Characteristic function estimation of non-Gaussian Ornstein-Uhlenbeck processes.

Continuous non-Gaussian stationary processes of the OU-type are becoming increasingly popular given their flexibility in modelling stylized features of financial series such as asymmetry, heavy tails and jumps. The use of non-Gaussian marginal distributions makes likelihood analysis of these processes unfeasible for virtually all cases of interest. This paper exploits the self-decomposability of the marginal laws of OU processes to provide explicit expressions of the characteristic function which can be applied to several models as well as to develop e±cient estimation techniques based on the empirical characteristic function. Extensions to OU-based stochastic volatility models are provided.Ornstein-Uhlenbeck process; Lévy process; self-decomposable distribution; characteristic function; estimation

Efficient maximum likelihood estimation for L\'{e}vy-driven Ornstein-Uhlenbeck processes

We consider the problem of efficient estimation of the drift parameter of an Ornstein-Uhlenbeck type process driven by a L\'{e}vy process when high-frequency observations are given. The estimator is constructed from the time-continuous likelihood function that leads to an explicit maximum likelihood estimator and requires knowledge of the continuous martingale part. We use a thresholding technique to approximate the continuous part of the process. Under suitable conditions, we prove asymptotic normality and efficiency in the H\'{a}jek-Le Cam sense for the resulting drift estimator. Finally, we investigate the finite sample behavior of the method and compare our approach to least squares estimation.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ510 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Joint Modelling of Gas and Electricity spot prices

The recent liberalization of the electricity and gas markets has resulted in the growth of energy exchanges and modelling problems. In this paper, we modelize jointly gas and electricity spot prices using a mean-reverting model which fits the correlations structures for the two commodities. The dynamics are based on Ornstein processes with parameterized diffusion coefficients. Moreover, using the empirical distributions of the spot prices, we derive a class of such parameterized diffusions which captures the most salient statistical properties: stationarity, spikes and heavy-tailed distributions. The associated calibration procedure is based on standard and efficient statistical tools. We calibrate the model on French market for electricity and on UK market for gas, and then simulate some trajectories which reproduce well the observed prices behavior. Finally, we illustrate the importance of the correlation structure and of the presence of spikes by measuring the risk on a power plant portfolio

Exact Scenario Simulation for Selected Multi-dimensional Stochastic Processes

Accurate scenario simulation methods for solutions of multi-dimensional stochastic differential equations find application in stochastic analysis, the statistics of stochastic processes and many other areas, for instance, in finance. They have been playing a crucial role as standard models in various areas and dominate often the communication and thinking in a particular field of application, even that they may be too simple for more advanced tasks. Various discrete time simulation methods have been developed over the years. However, the simulation of solutions of some stochastic differential equations can be problematic due to systematic errors and numerical instabilities. Therefore, it is valuable to identify multi-dimensional stochastic differential equations with solutions that can be simulated exactly. This avoids several of the theoretical and practical problems encountered by those simulation methods that use discrete time approximations. This paper provides a survey of methods for the exact simulation of paths of some multi-dimensional solutions of stochastic differential equations including Ornstein-Uhlenbeck, square root, squared Bessel, Wishart and Levy type processes.exact scenario simulation; multi-dimensional stochastic differential equations; multi-dimensional Ornstein-Uhlenbeck process; multi-dimensional square root process; multi-dimensional squared Bessel process; Wishart process; multi-dimensional Levy process

A Pathwise Fractional one Compartment Intra-Veinous Bolus Model

Extending deterministic compartments pharmacokinetic models as diffusions seems not realistic on biological side because paths of these stochastic processes are not smooth enough. In order to extend one compartment intra-veinous bolus models, this paper suggests to modelize the concentration process $C$ by a class of stochastic differential equations driven by a fractional Brownian motion of Hurst parameter belonging to $]1/2,1[$. The first part of the paper provides probabilistic and statistical results on the concentration process $C$ : the distribution of $C$, a control of the uniform distance between $C$ and the solution of the associated ordinary differential equation, an ergodic theorem for the concentration process and its application to the estimation of the elimination constant, and consistent estimators of the driving signal's Hurst parameter and of the volatility constant. The second part of the paper provides applications of these theoretical results on simulated concentration datas : a qualitative procedure for choosing parameters on small sets of observations, and simulations of the estimators of the elimination constant and of the driving signal's Hurst parameter. The relationship between the estimations quality and the size/length of the sample is discussed.Comment: 16 pages, 6 figure