The Annals of Applied Probability
- Ann. Appl. Probab.
- Volume 23, Number 2 (2013), 834-857.
Error distributions for random grid approximations of multidimensional stochastic integrals
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.
Ann. Appl. Probab., Volume 23, Number 2 (2013), 834-857.
First available in Project Euclid: 12 February 2013
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
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. https://projecteuclid.org/euclid.aoap/1360682031