Numerical methods for nonlinear stochastic differential equations with jumps

We present and analyse two implicit methods for Ito stochastic differential equations (SDEs) with Poisson-driven jumps. The first method, SSBE, is a split-step extension of the backward Euler method. The second method, CSSBE, arises from the introduction of a compensated, martingale, form of the Poisson process. We show that both methods are amenable to rigorous analysis when a one-sided Lipschitz condition, rather than a more restrictive global Lipschitz condition, holds for the drift. Our analysis covers strong convergence and nonlinear stability. We prove that both methods give strong convergence when the drift coefficient is one-sided Lipschitz and the diffusion and jump coefficients are globally Lipschitz. On the way to proving these results, we show that a compensated form of the Euler–Maruyama method converges strongly when the SDE coefficients satisfy a local Lipschitz condition and the pth moment of the exact and numerical solution are bounded for some p>2. Under our assumptions, both SSBE and CSSBE give well-defined, unique solutions for sufficiently small stepsizes, and SSBE has the advantage that the restriction is independent of the jump intensity. We also study the ability of the methods to reproduce exponential mean-square stability in the case where the drift has a negative one-sided Lipschitz constant. This work extends the deterministic nonlinear stability theory in numerical analysis. We find that SSBE preserves stability under a stepsize constraint that is independent of the initial data. CSSBE satisfies an even stronger condition, and gives a generalization of B-stability. Finally, we specialize to a linear test problem and show that CSSBE has a natural extension of deterministic A-stability. The difference in stability properties of the SSBE and CSSBE methods emphasizes that the addition of a jump term has a significant effect that cannot be deduced directly from the non-jump literature.This work was supported by Engineering and Physical Sciences Research Council grant GR/T19100 and by a Research Fellowship from The Royal Society of Edinburgh/Scottish Executive Education and Lifelong Learning Department.     

The spectrum of relativistic one-electron atoms according to Bethe and Salpeter

Bethe and Salpeter introduced a relativistic equation — different from the Bethe-Salpeter equation — which describes relativistic multi-particle systems. Here we will begin some basic work concerning its mathematical structure. In particular we show self-adjointness of the one-particle operator which will be a consequence of a sharp Sobolev type inequality yielding semi-boundedness of the corresponding sesquilinear form. Moreover we locate the essential spectrum of the operator and show the absence of singular continuous spectrum.     

Numerical simulation of a strongly nonlinear Ait-Sahalia-type interest rate model

We are interested in the strong convergence of Euler-Maruyama type approximations to the solution of a class of stochastic differential equations models with highly nonlinear coefficients, arising in mathematical finance. Results in this area can be used to justify Monte Carlo simulations for calibration and valuation. The equations that we study include the Ait-Sahalia type model of the spot interest rate, which has a polynomial drift term that blows up at the origin and a diffusion term with superlinear growth. After establishing existence and uniqueness for the solution, we show that an appropriate implicit numerical method preserves positivity and boundedness of moments, and converges strongly to the true solution.     

Score tests for inverse Gaussian mixtures

The mixed inverse Gaussian given by Whitmore (biScand. J. Statist., 13 , 1986, 211–220) provides a convenient way for testing the goodness‐of‐fit of a pure inverse Gaussian distribution. The test is a one‐sided score test with the null hypothesis being the pure inverse Gaussian (i.e. the mixing parameter is zero) and the alternative a mixture. We devise a simple score test and study its finite sample properties. Monte Carlo results show that it compares favourably with the smooth test of Ducharme ( Test , 10 , 2001, 271‐290). In practical applications, when the pure inverse Gaussian distribution is rejected, one is interested in making inference about the general values of the mixing parameter. However, as it is well known that the inverse Gaussian mixture is a defective distribution; hence, the standard likelihood inference cannot be applied. We propose several alternatives and provide score tests for the mixing parameter. Finite sample properties of these tests are examined by Monte Carlo simulation. Copyright © 2010 John Wiley & Sons, Ltd.     

N,N′ and N,O chelated phosphenium cations containing aminotroponiminate or aminotroponate units

New phosphenium cations stabilized by bidentate monoanionic N-isopropyl-2-(isopropylamino)troponiminate or 2-(isopropylamino)troponate units have been synthesized. These complexes were characterized by ³¹P, ¹H and ¹³C NMR spectroscopies, and the molecular structures were determined by X-ray crystallography. These data indicate the formation of N,N′- and N,O-chelate derivatives having three-coordinate phosphorous atoms included in planar heterobicycles. Moreover, computational studies support the presence of high delocalization of the positive charge into the π-conjugated carbon backbone and of the high-lying phosphorus lone-pair orbital.     

catena‐Poly­[[di­aqua­cobalt(II)]‐μ‐oxalato]

Single crystals of the title compound, [Co(C₂O₄)(H₂O)₂]_n, have been prepared by hydro­thermal methods and characterized by X‐ray diffraction analysis. The crystal structure consists of infinite one‐dimensional chains of di­aqua­cobalt(II) units bridged by oxalate groups. These chains lie on twofold symmetry axes parallel to the b axis, and the [Co(C₂O₄)]_n system is nearly planar within experimental error. The cobalt(II) coordination polyhedra are irregular octahedra, with oxalate O atoms at the equatorial positions and water mol­ecules at the axial positions. The chains are linked by hydrogen bonds via the water mol­ecules.     

Universal localization of triangular matrix rings

If is a triangular matrix ring, the columns and are f.g. projective -modules. We describe the universal localization of which makes invertible an -module morphism , generalizing a theorem of A. Schofield. We also describe the universal localization of -modules.

     

An answer to A.D.Wallace's question about countably compact cancellative semigroups

It is shown under CH that there exists a countably compact topological semigroup with two-sided cancellation which is not a topological group. Wallace's question" of 40 years standing is thus settled in the negative unless CH is explicitly denied. The example is a topological subsemigroup of an uncountable product of circle groups.

     

Preparation of a 4-monoketal of juglone methyl ether

Oxidation of 5-methoxy-1-naphthol with thallium trinitrate in ethylene glycol/trimethylorthoformate gives the 4-ethylene ketal of juglone methyl ether.