13,631 research outputs found
A Penalty Method for the Numerical Solution of Hamilton-Jacobi-Bellman (HJB) Equations in Finance
We present a simple and easy to implement method for the numerical solution
of a rather general class of Hamilton-Jacobi-Bellman (HJB) equations. In many
cases, the considered problems have only a viscosity solution, to which,
fortunately, many intuitive (e.g. finite difference based) discretisations can
be shown to converge. However, especially when using fully implicit time
stepping schemes with their desirable stability properties, one is still faced
with the considerable task of solving the resulting nonlinear discrete system.
In this paper, we introduce a penalty method which approximates the nonlinear
discrete system to first order in the penalty parameter, and we show that an
iterative scheme can be used to solve the penalised discrete problem in
finitely many steps. We include a number of examples from mathematical finance
for which the described approach yields a rigorous numerical scheme and present
numerical results.Comment: 18 Pages, 4 Figures. This updated version has a slightly more
detailed introduction. In the current form, the paper will appear in SIAM
Journal on Numerical Analysi
Evanescence in Coined Quantum Walks
In this paper we complete the analysis begun by two of the authors in a
previous work on the discrete quantum walk on the line [J. Phys. A 36:8775-8795
(2003) quant-ph/0303105 ]. We obtain uniformly convergent asymptotics for the
"exponential decay'' regions at the leading edges of the main peaks in the
Schr{\"o}dinger (or wave-mechanics) picture. This calculation required us to
generalise the method of stationary phase and we describe this extension in
some detail, including self-contained proofs of all the technical lemmas
required. We also rigorously establish the exact Feynman equivalence between
the path-integral and wave-mechanics representations for this system using some
techniques from the theory of special functions. Taken together with the
previous work, we can now prove every theorem by both routes.Comment: 32 pages AMS LaTeX, 5 figures in .eps format. Rewritten in response
to referee comments, including some additional references. v3: typos fixed in
equations (131), (133) and (134). v5: published versio
Hawking radiation as tunneling from charged black holes in 0A string theory
There has been much work on explaining Hawking radiation as a quantum
tunneling process through horizons. Basically, this intuitive picture requires
the calculation of the imaginary part of the action for outgoing particle. And
two ways are known for achieving this goal: the null-geodesic method and the
Hamilton-Jacobi method. We apply these methods to the charged black holes in 2D
dilaton gravity which is originated from the low energy effective theory of
type 0A string theory. We derive the correct Hawking temperature of the black
holes including the effect of the back reaction of the radiation, and obtain
the entropy by using the 1st law of black hole thermodynamics. For fixed-charge
ensemble, the 0A black holes are free of phase transition and thermodynamically
stable regardless of mass-charge ratio. We show this by interpreting the back
reaction term as the inverse of the heat capacity of the black holes. Finally,
the possibility of the phase transition in the fixed-potential ensemble is
discussed.Comment: 12 pages; v2: references added, revised with some changes in formula
and unaltered conclusions, to be published in Phys. Lett.
Nonlinear Scattering of a Bose-Einstein Condensate on a Rectangular Barrier
We consider the nonlinear scattering and transmission of an atom laser, or
Bose-Einstein condensate (BEC) on a finite rectangular potential barrier. The
nonlinearity inherent in this problem leads to several new physical features
beyond the well-known picture from single-particle quantum mechanics. We find
numerical evidence for a denumerably infinite string of bifurcations in the
transmission resonances as a function of nonlinearity and chemical potential,
when the potential barrier is wide compared to the wavelength of oscillations
in the condensate. Near the bifurcations, we observe extended regions of
near-perfect resonance, in which the barrier is effectively invisible to the
BEC. Unlike in the linear case, it is mainly the barrier width, not the height,
that controls the transmission behavior. We show that the potential barrier can
be used to create and localize a dark soliton or dark soliton train from a
phonon-like standing wave.Comment: 15 pages, 15 figures, new version includes clarification of
definition of transmission coefficient in general nonlinear vs. linear cas
Improving Tatonnement Methods for Solving Heterogeneous Agent Models
This paper modifies standard block Gauss-Seidel iterations used by tatonnement methods for solving large scale deterministic heterogeneous agent models. The composite method between first- and second-order tatonnement methods is shown to considerably improve convergence both in terms of speed as well as robustness relative to conventional first-order tatonnement methods. In addition, the relative advantage of the modified algorithm increases in the size and complexity of the economic model. Therefore, the algorithm allows significant reductions in computational time when solving large models. The algorithm is particularly attractive since it is easy to implement - it only augments conventional and intuitive tatonnement iterations with standard numerical methods.
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