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Design of high-order short-time approximations as a problem of matching the covariance of a Brownian motion

Cristian Predescu
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Abstract

One of the outstanding problems in the numerical discretization of the Feynman-Kac formula calls for the design of arbitrary-order short-time approximations that are constructed in a stable way, yet only require knowledge of the potential function. In essence, the problem asks for the development of a functional analogue to the Gauss quadrature technique for one-dimensional functions. In PRE 69, 056701 (2004), it has been argued that the problem of designing an approximation of order νis equivalent to the problem of constructing discrete-time Gaussian processes that are supported on finite-dimensional probability spaces and match certain generalized moments of the Brownian motion. Since Gaussian processes are uniquely determined by their covariance matrix, it is tempting to reformulate the moment-matching problem in terms of the covariance matrix alone. Here, we show how this can be accomplished.

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Cristian Predescu. 2005-08-28. Design of high-order short-time approximations as a problem of matching the covariance of a Brownian motion. https://arxiv.org/abs/math-ph/0508017

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