The Annals of Applied Probability

Error distributions for random grid approximations of multidimensional stochastic integrals

Carl Lindberg and Holger Rootzén

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This paper proves joint convergence of the approximation error for several stochastic integrals with respect to local Brownian semimartingales, for nonequidistant 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 SDEs, 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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Ann. Appl. Probab., Volume 23, Number 2 (2013), 834-857.

First available in Project Euclid: 12 February 2013

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Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems 60H05: Stochastic integrals 91G20: Derivative securities
Secondary: 60G44: Martingales with continuous parameter 60H35: Computational methods for stochastic equations [See also 65C30]

Approximation error random grid joint weak convergence multidimensional stochastic differential equation stochastic integrals random evaluation times discrete option hedging portfolio tracking error


Lindberg, Carl; Rootzén, Holger. Error distributions for random grid approximations of multidimensional stochastic integrals. Ann. Appl. Probab. 23 (2013), no. 2, 834--857. doi:10.1214/12-AAP858.

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