3,027 research outputs found
The exponentially convergent trapezoidal rule
It is well known that the trapezoidal rule converges geometrically when applied to analytic functions on periodic intervals or the real line. The mathematics and history of this phenomenon are reviewed and it is shown that far from being a curiosity, it is linked with computational methods all across scientific computing, including algorithms related to inverse Laplace transforms, special functions, complex analysis, rational approximation, integral equations, and the computation of functions and eigenvalues of matrices and operators
Perspectives on Beam-Shaping Optimization for Thermal-Noise Reduction in Advanced Gravitational-Wave Interferometric Detectors: Bounds, Profiles, and Critical Parameters
Suitable shaping (in particular, flattening and broadening) of the laser beam
has recently been proposed as an effective device to reduce internal (mirror)
thermal noise in advanced gravitational wave interferometric detectors. Based
on some recently published analytic approximations (valid in the
infinite-test-mass limit) for the Brownian and thermoelastic mirror noises in
the presence of arbitrary-shaped beams, this paper addresses certain
preliminary issues related to the optimal beam-shaping problem. In particular,
with specific reference to the Laser Interferometer Gravitational-wave
Observatory (LIGO) experiment, absolute and realistic lower-bounds for the
various thermal noise constituents are obtained and compared with the current
status (Gaussian beams) and trends ("mesa" beams), indicating fairly ample
margins for further reduction. In this framework, the effective dimension of
the related optimization problem, and its relationship to the critical design
parameters are identified, physical-feasibility and model-consistency issues
are considered, and possible additional requirements and/or prior information
exploitable to drive the subsequent optimization process are highlighted.Comment: 12 pages, 9 figures, 2 table
Stochastic expansions using continuous dictionaries: L\'{e}vy adaptive regression kernels
This article describes a new class of prior distributions for nonparametric
function estimation. The unknown function is modeled as a limit of weighted
sums of kernels or generator functions indexed by continuous parameters that
control local and global features such as their translation, dilation,
modulation and shape. L\'{e}vy random fields and their stochastic integrals are
employed to induce prior distributions for the unknown functions or,
equivalently, for the number of kernels and for the parameters governing their
features. Scaling, shape, and other features of the generating functions are
location-specific to allow quite different function properties in different
parts of the space, as with wavelet bases and other methods employing
overcomplete dictionaries. We provide conditions under which the stochastic
expansions converge in specified Besov or Sobolev norms. Under a Gaussian error
model, this may be viewed as a sparse regression problem, with regularization
induced via the L\'{e}vy random field prior distribution. Posterior inference
for the unknown functions is based on a reversible jump Markov chain Monte
Carlo algorithm. We compare the L\'{e}vy Adaptive Regression Kernel (LARK)
method to wavelet-based methods using some of the standard test functions, and
illustrate its flexibility and adaptability in nonstationary applications.Comment: Published in at http://dx.doi.org/10.1214/11-AOS889 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
On accurate and efficient valuation of financial contracts under models with jumps
The aim of this thesis is to develop efficient valuation methods for nancial contracts
under models with jumps and stochastic volatility, and to present their rigorous mathematical underpinning. For efficient risk management, large books of exotic options need
to be priced and hedged under models that are exible enough to describe the observed
option prices at speeds close to real time. To do so, hundreds of vanilla options, which
are quoted in terms of implied volatility, need to be calibrated to market prices quickly
and accurately on a regular basis. With this in mind we develop efficient methods for the
evaluation of (i) vanilla options, (ii) implied volatility and (iii) common path-dependent
options.
Firstly, we derive a new numerical method for the classical problem of pricing vanilla
options quickly in time-changed Brownian motion models. The method is based on ra-
tional function approximations of the Black-Scholes formula. Detailed numerical results
are given for a number of widely used models. In particular, we use the variance-gamma
model, the CGMY model and the Heston model without correlation to illustrate our results. Comparison to the standard fast Fourier option pricing method with respect to
speed appears to favour our newly developed method in the cases considered. Secondly,
we use this method to derive a procedure to compute, for a given set of arbitrage-free
European call option prices, the corresponding Black-Scholes implied volatility surface. In
order to achieve this, rational function approximations of the inverse of the Black-Scholes
formula are used. We are thus able to work out implied volatilities more efficiently than
is possible using other common methods. Error estimates are presented for a wide range
of parameters. Thirdly, we develop a new Monte Carlo variance reduction method to
estimate the expectations of path-dependent functionals, such as first-passage times and
occupation times, under a class of stochastic volatility models with jumps. The method is
based on a recursive approximation of the rst-passage time probabilities and expected oc-
cupation times of Levy bridge processes that relies in part on a randomisation of the time-
parameter. We derive the explicit form of the recursive approximation in the case of bridge
processes corresponding to the class of Levy processes with mixed-exponential jumps, and present a highly accurate numerical realisation. This class includes the linear Brownian
motion, Kou's double-exponential jump-di usion model and the hyper-exponential jump-
difusion model, and it is dense in the class of all Levy processes. We determine the rate
of convergence of the randomisation method and con rm it numerically. Subsequently,
we combine the randomisation method with a continuous Euler-Maruyama scheme to es-
timate path-functionals under stochastic volatility models with jumps. Compared with
standard Monte Carlo methods, we nd that the method is signi cantly more efficient. To
illustrate the efficiency of the method, it is applied to the valuation of range accruals and
barrier options.Open Acces
General Semiparametric Shared Frailty Model Estimation and Simulation with frailtySurv
The R package frailtySurv for simulating and fitting semi-parametric shared
frailty models is introduced. Package frailtySurv implements semi-parametric
consistent estimators for a variety of frailty distributions, including gamma,
log-normal, inverse Gaussian and power variance function, and provides
consistent estimators of the standard errors of the parameters' estimators. The
parameters' estimators are asymptotically normally distributed, and therefore
statistical inference based on the results of this package, such as hypothesis
testing and confidence intervals, can be performed using the normal
distribution. Extensive simulations demonstrate the flexibility and correct
implementation of the estimator. Two case studies performed with publicly
available datasets demonstrate applicability of the package. In the Diabetic
Retinopathy Study, the onset of blindness is clustered by patient, and in a
large hard drive failure dataset, failure times are thought to be clustered by
the hard drive manufacturer and model
- …