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\(\varepsilon\)-strong simulation for multidimensional stochastic differential equations via rough path analysis. (English) Zbl 1436.65012
Given a multidimensional Itô problem with continuous drift and variance that defines a stochastic process \(X\), the authors use rough path analysis to construct a family of processes \(Y\) that depend on a tolerance parameter \(E\) lying in the interval \((0,1)\). This is known as tolerance enforced simulation. Here \(Y\) is piecewise constant with a finite number of discontinuities and where \(\sup ||X-Y|| < E\) in the infinity norm. The approach uses the Itô-Lyons map and its continuity properties are studied through Lyon’s rough path theory. This theory allows us to characterise the solution of the stochastic differential equation on a path by path basis, free of probability, by imposing constraints on the iterated integrals that arise with respect to the underlying Wiener processes that define the problem. The Itô-Lyons map is known to be continuous under a suitable Hölder metric defined on the space of rough paths and the size of the tolerance \(E\) is related to size of the index of this Hölder metric.

65C30 Numerical solutions to stochastic differential and integral equations
60L20 Rough paths
65C05 Monte Carlo methods
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