Some problems of statistics and optimal control for stochastic processes in the field of electricity markets prices modeling.

Summary In this thesis, we study mathematical models for the representation of prices on the electricity markets, from the viewpoints of statistics of random processes and optimal stochastic control. In a first part, we perform estimation of the components of the volatility coefficient of a multidimensional diffusion process, which represents the evolution of prices in the electricity forward market. It is driven by two Brownian motions. We aim at achieving estimation efficiently in terms of convergence rate and, concerning the parametric part of those components, in terms of limit law. To do so, we must extend the usual notion of efficiency in the Cramér-Rao sense. Our estimation methods are based on realized quadratic variation of the observed process. In a second part, we add model error terms to the previous model, in order to care for some kind of degeneration occurring in it as soon as the dimension of the observed process is greater than two. Our estimation methods are still based on realized quadratic variation, and we give other tools in order to keep on estimating the volatility components with the optimal rate when error terms are present. Then, numerical tests provide us with some evidence that such errors are present in the data. Finally, we solve the problem of a producer, which trades on the electricity intraday market in order to cope with the uncertainties on the outputs of his production units. We assume that there is market impact, so that the producer influences prices as he trades. The price and the forecast of the consumers’ demand are modelled by jump diffusions. We use the tools of optimal stochastic control to determine the strategy of the producer in an approximate problem. We give conditions so that this strategy is close to optimality in the original problem, as well as numerical illustrations of that strategy.