13,664 research outputs found
A General Framework for the Construction and the Smoothing of Forward Rate Curves
This paper establishes a general theoretical and numerical framework for the construction and the smoothing of instantaneous forward rate curves. It is shown that if the smoothness of a curve is defined as an integral of a function in the derivatives of the curve, then the optimal curves are splines that satisfy certain ordinary differential equations. For such curves, and efficient numerical method is given for the determination of the spline parameters subject to mild assumptions. The resulting forward rate curves do not generally possess the desired degree of smoothness due mainly to the constraints imposed on the curves by the various market observed prices. A Partial solution to this problem is then introduced which achieves additional smoothing by taking into account the bid-ask ranges of each market rate. This eliminates much of the oscillatory patterns and the points of high curvature, and produces curves that are ideal for applications such as the estimation of interest rate models, and the pricing and risk management of interest rate derivatives, which are sensitive to forward rate curves.
Normal form approach to unconditional well-posedness of nonlinear dispersive PDEs on the real line
In this paper, we revisit the infinite iteration scheme of normal form
reductions, introduced by the first and second authors (with Z. Guo), in
constructing solutions to nonlinear dispersive PDEs. Our main goal is to
present a simplified approach to this method. More precisely, we study normal
form reductions in an abstract form and reduce multilinear estimates of
arbitrarily high degrees to successive applications of basic trilinear
estimates. As an application, we prove unconditional well-posedness of
canonical nonlinear dispersive equations on the real line. In particular, we
implement this simplified approach to an infinite iteration of normal form
reductions in the context of the cubic nonlinear Schr\"odinger equation (NLS)
and the modified KdV equation (mKdV) on the real line and prove unconditional
well-posedness in with (i) for the cubic NLS
and (ii) for the mKdV. Our normal form approach also allows us
to construct weak solutions to the cubic NLS in , , and distributional solutions to the mKdV in (with some uniqueness statements).Comment: 60 pages. Typos corrected. To appear in Ann. Fac. Sci. Toulouse Mat
A remark on normal forms and the "upside-down" I-method for periodic NLS: growth of higher Sobolev norms
We study growth of higher Sobolev norms of solutions to the one-dimensional
periodic nonlinear Schrodinger equation (NLS). By a combination of the normal
form reduction and the upside-down I-method, we establish \|u(t)\|_{H^s}
\lesssim (1+|t|)^{\alpha (s-1)+} with \alpha = 1 for a general power
nonlinearity. In the quintic case, we obtain the above estimate with \alpha =
1/2 via the space-time estimate due to Bourgain [4], [5]. In the cubic case, we
concretely compute the terms arising in the first few steps of the normal form
reduction and prove the above estimate with \alpha = 4/9. These results improve
the previously known results (except for the quintic case.) In Appendix, we
also show how Bourgain's idea in [4] on the normal form reduction for the
quintic nonlinearity can be applied to other powers.Comment: 24 pages. Small modification in Section 1, to appear in J. Anal. Mat
Reply to CatalĂĄn : double-proton-transfer dynamics of photo-excited 7-azaindole dimers
The letter by CatalĂĄn (1) is concerned with the nature of double-proton transfer in dimers of 7-azaindole (7-AI), and the pertinent issue is whether or not the reaction is concerted. The subject is not new, and, for organic reactions, it has been discussed in the literature for decades. It is now understood that the concerted/consecutive mechanism has to be defined based on the timescale of the vibration motions and the family of coherent trajectories involved (ref. 2 and references therein). For double-proton transfer in isolated dimers, this timescale criterion has been invoked, and, as supported by a variety of time-resolved experiments in several groups and also by theory (ref. 3 and references therein), the conclusion is that the reaction is not concerted on the timescale of the vibrations involved; CatalĂĄn and some researchers (see ref. 3) assert that the two protons move in exact concert, maintaining the C_(2h) symmetry at all times
Late Time Behaviors of an Inhomogeneous Rolling Tachyon
We study an inhomogeneous decay of an unstable D-brane in the context of
Dirac-Born-Infeld~(DBI)-type effective action. We consider tachyon and
electromagnetic fields with dependence of time and one spatial coordinate, and
an exact solution is found under an exponentially decreasing tachyon potential,
, which is valid for the description of the late time
behavior of an unstable D-brane. Though the obtained solution contains both
time and spatial dependence, the corresponding momentum density vanishes over
the entire spacetime region. The solution is governed by two parameters. One
adjusts the distribution of energy density in the inhomogeneous direction, and
the other interpolates between the homogeneous rolling tachyon and static
configuration. As time evolves, the energy of the unstable D-brane is converted
into the electric flux and tachyon matter.Comment: 17 pages, 1 figure, version to appear in PR
State Variables and the Affine Nature of Markovian HJM Term Structure Models
Finite dimensional Markovian HJM term structure models provide an ideal setting for the study of term structure dynamics and interest rate derivatives where the flexibility of the HJM framework and the tractability of Markovian models coexist. Consequently, these models became the focus of a series of papers including Carverhill (1994), Ritchken and Sankarasuramanian (1995), Bhar and Chiarella (1997), Inui and Kijima (1998) and de Jong and Santa-Clara (1999). In Chiarella and Kwon (2001b), a common generalisation of these models was obtained in which the components of the forward rate volatility process satisfied ordinary differential equations in the maturity variable. However, the generalised models require the introduction of a large number of state variables which, at first sight, do not appear to have clear links to market observed quantities. In this paper, it is shown that the forward rate curves for these models can often be expressed as affine functions of the state variables, and conversely that the state variables in these models can often be expressed as affine functions of a finite number of benchmark forward rates. Consequently, for these models, the entire forward rate curve is not only Markov but affine with respect to a finite number of benchmark forward rates. It is also shown that the forward rate curve can be expressed as an affine function of a finite number of yields which are directly observed in the market. This property is useful, for example, in the estimation of model parameters. Finally, an explicit formula for the bond price in terms of the state variables, generalising the formula given in Inui and Kijima (1998), is provided for the models considered in this paper.
A Complete Stochastic Volatility Model in the HJM Framework
This paper considers a stochastic volatility version of the Heath, Jarrow and Morton (1992) term structure model. Market completeness is obtained by adapting the Hobson and Rogers (1998) complete stochastic volatility stock market model to the interest rate setting. Numerical simulation for a special case is used to compare the stochastic volatility model against the traditional Vasicek (1977) model.
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