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April, 1980 Limit Distributions for the Error in Approximations of Stochastic Integrals
Holger Rootzen
Ann. Probab. 8(2): 241-251 (April, 1980). DOI: 10.1214/aop/1176994774

Abstract

We consider the approximation of an Ito integral $\int^t_0 \phi(s)dB(s)$ by a sequence of integrals $\int^t_0 \phi_n(s)dB(s)$ of simpler integrands. It is proved that if, for a sequence $\{\psi_n\}$ of adapted integrands, $\sup_{0\leqslant t\leqslant 1}|\int^t_0 \psi_n ds| \rightarrow_p 0$ and $\int^t_0 \psi^2_n ds\rightarrow_p\tau(t)$, for some continuous stochastic process $\{\tau(t); t \in\lbrack 0, 1\rbrack\}$, then $\int_0^{(\cdot)}\psi_ndB \rightarrow_d W \circ \tau$ in $C(0, 1)$, where $W$ is a Brownian motion independent of $\tau$. Further, if one is only interested in the limit distribution of functionals like $\int^1_0\psi_n dB$ or $\sup_{0\leqslant t\leqslant 1}| \int^t_0\psi_n dB|$, then in the second condition it is enough to require that $\int^1_0\psi^2_nds \rightarrow_p\tau(1)$. The convergence is stable in the sense of Renyi, and from this follow results on the fluctuations of the sample paths of the integrals. As an example we consider the case $\phi(t) = f(B(t), t)$ and $\phi_n(t) = \sum^n_{i=1} f(B(i/n), i/n)I(i/n \leqslant t < (i + 1)/n)$. Denoting the approximation error $\int^t_0(\phi - \phi_n)dB$ by $d_n(t)$, it follows from the above results that if $f$ is smooth enough then $n^{\frac{1}{2}}d_n \rightarrow_d W \circ \tau$, with $\tau(t) = 2^{-1} \int^t_0 f_1(B(s), s)^2ds$ where $f_1(x, t) = \frac{\partial f(x, t)}{\partial x}$. Similar results are obtained for approximations of the Stratonovich integral and for higher order approximations.

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Holger Rootzen. "Limit Distributions for the Error in Approximations of Stochastic Integrals." Ann. Probab. 8 (2) 241 - 251, April, 1980. https://doi.org/10.1214/aop/1176994774

Information

Published: April, 1980
First available in Project Euclid: 19 April 2007

zbMATH: 0428.60068
MathSciNet: MR566591
Digital Object Identifier: 10.1214/aop/1176994774

Subjects:
Primary: 60H05
Secondary: 60B10, 60G45

Rights: Copyright © 1980 Institute of Mathematical Statistics

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Vol.8 • No. 2 • April, 1980
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