Error distributions for random grid approximations of multidimensional stochastic integrals

Details Error distributions for random grid approximations of multidimensional stochastic integrals Error distributions for random grid approximations of multidimensional stochastic integrals

2011-08-25

arxiv.org/abs/1108.5063v1

Mathematics

Probability

Scientific paper

This paper proves joint convergence of the approximation error for several stochastic integrals with respect to local Brownian semimartingales, for non-equidistant and random grids. The conditions needed for convergence are that the Lebesgue integrals of the integrands tend uniformly to zero and that the squared variation and covariation processes converge. The paper also provides tools which simplify checking these conditions and which extend the range for the results. These results are used to prove an explicit limit theorem for random grid approximations of integrals based on solutions of multidimensional SDE-s, and to find ways to "design" and optimize the distribution of the approximation error. As examples we briefly discuss strategies for discrete option hedging.

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