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

On the magnetospheric ULF wave counterpart of substorm onset

One near‐ubiquitous signature of substorms observed on the ground is the azimuthal structuring of the onset auroral arc in the minutes prior to onset. Termed auroral beads, these optical signatures correspond to concurrent exponential increases in ground ultralow frequency (ULF) wave power and are likely the result of a plasma instability in the magnetosphere. Here, we present a case study showing the development of auroral beads from a Time History of Events and Macroscale Interactions during Substorms (THEMIS) all‐sky camera with near simultaneous exponential increases in auroral brightness, ionospheric and conjugate magnetotail ULF wave power, evidencing their intrinsic link. We further present a survey of magnetic field fluctuations in the magnetotail around substorm onset. We find remarkably similar superposed epoch analyses of ULF power around substorm onset from space and conjugate ionospheric observations. Examining periods of exponential wave growth, we find the ground‐ and space‐based observations to be consistent, with average growth rates of ∼0.01 s−1, lasting for ∼4 min. Cross‐correlation suggests that the space‐based observations lead those on the ground by approximately 1–1.5 min. Meanwhile, spacecraft located premidnight and ∼10 RE downtail are more likely to observe enhanced wave power. These combined observations lead us to conclude that there is a magnetospheric counterpart of auroral beads and exponentially increasing ground ULF wave power. This is likely the result of the linear phase of a magnetospheric instability, active in the magnetotail for several minutes prior to auroral breakup

Large Chiral Diffeomorphisms on Riemann Surfaces and W-algebras

The diffeomorphism action lifted on truncated (chiral) Taylor expansion of a complex scalar field over a Riemann surface is presented in the paper under the name of large diffeomorphisms. After an heuristic approach, we show how a linear truncation in the Taylor expansion can generate an algebra of symmetry characterized by some structure functions. Such a linear truncation is explicitly realized by introducing the notion of Forsyth frame over the Riemann surface with the help of a conformally covariant algebraic differential equation. The large chiral diffeomorphism action is then implemented through a B.R.S. formulation (for a given order of truncation) leading to a more algebraic set up. In this context the ghost fields behave as holomorphically covariant jets. Subsequently, the link with the so called W-algebras is made explicit once the ghost parameters are turned from jets into tensorial ghost ones. We give a general solution with the help of the structure functions pertaining to all the possible truncations lower or equal to the given order. This provides another contribution to the relationship between KdV flows and W-diffeomorphimsComment: LaTeX file, 31 pages, no figure. Version to appear in J. Math. Phys. Work partly supported by Region PACA and INF

Invariants of differential equations defined by vector fields

We determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold. We also find all invariants and differential invariants for this group up to the second order. A result on the characterization of classes of these equations by the invariant functions is also given.Comment: 13 page

Anomalous phase of MnP at very low field

Manganese phosphide MnP has been investigated for decades because of its rich magnetic phase diagram. It is well known that the MnP exhibits the ferromagnetic phase transition at \Tc=292 K and the helical magnetic phase below \TN=47 K at zero field. Recently, a novel magnetic phase transition was observed at $T^* = 282$ K when the magnetic field is lower than 5 Oe. However, the nature of the new phase has not been illuminated yet. In order to reveal it, we performed the AC and the DC magnetization measurements for a single crystal MnP at very low field. A divergent behavior of the real and the imaginary part of the AC susceptibility and a sharp increase of the DC magnetization was observed at $T^*$, indicating the magnetic phase transition at $T^*$. Furthermore a peculiar temperature hysteresis was observed: namely, the magnetization depends on whether cooling sample to the temperature lower than \TN or not before the measurements. This hysteresis phenomenon suggests the complicated nature of the new phase and a strong relation between the magnetic state of the new phase and the helical structure.Comment: 4 pages, 2 figure

Assessing the role that entertainers play in alcohol marketing and the maintenance of good order within on-trade licensed premises

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N-methyl-N-alkylpyrrolidinium nonafluoro-1-butanesulfonate salts : Ionic liquid properties and plastic crystal behaviour

A series of N-methyl-N-alkylpyrrolidinium nonafluoro-1-butanesulfonate salts were synthesised and characterised. The thermophysical characteristics of this family of salts have been investigated with respect to potential use as ionic liquids and solid electrolytes. N-Methyl-N-butylpyrrolidinium nonafluoro-1-butanesulfonate (p1,4NfO) has the lowest melting point of the family, at 94 &deg;C. Electrochemical analysis of p1,4 NfO in the liquid state shows an electrochemical window of ~6 V. All compounds exhibit one or more solid&ndash;solid transitions at sub-ambient temperatures, indicating the existence of plastic crystal phases.<br /

Penalty Methods for the Solution of Discrete HJB Equations -- Continuous Control and Obstacle Problems

In this paper, we present a novel penalty approach for the numerical solution of continuously controlled HJB equations and HJB obstacle problems. Our results include estimates of the penalisation error for a class of penalty terms, and we show that variations of Newton's method can be used to obtain globally convergent iterative solvers for the penalised equations. Furthermore, we discuss under what conditions local quadratic convergence of the iterative solvers can be expected. We include numerical results demonstrating the competitiveness of our methods.Comment: 31 Pages, 7 Figure

Isospin Splitting in the Baryon Octet and Decuplet

Baryon mass splittings are analyzed in terms of a simple model with general pairwise interactions. At present, the $\Delta$ masses are poorly known from experiments. Improvement of these data would provide an opportunity to make a significant test of our understanding of electromagnetic and quark-mass contributions to hadronic masses. The problem of determining resonance masses from scattering and production data is discussed.Comment: 9 pages, LATEX inc. 2 LATEX "pictures", CMU-HEP91-24-R9