Optimal approximation rate of certain stochastic integrals

Abstract

Given an increasing function $H:[0,1)\to [0,\infty )$ and $$A_n(H):=\underset{\tau \in {\mathcal{T}}_n}{\inf }(\sum _{i=1}^n{\int }_{t_{i-1}}^{t_i}(t_i-t)H^2(t)dt{)}^{1/2},$$ where ${\mathcal{T}}_n:=\{\tau =(t_i{)}_{i=0}^n:0=t_0<t_1<...<t_n=1\}$, we characterize the property $A_n(H)\le \frac{c}{\sqrt{n}}$, and give conditions for $A_n(H)\le \frac{c}{\sqrt{n^{\beta }}}$ and $A_n(H)\ge \frac{1}{c\sqrt{n^{\beta }}}$ for $\beta \in (0,1)$, both in terms of integrability properties of $H$. These results are applied to the approximation of certain stochastic integrals.

23 pages; typos corrected, minor changes

References in corpus (1)

• On discrete time hedging in d-dimensional option pricing models