An Implicit Euler Scheme with Non-uniform Time Discretization for Heat Equations with Multiplicative Noise

Details An Implicit Euler Scheme with Non-uniform Time Discretization for Heat Equations with Multiplicative Noise An Implicit Euler Scheme with Non-uniform Time Discretization for Heat Equations with Multiplicative Noise

2006-04-27

arxiv.org/abs/math/0604600v1

Mathematics

Probability

22 pages

Scientific paper

We present an algorithm for solving stochastic heat equations, whose key ingredient is a non-uniform time discretization of the driving Brownian motion $W$. For this algorithm we derive an error bound in terms of its number of evaluations of one-dimensional components of $W$. The rate of convergence depends on the spatial dimension of the heat equation and on the decay of the eigenfunctions of the covariance of $W$. According to known lower bounds, our algorithm is optimal, up to a constant, and this optimality cannot be achieved by uniform time discretizations.

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