409 research outputs found
Convergence of numerical methods for stochastic differential equations in mathematical finance
Many stochastic differential equations that occur in financial modelling do
not satisfy the standard assumptions made in convergence proofs of numerical
schemes that are given in textbooks, i.e., their coefficients and the
corresponding derivatives appearing in the proofs are not uniformly bounded and
hence, in particular, not globally Lipschitz. Specific examples are the Heston
and Cox-Ingersoll-Ross models with square root coefficients and the Ait-Sahalia
model with rational coefficient functions. Simple examples show that, for
example, the Euler-Maruyama scheme may not converge either in the strong or
weak sense when the standard assumptions do not hold. Nevertheless, new
convergence results have been obtained recently for many such models in
financial mathematics. These are reviewed here. Although weak convergence is of
traditional importance in financial mathematics with its emphasis on
expectations of functionals of the solutions, strong convergence plays a
crucial role in Multi Level Monte Carlo methods, so it and also pathwise
convergence will be considered along with methods which preserve the positivity
of the solutions.Comment: Review Pape
Heavy pseudoscalar mesons in a Schwinger-Dyson--Bethe-Salpeter approach
The mass spectrum of heavy pseudoscalar mesons, described as quark-antiquark
bound systems, is considered within the Bethe-Salpeter formalism with
momentum-dependent masses of the constituents. This dependence is found by
solving the Schwinger-Dyson equation for quark propagators in rainbow-ladder
approximation. Such an approximation is known to provide both a fast
convergence of numerical methods and accurate results for lightest mesons.
However, as the meson mass increases, the method becomes less stable and
special attention must be devoted to details of numerical means of solving the
corresponding equations. We focus on the pseudoscalar sector and show that our
numerical scheme describes fairly accurately the , , , and
ground states. Excited states are considered as well. Our calculations
are directly related to the future physics programme at FAIR.Comment: 9 pages, 3 figures; Based on materials of the contribution
"Relativistic Description of Two- and Three-Body Systems in Nuclear Physics",
ECT*, October 19-23, 200
Accelerating Extremum Seeking Convergence by Richardson Extrapolation Methods
In this paper, we propose the concept of accelerated convergence that has originally been developed to speed up the convergence of numerical methods for extremum seeking (ES) loops. We demonstrate how the dynamics of ES loops may be analyzed to extract structural information about the generated output of the loop. This information is then used to distil the limit of the loop without having to wait for the system to converge to it
Using cylindrical algebraic decomposition and local Fourier analysis to study numerical methods: two examples
Local Fourier analysis is a strong and well-established tool for analyzing
the convergence of numerical methods for partial differential equations. The
key idea of local Fourier analysis is to represent the occurring functions in
terms of a Fourier series and to use this representation to study certain
properties of the particular numerical method, like the convergence rate or an
error estimate.
In the process of applying a local Fourier analysis, it is typically
necessary to determine the supremum of a more or less complicated term with
respect to all frequencies and, potentially, other variables. The problem of
computing such a supremum can be rewritten as a quantifier elimination problem,
which can be solved with cylindrical algebraic decomposition, a well-known tool
from symbolic computation.
The combination of local Fourier analysis and cylindrical algebraic
decomposition is a machinery that can be applied to a wide class of problems.
In the present paper, we will discuss two examples. The first example is to
compute the convergence rate of a multigrid method. As second example we will
see that the machinery can also be used to do something rather different: We
will compare approximation error estimates for different kinds of
discretizations.Comment: The research was funded by the Austrian Science Fund (FWF): J3362-N2
On the efficient application of the repeated Richardson extrapolation technique to option pricing
Richardson extrapolation (RE) is a commonly used technique in financial applications for accelerating the convergence of numerical methods. Particularly in option pricing, it is possible to refine the results of several approaches by applying RE, in order to avoid the difficulties of employing slowly converging schemes. But the effectiveness of such a technique is fully achieved when its repeated version (RRE) is applied. Nevertheless, its application in financial literature is pretty rare. This is probably due to the necessity to pay special attention to the numerical aspects of its implementation, such as the choice of both the sequence of the stepsizes and the order of the method. In this contribution, we consider several numerical schemes for the valuation of American options and investigate the possibility of an appropriate application of RRE. As a result, we find that, in the analyzed approaches in which the convergence is monotonic, RRE can be used as an effective tool for improving significantly the accuracy.Richardson extrapolation, repeated Richardson extrapolation, American options, randomization technique, flexible binomial method
Weak Convergence in the Prokhorov Metric of Methods for Stochastic Differential Equations
We consider the weak convergence of numerical methods for stochastic
differential equations (SDEs). Weak convergence is usually expressed in terms
of the convergence of expected values of test functions of the trajectories.
Here we present an alternative formulation of weak convergence in terms of the
well-known Prokhorov metric on spaces of random variables. For a general class
of methods, we establish bounds on the rates of convergence in terms of the
Prokhorov metric. In doing so, we revisit the original proofs of weak
convergence and show explicitly how the bounds on the error depend on the
smoothness of the test functions. As an application of our result, we use the
Strassen - Dudley theorem to show that the numerical approximation and the true
solution to the system of SDEs can be re-embedded in a probability space in
such a way that the method converges there in a strong sense. One corollary of
this last result is that the method converges in the Wasserstein distance,
another metric on spaces of random variables. Another corollary establishes
rates of convergence for expected values of test functions assuming only local
Lipschitz continuity. We conclude with a review of the existing results for
pathwise convergence of weakly converging methods and the corresponding strong
results available under re-embedding.Comment: 12 pages, 2nd revision for IMA J Numerical Analysis. Further minor
errors correcte
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