Approximation of Evolution Equations

Description

We study algorithms for approximation of stochastic evolution equations. The latter are used, e.g., for modelling population dynamics, kinetics of chemical reactions, and interest rate dynamics in mathematical finance, and they may be considered as infinite-dimensional counterparts to systems of ordinary stochastic differential equations. We wish to determine algorithms that optimally relate error and cost. In order to establish optimality lower bounds, i.e., the following kind of results, are needed: the error of any algorithm with computational cost at most N is at least e(N). Here e(N) does not depend on the specific algorithm but only on the evolution equation under consideration. The use of non-uniform time discrectizations, or more generally adaptive time discretizations, is crucial for construction of almost optimal algorithms, i.e., for algorithms with cost at most N and error close to e(N). The asymptotic analysis is supplemented by simulation experiments.

For illustration we show a single trajectory of the solution of a heat equation

dX(t) = Δ X(t)dt + BdW(t)

with additive noise in spatial dimension one. The noise is given by a trace-class Brownian motion W in L2 ( [0,1] ) and B(h) = f ⋅ h with f(u) = u if
0 ≤ u ≤ 1/2 and f(u) = 0 if 1/2 < u ≤ 1.Click here for the same trajectory from a different perspective.

Publications

Sponsor

Deutsche Forschungsgemeinschaft (DFG RI599/3)

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