3,305 research outputs found
The Laplace transform of the lognormal distribution
We study the analytical properties of the Laplace transform of the lognormal
distribution. Two integral expressions for the analytic continuation of the
Laplace transform of the lognormal distribution are provided, one of which
takes the form of a Mellin-Barnes integral. As a corollary, we obtain an
integral expression for the characteristic function; we show that the integral
expression derived by Leipnik in [11] is incorrect. We present two
approximations for the Laplace transform of the lognormal distribution, both
valid in . In the last section, we discuss how
one may use our results to compute the density of a sum of independent
lognormal random variables.Comment: 18 pages, 2 figure
Earthquake Forecasting Based on Data Assimilation: Sequential Monte Carlo Methods for Renewal Processes
In meteorology, engineering and computer sciences, data assimilation is
routinely employed as the optimal way to combine noisy observations with prior
model information for obtaining better estimates of a state, and thus better
forecasts, than can be achieved by ignoring data uncertainties. Earthquake
forecasting, too, suffers from measurement errors and partial model information
and may thus gain significantly from data assimilation. We present perhaps the
first fully implementable data assimilation method for earthquake forecasts
generated by a point-process model of seismicity. We test the method on a
synthetic and pedagogical example of a renewal process observed in noise, which
is relevant to the seismic gap hypothesis, models of characteristic earthquakes
and to recurrence statistics of large quakes inferred from paleoseismic data
records. To address the non-Gaussian statistics of earthquakes, we use
sequential Monte Carlo methods, a set of flexible simulation-based methods for
recursively estimating arbitrary posterior distributions. We perform extensive
numerical simulations to demonstrate the feasibility and benefits of
forecasting earthquakes based on data assimilation. In particular, we show that
the forecasts based on the Optimal Sampling Importance Resampling (OSIR)
particle filter are significantly better than those of a benchmark forecast
that ignores uncertainties in the observed event times. We use the marginal
data likelihood, a measure of the explanatory power of a model in the presence
of data errors, to estimate parameters and compare models.Comment: 55 pages, 15 figure
Evolution of the cosmological density distribution function
We present a new calculation for the evolution of the 1-point Probability
Distribution Function (PDF) of the cosmological density field based on an exact
statistical treatment. Using the Chapman-Kolmogorov equation and second-order
Eulerian perturbation theory we propagate the initial density distribution into
the nonlinear regime. Our calculations yield the moment generating function,
allowing a straightforward derivation of the skewness of the PDF to second
order. We find a new dependency on the initial perturbation spectrum. We
compare our results with other approximations to the 1-pt PDF, and with N-body
simulations. We find that our distribution accurately models the evolution of
the 1-pt PDF of dark matter.Comment: 7 pages, 7 figures, accepted for publication in MNRA
Efficient and accurate log-L\'evy approximations to L\'evy driven LIBOR models
The LIBOR market model is very popular for pricing interest rate derivatives,
but is known to have several pitfalls. In addition, if the model is driven by a
jump process, then the complexity of the drift term is growing exponentially
fast (as a function of the tenor length). In this work, we consider a
L\'evy-driven LIBOR model and aim at developing accurate and efficient
log-L\'evy approximations for the dynamics of the rates. The approximations are
based on truncation of the drift term and Picard approximation of suitable
processes. Numerical experiments for FRAs, caps, swaptions and sticky ratchet
caps show that the approximations perform very well. In addition, we also
consider the log-L\'evy approximation of annuities, which offers good
approximations for high volatility regimes.Comment: 32 pages, 21 figures. Added an example of a path-dependent option
(sticky ratchet caplet). Forthcoming in the Journal of Computational Financ
A New Asymptotic Analysis Technique for Diversity Receptions Over Correlated Lognormal Fading Channels
Prior asymptotic performance analyses are based on the series expansion of
the moment-generating function (MGF) or the probability density function (PDF)
of channel coefficients. However, these techniques fail for lognormal fading
channels because the Taylor series of the PDF of a lognormal random variable is
zero at the origin and the MGF does not have an explicit form. Although
lognormal fading model has been widely applied in wireless communications and
free-space optical communications, few analytical tools are available to
provide elegant performance expressions for correlated lognormal channels. In
this work, we propose a novel framework to analyze the asymptotic outage
probabilities of selection combining (SC), equal-gain combining (EGC) and
maximum-ratio combining (MRC) over equally correlated lognormal fading
channels. Based on these closed-form results, we reveal the followings: i) the
outage probability of EGC or MRC becomes an infinitely small quantity compared
to that of SC at large signal-to-noise ratio (SNR); ii) channel correlation can
result in an infinite performance loss at large SNR. More importantly, the
analyses reveal insights into the long-standing problem of performance analyses
over correlated lognormal channels at high SNR, and circumvent the
time-consuming Monte Carlo simulation and numerical integration
Mixed Finite Element Analysis of Lognormal Diffusion and Multilevel Monte Carlo Methods
This work is motivated by the need to develop efficient tools for uncertainty
quantification in subsurface flows associated with radioactive waste disposal
studies. We consider single phase flow problems in random porous media
described by correlated lognormal distributions. We are interested in the error
introduced by a finite element discretisation of these problems. In contrast to
several recent works on the analysis of standard nodal finite element
discretisations, we consider here mass-conservative lowest order Raviart-Thomas
mixed finite elements. This is very important since local mass conservation is
highly desirable in realistic groundwater flow problems. Due to the limited
spatial regularity and the lack of uniform ellipticity and boundedness of the
operator the analysis is non-trivial in the presence of lognormal random
fields. We establish finite element error bounds for Darcy velocity and
pressure, as well as for a more accurate recovered pressure approximation. We
then apply the error bounds to prove convergence of the multilevel Monte Carlo
algorithm for estimating statistics of these quantities. Moreover, we prove
convergence for a class of bounded, linear functionals of the Darcy velocity.
An important special case is the approximation of the effective permeability in
a 2D flow cell. We perform numerical experiments to confirm the convergence
results
Multilevel path simulation for weak approximation schemes
In this paper we discuss the possibility of using multilevel Monte Carlo
(MLMC) methods for weak approximation schemes. It turns out that by means of a
simple coupling between consecutive time discretisation levels, one can achieve
the same complexity gain as under the presence of a strong convergence. We
exemplify this general idea in the case of weak Euler scheme for L\'evy driven
stochastic differential equations, and show that, given a weak convergence of
order the complexity of the corresponding "weak" MLMC
estimate is of order The numerical
performance of the new "weak" MLMC method is illustrated by several numerical
examples
Weighted Sum of Correlated Lognormals: Convolution Integral Solution
Probability density function (pdf) for sum of n
correlated lognormal variables is deducted as a special
convolution integral. Pdf for weighted sums (where weights can
be any real numbers) is also presented. The result for four
dimensions was checked by Monte Carlo simulation
Interest Rate Model Calibration Using Semidefinite Programming
We show that, for the purpose of pricing Swaptions, the Swap rate and the
corresponding Forward rates can be considered lognormal under a single
martingale measure. Swaptions can then be priced as options on a basket of
lognormal assets and an approximation formula is derived for such options. This
formula is centered around a Black-Scholes price with an appropriate
volatility, plus a correction term that can be interpreted as the expected
tracking error. The calibration problem can then be solved very efficiently
using semidefinite programming
How Much Data Do You Need? An Operational, Pre-Asymptotic Metric for Fat-tailedness
This note presents an operational measure of fat-tailedness for univariate
probability distributions, in where 0 is maximally thin-tailed
(Gaussian) and 1 is maximally fat-tailed. Among others,1) it helps assess the
sample size needed to establish a comparative needed for statistical
significance, 2) allows practical comparisons across classes of fat-tailed
distributions, 3) helps understand some inconsistent attributes of the
lognormal, pending on the parametrization of its scale parameter. The
literature is rich for what concerns asymptotic behavior, but there is a large
void for finite values of , those needed for operational purposes.
Conventional measures of fat-tailedness, namely 1) the tail index for the power
law class, and 2) Kurtosis for finite moment distributions fail to apply to
some distributions, and do not allow comparisons across classes and
parametrization, that is between power laws outside the Levy-Stable basin, or
power laws to distributions in other classes, or power laws for different
number of summands. How can one compare a sum of 100 Student T distributed
random variables with 3 degrees of freedom to one in a Levy-Stable or a
Lognormal class? How can one compare a sum of 100 Student T with 3 degrees of
freedom to a single Student T with 2 degrees of freedom? We propose an
operational and heuristic measure that allow us to compare -summed
independent variables under all distributions with finite first moment. The
method is based on the rate of convergence of the Law of Large numbers for
finite sums, -summands specifically. We get either explicit expressions or
simulation results and bounds for the lognormal, exponential, Pareto, and the
Student T distributions in their various calibrations --in addition to the
general Pearson classes
- …