64,004 research outputs found

    A comparative linear mean-square stability analysis of Maruyama- and Milstein-type methods

    Full text link
    In this article we compare the mean-square stability properties of the Theta-Maruyama and Theta-Milstein method that are used to solve stochastic differential equations. For the linear stability analysis, we propose an extension of the standard geometric Brownian motion as a test equation and consider a scalar linear test equation with several multiplicative noise terms. This test equation allows to begin investigating the influence of multi-dimensional noise on the stability behaviour of the methods while the analysis is still tractable. Our findings include: (i) the stability condition for the Theta-Milstein method and thus, for some choices of Theta, the conditions on the step-size, are much more restrictive than those for the Theta-Maruyama method; (ii) the precise stability region of the Theta-Milstein method explicitly depends on the noise terms. Further, we investigate the effect of introducing partially implicitness in the diffusion approximation terms of Milstein-type methods, thus obtaining the possibility to control the stability properties of these methods with a further method parameter Sigma. Numerical examples illustrate the results and provide a comparison of the stability behaviour of the different methods.Comment: 19 pages, 10 figure

    Unified Solution of the Expected Maximum of a Random Walk and the Discrete Flux to a Spherical Trap

    Full text link
    Two random-walk related problems which have been studied independently in the past, the expected maximum of a random walker in one dimension and the flux to a spherical trap of particles undergoing discrete jumps in three dimensions, are shown to be closely related to each other and are studied using a unified approach as a solution to a Wiener-Hopf problem. For the flux problem, this work shows that a constant c = 0.29795219 which appeared in the context of the boundary extrapolation length, and was previously found only numerically, can be derived explicitly. The same constant enters in higher-order corrections to the expected-maximum asymptotics. As a byproduct, we also prove a new universal result in the context of the flux problem which is an analogue of the Sparre Andersen theorem proved in the context of the random walker's maximum.Comment: Two figs. Accepted for publication, Journal of Statistical Physic
    corecore