2,581 research outputs found
Markov chain Monte Carlo for exact inference for diffusions
We develop exact Markov chain Monte Carlo methods for discretely-sampled,
directly and indirectly observed diffusions. The qualification "exact" refers
to the fact that the invariant and limiting distribution of the Markov chains
is the posterior distribution of the parameters free of any discretisation
error. The class of processes to which our methods directly apply are those
which can be simulated using the most general to date exact simulation
algorithm. The article introduces various methods to boost the performance of
the basic scheme, including reparametrisations and auxiliary Poisson sampling.
We contrast both theoretically and empirically how this new approach compares
to irreducible high frequency imputation, which is the state-of-the-art
alternative for the class of processes we consider, and we uncover intriguing
connections. All methods discussed in the article are tested on typical
examples.Comment: 23 pages, 6 Figures, 3 Table
Simulation of diffusions by means of importance sampling paradigm
The aim of this paper is to introduce a new Monte Carlo method based on
importance sampling techniques for the simulation of stochastic differential
equations. The main idea is to combine random walk on squares or rectangles
methods with importance sampling techniques. The first interest of this
approach is that the weights can be easily computed from the density of the
one-dimensional Brownian motion. Compared to the Euler scheme this method
allows one to obtain a more accurate approximation of diffusions when one has
to consider complex boundary conditions. The method provides also an
interesting alternative to performing variance reduction techniques and
simulating rare events.Comment: Published in at http://dx.doi.org/10.1214/09-AAP659 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Statistical Romberg extrapolation: A new variance reduction method and applications to option pricing
We study the approximation of by a Monte Carlo algorithm,
where is the solution of a stochastic differential equation and is a
given function. We introduce a new variance reduction method, which can be
viewed as a statistical analogue of Romberg extrapolation method. Namely, we
use two Euler schemes with steps and . This
leads to an algorithm which, for a given level of the statistical error, has a
complexity significantly lower than the complexity of the standard Monte Carlo
method. We analyze the asymptotic error of this algorithm in the context of
general (possibly degenerate) diffusions. In order to find the optimal
(which turns out to be ), we establish a central limit type theorem,
based on a result of Jacod and Protter for the asymptotic distribution of the
error in the Euler scheme. We test our method on various examples. In
particular, we adapt it to Asian options. In this setting, we have a CLT and,
as a by-product, an explicit expansion of the discretization error.Comment: Published at http://dx.doi.org/10.1214/105051605000000511 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Spatial Smoothing for Diffusion Tensor Imaging with low Signal to Noise Ratios
Though low signal to noise ratio (SNR) experiments in DTI give key information about tracking and anisotropy, e.g. by measurements with very small voxel sizes, due to the complicated impact of thermal noise such experiments are up to now seldom analysed. In this paper Monte Carlo simulations are presented which investigate the random fields of noise for different DTI variables in low SNR situations. Based on this study a strategy for spatial smoothing, which demands essentially uniform noise, is derived. To construct a convenient filter the weights of the nonlinear Aurich chain are adapted to DTI. This edge preserving three dimensional filter is then validated in different variants via a quasi realistic model and is applied to very new data with isotropic voxels of the size 1x1x1 mm3 which correspond to a spatial mean SNR of approximately 3
A selective overview of nonparametric methods in financial econometrics
This paper gives a brief overview on the nonparametric techniques that are
useful for financial econometric problems. The problems include estimation and
inferences of instantaneous returns and volatility functions of
time-homogeneous and time-dependent diffusion processes, and estimation of
transition densities and state price densities. We first briefly describe the
problems and then outline main techniques and main results. Some useful
probabilistic aspects of diffusion processes are also briefly summarized to
facilitate our presentation and applications.Comment: 32 pages include 7 figure
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
Marginalization for rare event simulation in switching diffusions
In this paper we use splitting technique to estimate the probability of
hitting a rare but critical set by the continuous component of a switching
diffusion. Instead of following classical approach we use Wonham filter to
achieve multiple goals including reduction of asymptotic variance and exemption
from sampling the discrete components
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