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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.

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