Expectation of $1/V(t)$ finite, $V(t)$ strictly positive stochastic process

Question

How can we prove that the expectation of the stochastic process $1/V(t)$, $\forall t \in [0,T]$, is finite? \begin{eqnarray}\nonumber \mathbb{E} \left[ 1 / V(t) \right] &<& \infty, \end{eqnarray} where $V(t)$ is an stochastic process (mean-reverting): \begin{eqnarray}\nonumber dV(t) &=& \kappa \, (\theta - V(t)) \, dt + \eta \, \sqrt{V(t)} \, dW(t) \\ V(0) &=& v_0 > 0, \end{eqnarray} and the parameters are such that the process $V(t)$ has a unique global strong solution (Watanabe-Yamada theorem) and it is strictly positive (Feller condition). We do also assume that: \begin{eqnarray}\nonumber \mathbb{E} \left[ \int_0^T V(t)^2\, dt \right] < \infty. \end{eqnarray} (Main idea of the proof). Using Itô's lemma and Dynkin's formula, where $f(x)=1/x$ is a function of class $C^2$ on the interval $\left(0, \infty\right)$, we have \begin{eqnarray}\nonumber \mathbb{E}\left[ f(V(t)) \right] &=& f(V(0)) + \mathbb{E} \int_0^t \Delta f(V(s)) ds \\ &=& \frac{1}{V(0)} + \mathbb{E} \int_0^t \left( \frac{-\kappa( \theta - V(s))}{V^2(s)} + \frac{\eta^2}{V^2(s)} \right) ds \\ &=& \frac{1}{V(0)} + \kappa \, \mathbb{E} \int_0^t \frac{1}{V(s)} ds + \left(\eta^2 - \kappa \,\theta \right) \mathbb{E} \int_0^t \frac{1}{V^2(s)} ds. \end{eqnarray} By Fubini-Tonelli theorem, we change the order of integration to get \begin{eqnarray}\nonumber \mathbb{E}\left[ f(V(t)) \right] &=& \frac{1}{V(0)} + \kappa\int_0^t \mathbb{E} \left[ \frac{1}{V(s)} \right] ds + \left(\eta^2 - \kappa \, \theta \right) \int_0^t \mathbb{E} \left[ \frac{1}{V^2(s)} \right] ds. \end{eqnarray} If $\kappa \, \theta \geq \eta^2$, the third term is non-positive and upper bounded by zero, and we have \begin{eqnarray}\nonumber \mathbb{E}\left[ \frac{1}{V(t)} \right] &\leq& \frac{1}{v_0} + \kappa \int_0^t \mathbb{E} \left[ \frac{1}{V(s)} \right] ds. \end{eqnarray} By the Gronwall's inequality, we find the following upper bound, and the claim follows immediately for all $t \in [0,T]$ \begin{eqnarray}\nonumber \mathbb{E}\left[ \frac{1}{V(t)} \right] &\leq& \frac{1}{v_0} \, e^{\kappa t} \leq \frac{1}{v_0} \, e^{\kappa T} < \infty. \end{eqnarray} Therefore, the expectation of $1/V(t)$, $\forall t \in [0,T]$, is finite if the parameters $\kappa, \theta, \eta$ of the stochastic process $V(t)$ satisfy $\eta^2 \leq \kappa \, \theta$, which is stronger than the Feller condition ($\eta^2 \leq 2 \, \kappa \, \theta$). Otherwise, $\mathbb{E}\left[ 1/V(t) \right]$ may not be finite? Is it possible to find an upper bound when the inequality $\eta^2 \leq \kappa \, \theta$ is not satisfied? Or at least when the parameters obey the condition $\eta^2 \leq 2 \, \kappa \, \theta$ (weaker condition)?