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__Computing deterministic quadrature rules for marginals of SDEs.__

L. Yaroslavsteva and T. Müller-Gronbach

**Abstract**

We consider the problem of approximating the marginal distribution of the solution of an SDE by probability measures with finite support. We study deterministic algorithms in a worst case analysis w.r.t. classes of SDEs, defined in terms of smoothness constraints on their coefficients. The cost of an
approximation is given by the sum of the size of its support and the number of evaluations of the coefficients used to compute the corresponding nodes and weights. The error is defined as the worst case
quadrature error of the resulting quadrature rule over a class of test functions. We present sharp asymptotic lower and upper bounds on the respective N-th minimal errors in terms of the smoothness of the coefficients
and the test functions. Moreover, we present an algorithm, which is based on a space discretization of a weak Taylor scheme and a support reduction strategy. It is easy to implement and it performs asymptotically optimal in many cases. We also show numerical examples.