Strong predictor-corrector euler methods for stochastic differential equations

This paper introduces a new class of numerical schemes for the pathwise approximation of solutions of stochastic differential equations (SDEs). The proposed family of strong predictor-corrector Euler methods are designed to handle scenario simulation of solutions of SDEs. It has the potential to overcome some of the numerical instabilities that are often experienced when using the explicit Euler method. This is of importance, for instance, in finance where martingale dynamics arise for solutions of SDEs with multiplicative diffusion coefficients. Numerical experiments demonstrate the improved asymptotic stability properties of the proposed symmetric predictor-corrector Euler methods. © 2008 World Scientific Publishing Company

Hawking-like radiation does not require a trapped region

We discuss the issue of quasi-particle production by ``analogue black holes'' with particular attention to the possibility of reproducing Hawking radiation in a laboratory. By constructing simple geometric acoustic models, we obtain a somewhat unexpected result: We show that in order to obtain a stationary and Planckian emission of quasi-particles, it is not necessary to create a trapped region in the acoustic spacetime (corresponding to a supersonic regime in the fluid flow). It is sufficient to set up a dynamically changing flow asymptotically approaching a sonic regime with sufficient rapidity in laboratory time.Comment: revtex4, 4 pages, 1 figur

Analogue Cosmological Particle Creation: Quantum Correlations in Expanding Bose Einstein Condensates

We investigate the structure of quantum correlations in an expanding Bose Einstein Condensate (BEC) through the analogue gravity framework. We consider both a 3+1 isotropically expanding BEC as well as the experimentally relevant case of an elongated, effectively 1+1 dimensional, expanding condensate. In this case we include the effects of inhomogeneities in the condensate, a feature rarely included in the analogue gravity literature. In both cases we link the BEC expansion to a simple model for an expanding spacetime and then study the correlation structure numerically and analytically (in suitable approximations). We also discuss the expected strength of such correlation patterns and experimentally feasible BEC systems in which these effects might be detected in the near future.Comment: Reference adde

A Hardware Generator of Multi-Point Distributed Random Numbers for Monte Carlo Simulation

Monte Carlo simulation of weak approximation of stochastic differential equations constitutes an intensive computational task. In applications such as finance, for instance, to achieve "real time" execution, as often required, one needs highly efficient implementations of the multi-point distributed random number generator underlying the simulations. In this paper, a fast and flexible dedicated hardware solution on a field programmable gate array is presented. A comparative performance analysis between a software-only and the poposed hardware solution demonstrated that the hardware solution is bottleneck-free, retains the flexibility of the software solution and significantly increases the computational efficiency. Moreover, simulations in Applications wuch as economics insurance, physics, population dynamics, epidemiology, structural mechanics, checmistry and biotechnology can benefit from the obtained speedups

Theory of gravitation theories: a no-progress report

Already in the 1970s there where attempts to present a set of ground rules, sometimes referred to as a theory of gravitation theories, which theories of gravity should satisfy in order to be considered viable in principle and, therefore, interesting enough to deserve further investigation. From this perspective, an alternative title of the present paper could be ``why are we still unable to write a guide on how to propose viable alternatives to general relativity?''. Attempting to answer this question, it is argued here that earlier efforts to turn qualitative statements, such as the Einstein Equivalence Principle, into quantitative ones, such as the metric postulates, stand on rather shaky grounds -- probably contrary to popular belief -- as they appear to depend strongly on particular representations of the theory. This includes ambiguities in the identification of matter and gravitational fields, dependence of frequently used definitions, such as those of the stress-energy tensor or classical vacuum, on the choice of variables, etc. Various examples are discussed and possible approaches to this problem are pointed out. In the course of this study, several common misconceptions related to the various forms of the Equivalence Principle, the use of conformal frames and equivalence between theories are clarified.Comment: Invited paper in the Gravity Research Foundation 2007 special issue to be published by Int. J. Mod. Phys.

Numerical solution of stochastic differential equations with jumps in finance

University of Technology, Sydney. Faculty of Business.This thesis concerns the design and analysis of new discrete time approximations for stochastic differential equations (SDEs) driven by Wiener processes and Poisson random measures. In financial modelling, SDEs with jumps are often used to describe the dynamics of state variables such as credit ratings, stock indices, interest rates, exchange rates and electricity prices. The jump component can capture event-driven uncertainties, such as corporate defaults, operational failures or central bank announcements. The thesis proposes new, efficient, and numerically stable strong and weak approximations. Strong approximations provide efficient tools for problems such as filtering, scenario analysis and hedge simulation, while weak approximations are useful for handling problems such as derivative pricing, the evaluation of moments, and the computation of risk measures and expected utilities. The discrete time approximations proposed are divided into regular and jump-adapted schemes. Regular schemes employ time discretizations that do not include the jump times of the Poisson measure. Jump-adapted time discretizations, on the other hand, include these jump times. The first part of the thesis introduces stochastic expansions for jump diffusions and proves new, powerful lemmas providing moment estimates of multiple stochastic integrals. The second part presents strong approximations with a new strong convergence theorem for higher order general approximations. Innovative strong derivative-free and predictor-corrector schemes are derived. Furthermore, the strong convergence of higher order schemes for pure jump SDEs is established under conditions weaker than those required for jump diffusions. The final part of the thesis presents a weak convergence theorem for jump-adapted higher order general approximations. These approximations include new derivative-free, predictor-corrector, and simplified schemes. Finally, highly efficient implementations of simplified weak schemes based on random bit generators and hardware accelerators are developed and tested

A New Approach to Black Hole Microstates

If one encodes the gravitational degrees of freedom in an orthonormal frame field there is a very natural first order action one can write down (which in four dimensions is known as the Goldberg action). In this essay we will show that this action contains a boundary action for certain microscopic degrees of freedom living at the horizon of a black hole, and argue that these degrees of freedom hold great promise for explaining the microstates responsible for black hole entropy, in any number of spacetime dimensions. This approach faces many interesting challenges, both technical and conceptual.Comment: 6 pages, 0 figures, LaTeX; submitted to Mod. Phys. Lett. A.; this essay received "honorable mention" from the Gravity Research Foundation, 199

Sonoluminescence as a QED vacuum effect. II: Finite Volume Effects

In a companion paper [quant-ph/9904013] we have investigated several variations of Schwinger's proposed mechanism for sonoluminescence. We demonstrated that any realistic version of Schwinger's mechanism must depend on extremely rapid (femtosecond) changes in refractive index, and discussed ways in which this might be physically plausible. To keep that discussion tractable, the technical computations in that paper were limited to the case of a homogeneous dielectric medium. In this paper we investigate the additional complications introduced by finite-volume effects. The basic physical scenario remains the same, but we now deal with finite spherical bubbles, and so must decompose the electromagnetic field into Spherical Harmonics and Bessel functions. We demonstrate how to set up the formalism for calculating Bogolubov coefficients in the sudden approximation, and show that we qualitatively retain the results previously obtained using the homogeneous-dielectric (infinite volume) approximation.Comment: 23 pages, LaTeX 209, ReV-TeX 3.2, five figure