\(\varepsilon\)-strong simulation for multidimensional stochastic differential equations via rough path analysis.

*(English)*Zbl 1436.65012Given 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.

Reviewer: Kevin Burrage (Brisbane)