Sharp error estimates for a discretisation of the 1D convection/diffusion equation with Dirac initial data

This paper derives sharp l$\infty$ and l1 estimates of the error arising from an explicit approximation of the constant coefficient 1D convection/diffusion equation with Dirac initial data. The analysis embeds the discrete equations within a semi-discrete system of equations which can be solved by Fourier analysis. The error estimates are then obtained through asymptotic approximation of the integrals resulting from the inverse Fourier transform. this research is motivated by the desire to prove convergence of approximations to adjoint partial differential equations

Convergence analysis of Crank-Nicolson and Rannacher time-marching

This paper presents a convergence analysis of Crank-Nicolson and Rannacher time-marching methods which are often used in finite difference discretisations of the Black-Scholes equations. Particular attention is paid to the important role of Rannacher's startup procedure, in which one or more initial timesteps use Backward Euler timestepping, to achieve second order convergence for approximations of the first and second derivatives. Numerical results confirm the sharpness of the error analysis which is based on asymptotic analysis of the behaviour of the Fourier transform. The relevance to Black-Scholes applications is discussed in detail, with numerical results supporting recommendations on how to maximise the accuracy for a given computational cost

Monte Carlo evaluation of sensitivities in computational finance

In computational finance, Monte Carlo simulation is used to compute the correct prices for financial options. More important, however, is the ability to compute the so-called "Greeks'', the first and second order derivatives of the prices with respect to input parameters such as the current asset price, interest rate and level of volatility.\ud \ud This paper discusses the three main approaches to computing Greeks: finite difference, likelihood ratio method (LRM) and pathwise sensitivity calculation. The last of these has an adjoint implementation with a computational cost which is independent of the number of first derivatives to be calculated. We explain how the practical development of adjoint codes is greatly assisted by using Algorithmic Differentiation, and in particular discuss the performance achieved by the FADBAD++ software package which is based on templates and operator overloading within C++.\ud \ud The pathwise approach is not applicable when the financial payoff function is not differentiable, and even when the payoff is differentiable, the use of scripting in real-world implementations means it can be very difficult in practice to evaluate the derivative of very complex financial products. A new idea is presented to address these limitations by combining the adjoint pathwise approach for the stochastic path evolution with LRM for the payoff evaluation

Preconditioned iterative solution of the 2D Helmholtz equation

Using a finite element method to solve the Helmholtz equation leads to a sparse system of equations which in three dimensions is too large to solve directly. It is also non-Hermitian and highly indefinite and consequently difficult to solve iteratively. The approach taken in this paper is to precondition this linear system with a new preconditioner and then solve it iteratively using a Krylov subspace method. Numerical analysis shows the preconditioner to be effective on a simple 1D test problem, and results are presented showing considerable convergence acceleration for a number of different Krylov methods for more complex problems in 2D, as well as for the more general problem of harmonic disturbances to a non-stagnant steady flow

Analysis of Adjoint Error Correction for Superconvergent Functional Estimates

Earlier work introduced the notion of adjoint error correction for obtaining superconvergent estimates of functional outputs from approximate PDE solutions. This idea is based on a posteriori error analysis suggesting that the leading order error term in the functional estimate can be removed by using an adjoint PDE solution to reveal the sensitivity of the functional to the residual error in the original PDE solution. The present work provides a priori error analysis that correctly predicts the behaviour of the remaining leading order error term. Furthermore, the discussion is extended from the case of homogeneous boundary conditions and bulk functionals, to encompass the possibilities of inhomogeneous boundary conditions and boundary functionals. Numerical illustrations are provided for both linear and nonlinear problems.\ud \ud This research was supported by EPSRC under grant GR/K91149, and by NASA/Ames Cooperative Agreement No. NCC 2-5431

Antithetic multilevel Monte Carlo estimation for multi-dimensional SDEs without L\'{e}vy area simulation

In this paper we introduce a new multilevel Monte Carlo (MLMC) estimator for multi-dimensional SDEs driven by Brownian motions. Giles has previously shown that if we combine a numerical approximation with strong order of convergence $O(\Delta t)$ with MLMC we can reduce the computational complexity to estimate expected values of functionals of SDE solutions with a root-mean-square error of $\epsilon$ from $O(\epsilon^{-3})$ to $O(\epsilon^{-2})$. However, in general, to obtain a rate of strong convergence higher than $O(\Delta t^{1/2})$ requires simulation, or approximation, of L\'{e}vy areas. In this paper, through the construction of a suitable antithetic multilevel correction estimator, we are able to avoid the simulation of L\'{e}vy areas and still achieve an $O(\Delta t^2)$ multilevel correction variance for smooth payoffs, and almost an $O(\Delta t^{3/2})$ variance for piecewise smooth payoffs, even though there is only $O(\Delta t^{1/2})$ strong convergence. This results in an $O(\epsilon^{-2})$ complexity for estimating the value of European and Asian put and call options.Comment: Published in at http://dx.doi.org/10.1214/13-AAP957 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

The optical counterpart of SAX J1808.4-3658, the transient bursting millisecond X-ray pulsar

A set of CCD images have been obtained during the decline of the X-ray transient SAX J1808.4-3658 during April-June 1998. The optical counterpart has been confirmed by several pieces of evidence. The optical flux shows a modulation on several nights which is consistent with the established X-ray binary orbit period of 2 hours. This optical variability is roughly in antiphase with the weak X-ray modulation. The source mean magnitude of V=16.7 on April 18 declined rapidly after April 22. From May 2 onwards the magnitude was more constant at around V=18.45 but by June 27 was below our sensitivity limit. The optical decline precedes the rapid second phase of the X-ray decrease by 3 +/- 1 days. The source has been identified on a 1974 UK Schmidt plate at an estimated magnitude of ~20. The nature of the optical companion is discussed.Comment: 5 pages, 3 figures; published in MNRAS, March 15th 199