process and which is discretely observed. An estimation procedure is proposed for the mean reversion parameter of the Poisson process in a high frequency framework with finite time horizon, assuming this parameter is large. Results are applied to the modeling of the spikes in electricity prices time series. In a third part, we consider a doubly stochastic Poisson process with stochastic intensity function of a continuous semimartingale. A local polynomial estimator is considered in order to infer the intensity function and a method is given to select the optimal bandwidth. An oracle inequality is derived. Furthermore, a test is proposed in order to determine if the intensity function belongs to some parametrical family. Using these results, we model the dependence between the intensity of electricity spikes and exogenous factors such as the wind production.

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.

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.