135 research outputs found
An SFPâFCC method for pricing and hedging early-exercise options under LĂ©vy processes
This paper extends the singular FourierâPadĂ© (SFP) method proposed by Chan [Singular FourierâPadĂ© series expansion of European option prices. Quant. Finance, 2018, 18, 1149â1171] for pricing/hedging early-exercise optionsâBermudan, American and discrete-monitored barrier optionsâunder a LĂ©vy process. The current SFP method is incorporated with the FilonâClenshawâCurtis (FCC) rules invented by DomĂnguez et al. [Stability and error estimates for FilonâClenshawâCurtis rules for highly oscillatory integrals. IMA J. Numer. Anal., 2011, 31, 1253â1280], and we call the new method SFPâFCC. The main purpose of using the SFPâFCC method is to require a small number of terms to yield fast error convergence and to formulate option pricing and option Greek curves rather than individual prices/Greek values. We also numerically show that the SFPâFCC method can retain a global spectral convergence rate in option pricing and hedging when the risk-free probability density function is piecewise smooth. Moreover, the computational complexity of the method is O((Lâ1)(N+1)( Ă log Ă)) with N, a (small) number of complex Fourier series terms, Ă, a number of Chebyshev series terms and L, the number of early-exercise/monitoring dates. Finally, we compare the accuracy and computational time of our method with those of existing techniques in numerical experiments
Nonnormality and the localized control of extended systems
The idea of controlling the dynamics of spatially extended systems using a
small number of localized perturbations is very appealing - such a setup is
easy to implement in practice. However, when the distance between controllers
generating the perturbations becomes large, control fails due to increasing
sensitivity of the system to noise and nonlinearities. We show that this
failure is due to the fact that the evolution operator for the controlled
system becomes increasingly nonnormal as the distance between controllers
grows. This nonnormality is the result of control and can arise even for
systems whose evolution operator is normal in the absence of control.Comment: 4 pages, 4 figure
Spectral theory for the failure of linear control in a nonlinear stochastic system
We consider the failure of localized control in a nonlinear spatially
extended system caused by extremely small amounts of noise. It is shown that
this failure occurs as a result of a nonlinear instability. Nonlinear
instabilities can occur in systems described by linearly stable but strongly
nonnormal evolution operators. In spatially extended systems the nonnormality
manifests itself in two different but complementary ways: transient
amplification and spectral focusing of disturbances. We show that temporal and
spatial aspects of the nonnormality and the type of nonlinearity are all
crucially important to understanding and describing the mechanism of nonlinear
instability. Presented results are expected to apply equally to other physical
systems where strong nonnormality is due to the presence of mean flow rather
than the action of control.Comment: Submitted to Physical Review
Population Dynamics and Non-Hermitian Localization
We review localization with non-Hermitian time evolution as applied to simple
models of population biology with spatially varying growth profiles and
convection. Convection leads to a constant imaginary vector potential in the
Schroedinger-like operator which appears in linearized growth models. We
illustrate the basic ideas by reviewing how convection affects the evolution of
a population influenced by a simple square well growth profile. Results from
discrete lattice growth models in both one and two dimensions are presented. A
set of similarity transformations which lead to exact results for the spectrum
and winding numbers of eigenfunctions for random growth rates in one dimension
is described in detail. We discuss the influence of boundary conditions, and
argue that periodic boundary conditions lead to results which are in fact
typical of a broad class of growth problems with convection.Comment: 19 pages, 11 figure
Sensitivity analysis of reactive ecological dynamics
Author Posting. © Springer, 2008. This is the author's version of the work. It is posted here by permission of Springer for personal use, not for redistribution. The definitive version was published in Bulletin of Mathematical Biology 70 (2008): 1634-1659, doi:10.1007/s11538-008-9312-7.Ecological systems with asymptotically stable equilibria may exhibit significant transient
dynamics following perturbations. In some cases, these transient dynamics include
the possibility of excursions away from the equilibrium before the eventual return; systems
that exhibit such amplification of perturbations are called reactive. Reactivity is
a common property of ecological systems, and the amplification can be large and long-lasting.
The transient response of a reactive ecosystem depends on the parameters of
the underlying model. To investigate this dependence, we develop sensitivity analyses
for indices of transient dynamics (reactivity, the amplification envelope, and the optimal
perturbation) in both continuous- and discrete-time models written in matrix form.
The sensitivity calculations require expressions, some of them new, for the derivatives
of equilibria, eigenvalues, singular values, and singular vectors, obtained using matrix
calculus. Sensitivity analysis provides a quantitative framework for investigating the
mechanisms leading to transient growth. We apply the methodology to a predator-prey
model and a size-structured food web model. The results suggest predator-driven and
prey-driven mechanisms for transient amplification resulting from multispecies interactions.Financial support provided by NSF grant DEB-0343820, NOAA grant NA03-NMF4720491,
the Ocean Life Institute of the Woods Hole Oceanographic Institution, and the Academic
Programs Office of the MIT-WHOI Joint Program in Oceanography
A bootstrap method for sum-of-poles approximations
A bootstrap method is presented for finding efficient sum-of-poles approximations of causal functions. The method is based on a recursive application of the nonlinear least squares optimization scheme developed in (Alpert et al. in SIAM J. Numer. Anal. 37:1138â1164, 2000), followed by the balanced truncation method for model reduction in computational control theory as a final optimization step. The method is expected to be useful for a fairly large class of causal functions encountered in engineering and applied physics. The performance of the method and its application to computational physics are illustrated via several numerical examples
Stabilization of Hydrodynamic Flows by Small Viscosity Variations
Motivated by the large effect of turbulent drag reduction by minute
concentrations of polymers we study the effects of a weakly space-dependent
viscosity on the stability of hydrodynamic flows. In a recent Letter [Phys.
Rev. Lett. {\bf 87}, 174501, (2001)] we exposed the crucial role played by a
localized region where the energy of fluctuations is produced by interactions
with the mean flow (the "critical layer"). We showed that a layer of weakly
space-dependent viscosity placed near the critical layer can have a very large
stabilizing effect on hydrodynamic fluctuations, retarding significantly the
onset of turbulence. In this paper we extend these observation in two
directions: first we show that the strong stabilization of the primary
instability is also obtained when the viscosity profile is realistic (inferred
from simulations of turbulent flows with a small concentration of polymers).
Second, we analyze the secondary instability (around the time-dependent primary
instability) and find similar strong stabilization. Since the secondary
instability develops around a time-dependent solution and is three-dimensional,
this brings us closer to the turbulent case. We reiterate that the large effect
is {\em not} due to a modified dissipation (as is assumed in some theories of
drag reduction), but due to reduced energy intake from the mean flow to the
fluctuations. We propose that similar physics act in turbulent drag reduction.Comment: 10 pages, 17 figs., REVTeX4, PRE, submitte
Nonlinear Instability in a Semiclassical Problem
We consider a nonlinear evolution problem with an asymptotic parameter and
construct examples in which the linearized operator has spectrum uniformly
bounded away from Re z >= 0 (that is, the problem is spectrally stable), yet
the nonlinear evolution blows up in short times for arbitrarily small initial
data.
We interpret the results in terms of semiclassical pseudospectrum of the
linearized operator: despite having the spectrum in Re z < -c < 0, the
resolvent of the linearized operator grows very quickly in parts of the region
Re z > 0. We also illustrate the results numerically
Interpolatory methods for model reduction of multi-input/multi-output systems
We develop here a computationally effective approach for producing
high-quality -approximations to large scale linear
dynamical systems having multiple inputs and multiple outputs (MIMO). We extend
an approach for model reduction introduced by Flagg,
Beattie, and Gugercin for the single-input/single-output (SISO) setting, which
combined ideas originating in interpolatory -optimal model
reduction with complex Chebyshev approximation. Retaining this framework, our
approach to the MIMO problem has its principal computational cost dominated by
(sparse) linear solves, and so it can remain an effective strategy in many
large-scale settings. We are able to avoid computationally demanding
norm calculations that are normally required to monitor
progress within each optimization cycle through the use of "data-driven"
rational approximations that are built upon previously computed function
samples. Numerical examples are included that illustrate our approach. We
produce high fidelity reduced models having consistently better
performance than models produced via balanced truncation;
these models often are as good as (and occasionally better than) models
produced using optimal Hankel norm approximation as well. In all cases
considered, the method described here produces reduced models at far lower cost
than is possible with either balanced truncation or optimal Hankel norm
approximation
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