I need a certain estimation from a paper in which it has been described to use the Cauchy-Schwarz inequality and Jensen's inequality. I did some calculations and had a few tries but I don't seem to find the right approach.
The situation:
- Two sub-probability measures $\nu_s$ and $\mu_s$ (They're processes where for every $s\in[0,t]$ each is a sub-probability measure)
- Two continuous functions $\psi$ and $\varphi:\mathbb{R}\mapsto\mathbb{R}$, with $\Vert\psi\Vert_{\text{Lip}}$,$\Vert\varphi\Vert_{\text{Lip}}\leq 1$ and $|\psi(x)|,|\varphi|\leq 1+|x|$
- $\varphi$ is supported on $[-\lambda, \lambda]$ for $\lambda\in\mathbb{R}$
- A sub-gaussian random variable $M^*$
The inequality using Cauchy-Schwarz and Jensen's inequality: \begin{align*} \mathbb{E}[M^*\int_0^t|\int_\lambda^\infty (\psi-\varphi)d(\nu_s-\mu_s)|ds]\leq C\mathbb{E}[\int_0^t\int_\lambda^\infty |x|^2 d(\nu_s(x)+\mu_s(x))]. \end{align*}
As mentioned above, I didn't have much success in getting the inequality above. $\varphi$ should vanish as it is not supported on $[\lambda,\infty)$ and the Cauchy-Schwarz inequality should bring the square of $|x|^2$. Never the less I often ended up with a root on the expectation, to mention one problem I had.