Quadratic variation process of $G$–Brownian motion

I would like to prove the inequality $$\hat{\mathbb{E}}\left[\left(\int^T_0 \eta_t d \langle B \rangle_t \right)^2\right] \leq C \hat{\mathbb{E}}\left[ \int^T_0 \eta^2_t dt \right],$$ where $\langle B \rangle_t$ is the quadratic variation process of a G–Brownian motion and $$\int^T_0 \eta_t d \langle B \rangle_t = \sum^{N-1}_{k = 0} \xi_k (\langle B \rangle_{t_{k+1}} - \langle B \rangle_{t_k}).$$

This inequality is often used in articles about $G$-expectations, like http://arxiv.org/pdf/0711.2834v1.pdf.

This is what I have already tried \begin{align*} & \hat{\mathbb{E}}\left[\left(\int^T_0 \eta_t d \langle B \rangle_t \right)^2\right] \\ = & \hat{\mathbb{E}}\left[\left(\sum^{N-1}_{k = 0} \xi_k (\langle B \rangle_{t_{k+1}} - \langle B \rangle_{t_k}) \right)^2\right] \\ = & \hat{\mathbb{E}}\left[\sum^{N-1}_{k = 0} \xi^2_k (\langle B \rangle_{t_{k+1}} - \langle B \rangle_{t_k})^2 + 2 \sum^{N-1}_{k = 0} \sum^{N-1}_{l = 0, l < k} \xi_k (\langle B \rangle_{t_{k+1}} - \langle B \rangle_{t_k}) \xi_l (\langle B \rangle_{t_{l+1}} - \langle B \rangle_{t_l}) \right] \\ \end{align*} Then I was able to prove for the first term $$... \leq \hat{\mathbb{E}}\left[\sum^{N-1}_{k = 0} \xi^2_k \overline{\sigma}^4 (t_{k+1} - t_k)^2 \right] \leq T \overline{\sigma}^4 \hat{\mathbb{E}}\left[\int^T_0 |\eta_t|^2 dt \right],$$ but I don't know what to do with the second term.

• Two: suggestions: provide the definition of G-expectations and G-Brownian motion; show what you did to solve the question. – Did Apr 3 '13 at 7:31