Bound of stochastic integral $\int^T_0 g_sf_s dW_s$

Question

Let $W_s$ be $n$-dimensional Wiener process and $f_s$ and $g_s$ are some adapted processes in $\mathbb{R}^{n\times n}$ such that $\max_s|g_s|<C$ a.s. and $\mathbb{E}\int^T_0|f_s|^2 ds<\infty$, is it possible to obtain the inequality $$ \mathbb{E}\left| \int^T_0 g_sf_s dW_s\right|^2<C^2 \mathbb{E}\left| \int^T_0f_s dW_s\right|^2?$$

Answer 1

We can show this result by using Itô's isometry to convert the variance of the stochastic integral into an ordinary integral, where we can then use basic integral properties:

Assume that $\int_0^t|f_s|^2ds<\infty$, which then immediately implies $\int_0^t|g_sf_s|^2ds<\infty$ due to $g_s$ being uniformly bounded. Then note that $$\mathbb E\bigg[\bigg|\int_0^t g_sf_sdW_s\bigg|^2\bigg]\stackrel{\text{Itô isom}}{=}\mathbb E\bigg[\int_0^t \text{tr}\big(g_sf_s\cdot f_s^Tg_s^T\big)ds\bigg]=\mathbb E\bigg[\int_0^t\text{tr}(f_sg_s\cdot g_s^Tf_s^T\big)ds\bigg]$$ where we can then use the bound $|g_s|\leq C$ (where $|g_s|$ either denotes the operator norm or the euclidian norm) to get: $$\leq C^2\mathbb E\bigg[\int_0^t\text{tr}(f_s\cdot f_s^T)ds\bigg]\stackrel{\text{Itô isom.}}{=}C^2\mathbb E\bigg[\bigg|\int_0^t f_sdW_s\bigg|^2\bigg].$$

For a reference for the multivariate Itô isometry, see e.g. Baldi's "Stochastic Calculus" Proposition 8.5: