## The Annals of Probability

### Limit Distributions for the Error in Approximations of Stochastic Integrals

Holger Rootzen

#### 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.

#### Article information

Source
Ann. Probab., Volume 8, Number 2 (1980), 241-251.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176994774

Digital Object Identifier
doi:10.1214/aop/1176994774

Mathematical Reviews number (MathSciNet)
MR566591

Zentralblatt MATH identifier
0428.60068

JSTOR