23,861 research outputs found
Smoothing the payoff for efficient computation of Basket option prices
We consider the problem of pricing basket options in a multivariate Black
Scholes or Variance Gamma model. From a numerical point of view, pricing such
options corresponds to moderate and high dimensional numerical integration
problems with non-smooth integrands. Due to this lack of regularity, higher
order numerical integration techniques may not be directly available, requiring
the use of methods like Monte Carlo specifically designed to work for
non-regular problems. We propose to use the inherent smoothing property of the
density of the underlying in the above models to mollify the payoff function by
means of an exact conditional expectation. The resulting conditional
expectation is unbiased and yields a smooth integrand, which is amenable to the
efficient use of adaptive sparse grid cubature. Numerical examples indicate
that the high-order method may perform orders of magnitude faster compared to
Monte Carlo or Quasi Monte Carlo in dimensions up to 35
An Optimal Control Derivation of Nonlinear Smoothing Equations
The purpose of this paper is to review and highlight some connections between
the problem of nonlinear smoothing and optimal control of the Liouville
equation. The latter has been an active area of recent research interest owing
to work in mean-field games and optimal transportation theory. The nonlinear
smoothing problem is considered here for continuous-time Markov processes. The
observation process is modeled as a nonlinear function of a hidden state with
an additive Gaussian measurement noise. A variational formulation is described
based upon the relative entropy formula introduced by Newton and Mitter. The
resulting optimal control problem is formulated on the space of probability
distributions. The Hamilton's equation of the optimal control are related to
the Zakai equation of nonlinear smoothing via the log transformation. The
overall procedure is shown to generalize the classical Mortensen's minimum
energy estimator for the linear Gaussian problem.Comment: 7 pages, 0 figures, under peer reviewin
A Multilevel Approach for Stochastic Nonlinear Optimal Control
We consider a class of finite time horizon nonlinear stochastic optimal
control problem, where the control acts additively on the dynamics and the
control cost is quadratic. This framework is flexible and has found
applications in many domains. Although the optimal control admits a path
integral representation for this class of control problems, efficient
computation of the associated path integrals remains a challenging Monte Carlo
task. The focus of this article is to propose a new Monte Carlo approach that
significantly improves upon existing methodology. Our proposed methodology
first tackles the issue of exponential growth in variance with the time horizon
by casting optimal control estimation as a smoothing problem for a state space
model associated with the control problem, and applying smoothing algorithms
based on particle Markov chain Monte Carlo. To further reduce computational
cost, we then develop a multilevel Monte Carlo method which allows us to obtain
an estimator of the optimal control with mean squared
error with a computational cost of
. In contrast, a computational cost
of is required for existing methodology to achieve
the same mean squared error. Our approach is illustrated on two numerical
examples, which validate our theory
Refraction-corrected ray-based inversion for three-dimensional ultrasound tomography of the breast
Ultrasound Tomography has seen a revival of interest in the past decade,
especially for breast imaging, due to improvements in both ultrasound and
computing hardware. In particular, three-dimensional ultrasound tomography, a
fully tomographic method in which the medium to be imaged is surrounded by
ultrasound transducers, has become feasible. In this paper, a comprehensive
derivation and study of a robust framework for large-scale bent-ray ultrasound
tomography in 3D for a hemispherical detector array is presented. Two
ray-tracing approaches are derived and compared. More significantly, the
problem of linking the rays between emitters and receivers, which is
challenging in 3D due to the high number of degrees of freedom for the
trajectory of rays, is analysed both as a minimisation and as a root-finding
problem. The ray-linking problem is parameterised for a convex detection
surface and three robust, accurate, and efficient ray-linking algorithms are
formulated and demonstrated. To stabilise these methods, novel
adaptive-smoothing approaches are proposed that control the conditioning of the
update matrices to ensure accurate linking. The nonlinear UST problem of
estimating the sound speed was recast as a series of linearised subproblems,
each solved using the above algorithms and within a steepest descent scheme.
The whole imaging algorithm was demonstrated to be robust and accurate on
realistic data simulated using a full-wave acoustic model and an anatomical
breast phantom, and incorporating the errors due to time-of-flight picking that
would be present with measured data. This method can used to provide a
low-artefact, quantitatively accurate, 3D sound speed maps. In addition to
being useful in their own right, such 3D sound speed maps can be used to
initialise full-wave inversion methods, or as an input to photoacoustic
tomography reconstructions
Controlled Sequential Monte Carlo
Sequential Monte Carlo methods, also known as particle methods, are a popular
set of techniques for approximating high-dimensional probability distributions
and their normalizing constants. These methods have found numerous applications
in statistics and related fields; e.g. for inference in non-linear non-Gaussian
state space models, and in complex static models. Like many Monte Carlo
sampling schemes, they rely on proposal distributions which crucially impact
their performance. We introduce here a class of controlled sequential Monte
Carlo algorithms, where the proposal distributions are determined by
approximating the solution to an associated optimal control problem using an
iterative scheme. This method builds upon a number of existing algorithms in
econometrics, physics, and statistics for inference in state space models, and
generalizes these methods so as to accommodate complex static models. We
provide a theoretical analysis concerning the fluctuation and stability of this
methodology that also provides insight into the properties of related
algorithms. We demonstrate significant gains over state-of-the-art methods at a
fixed computational complexity on a variety of applications
Sequential Bayesian inference for implicit hidden Markov models and current limitations
Hidden Markov models can describe time series arising in various fields of
science, by treating the data as noisy measurements of an arbitrarily complex
Markov process. Sequential Monte Carlo (SMC) methods have become standard tools
to estimate the hidden Markov process given the observations and a fixed
parameter value. We review some of the recent developments allowing the
inclusion of parameter uncertainty as well as model uncertainty. The
shortcomings of the currently available methodology are emphasised from an
algorithmic complexity perspective. The statistical objects of interest for
time series analysis are illustrated on a toy "Lotka-Volterra" model used in
population ecology. Some open challenges are discussed regarding the
scalability of the reviewed methodology to longer time series,
higher-dimensional state spaces and more flexible models.Comment: Review article written for ESAIM: proceedings and surveys. 25 pages,
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