Iam trying to understand the following inequality:$p>0$ Let $$T_{m}:=\sum_{i=1}^{m}\left(\left|\int\limits_{\frac{i}{n}}^{\frac{i-1}{n}}g(s)dW_{s}\right|^{p}-\left|g \left(\frac{i-1}{n}\right)\left(W_{\frac{i}{n}}-W_{\frac{i-1}{n}}\right)\right|^{p}\right)$$ Then wen have $$\left|T_{m}\right|\leq C(p)\sum_{i=1}^{m}\left|g\left(\frac{i-1}{n}\right)\left(W_{\frac{i}{n}}-W_{\frac{i-1}{n}}\right)\right|^{max\{p-1,0\}}\left|\int\limits_{\frac{i}{n}}^{\frac{i-1}{n}}g(s)dW_{s}-g \left(\frac{i-1}{n}\right)\left(W_{\frac{i}{n}}-W_{\frac{i-1}{n}}\right)\right|^{min\{p,1\}}+C(p)\sum_{i=1}^{m}\left|\int\limits_{\frac{i}{n}}^{\frac{i-1}{n}}g(s)dW_{s}-g \left(\frac{i-1}{n}\right)\left(W_{\frac{i}{n}}-W_{\frac{i-1}{n}}\right)\right|^{p}$$
Where $(W_{t})_{t\geq 0}$ is a Standard brownian motion and $C(p)$ is a constant which only depends on $p$. I thought that it is an application of the mean value theorem applied to the function $f$ given by $f(x)=\left|x-g \left(\frac{i-1}{n}\right)\left(W_{\frac{i}{n}}-W_{\frac{i-1}{n}}\right)\right|^{p}$ for $p>1$ and and an application of the elementary inequality $\left|\left|x\right|^p-\left|y\right|^p\right|\leq \left|x-y\right|^p$ for $p\leq 1 $. But unfortunately I cant really prove it.
I think that I have to apply the Mean Value Theorem in the random interval $$\left[\int_{\frac{i-1}{n}}^{\frac{i}{n}}g(s)dW_{s}, g\left(\frac{i-1}{n}\right)\left(W_{\frac{i}{n}}-W_{\frac{i-1}{n}}\right)\right].$$ But my function $f$, defined above doesn't seem to be the appropriate one. Maybe someone else has a better idea.