Some approximations of stochastic \(\theta\)-integrals (Q1568069)

The authors consider some approximations of stochastic \(\theta\)-integrals \((\theta)\int^t_0 f(B(s))dB(s)\) with respect to the Brownian motion \(B\), for \(f\) smooth enough, \(\theta\in [0,1]\), \(t\leq a\). Using the formula connecting Itô integral with \(\theta\)-integral, namely \[ (\theta) \int^t_0 f \bigl(B(s)\bigr)dB(s)= \int^t_0 f\bigl(B(s)\bigr)dB(s)+(\theta)\int^t_0 f'\bigl( B(s) \bigr)dB(s), \] they obtain that \(\theta\)-integral can be approximated by a sum of type \(\sum^p_{k=1}f_n (\overline B^\theta_n (t_{k-1})) [B_n^\theta (t_k)-B_n^\theta (t_{k-1})]\), where \(0=t_0< t_1< \cdots<t_m=a\) is a partition of the interval \([0,a]\), \(t_p\leq t<t_{p+1}\), \(\overline B^\theta_n (t_{k-1}) \in[B_n^\theta (t_{k-1}),B_n^\theta (t_k)]\) and \(f_n\) \((B^\theta_n)\) are convolution-type approximations of the function \(f\) (resp. of the Brownian motion \(B)\).