583 research outputs found
Multilevel Richardson-Romberg extrapolation
We propose and analyze a Multilevel Richardson-Romberg (MLRR) estimator which
combines the higher order bias cancellation of the Multistep Richardson-Romberg
method introduced in [Pa07] and the variance control resulting from the
stratification introduced in the Multilevel Monte Carlo (MLMC) method (see
[Hei01, Gi08]). Thus, in standard frameworks like discretization schemes of
diffusion processes, the root mean squared error (RMSE) can
be achieved with our MLRR estimator with a global complexity of
instead of with the standard MLMC method, at least when the weak
error of the biased implemented estimator
can be expanded at any order in and . The MLRR estimator is then halfway between a regular MLMC
and a virtual unbiased Monte Carlo. When the strong error , , the gain of MLRR over MLMC becomes even
more striking. We carry out numerical simulations to compare these estimators
in two settings: vanilla and path-dependent option pricing by Monte Carlo
simulation and the less classical Nested Monte Carlo simulation.Comment: 38 page
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
On some Non Asymptotic Bounds for the Euler Scheme
We obtain non asymptotic bounds for the Monte Carlo algorithm associated to
the Euler discretization of some diffusion processes. The key tool is the
Gaussian concentration satisfied by the density of the discretization scheme.
This Gaussian concentration is derived from a Gaussian upper bound of the
density of the scheme and a modification of the so-called "Herbst argument"
used to prove Logarithmic Sobolev inequalities. We eventually establish a
Gaussian lower bound for the density of the scheme that emphasizes the
concentration is sharp.Comment: 26 page
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.Electricity markets; spot price modelling; ergodic diffusion; saddlepoint
Behavior of the Euler scheme with decreasing step in a degenerate situation
The aim of this paper is to study the behavior of the weighted empirical
measures of the decreasing step Euler scheme of a one-dimensional diffusion
process having multiple invariant measures. This situation can occur when the
drift and the diffusion coefficient are vanish simultaneously. As a first step,
we give a brief description of the Feller's classification of the
one-dimensional process. We recall the concept of attractive and repulsive
boundary point and introduce the concept of strongly repulsive point. That
allows us to establish a classification of the ergodic behavior of the
diffusion. We conclude this section by giving necessary and sufficient
conditions on the nature of boundary points in terms of Lyapunov functions. In
the second section we use this characterization to study the decreasing step
Euler scheme. We give also an numerical example in higher dimension
Incremental Decision Tree based on order statistics
International audienceNew application domains generate data which are not persistent anymore but volatile: network management, web profile modeling... These data arrive quickly, massively and are visible just once. Thus they necessarily have to be learnt according to their arrival orders. For classification problems online decision trees are known to perform well and are widely used on streaming data. In this paper, we propose a new decision tree method based on order statistics. The construction of an online tree usually needs summaries in the leaves. Our solution uses bounded error quantiles summaries. A robust and performing discretization or grouping method uses these summaries to provide, at the same time, a criterion to find the best split and better density estimations. This estimation is then used to build a na¨ıve Bayes classifier in the leaves to improve the prediction in the early learning stage
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