Suppose that $|\Lambda_t|$ is an $\mathbb{R}^d$-stochastic process with differential of $|\Lambda_t|^2$ being \begin{align*} d |\Lambda_t|^2 & = 2 e^{\delta t } \langle G(U_t), \Lambda_t\rangle dt \end{align*} where $\delta>0$ and $U_t$ is a given process.
Now fix $\theta>0$, apply Ito's formula to $f(x)= \sqrt{x+ \theta }$ and use the previous inequality, then: \begin{align*} |\Lambda_t |& \leq \sqrt{|\Lambda_t |^2+ \theta } \\ & = \sqrt{|v|^2+\theta}+\int_0^t \frac{1}{\sqrt{|\Lambda_s|^2+\theta}} e^{\delta s} \langle G(U_s), \Lambda_s\rangle ds \end{align*}
Was the previous calculation correct? Could I apply Ito's formula to $f(x)= \sqrt{x+ \theta }$ even if it is not $\mathcal{C}^2(\mathbb{R})$?