I am trying to understand Feller’s square root condition which provides the existence of the positive solution of the Cox-Ingersoll-Ross which introduced by Gikhman, Ilya
please find link to the note feller_condition
what I don't really understand is how he deal with the stochastic term in the following context
$$v^{-n} (T_\varepsilon \wedge t) = v^{-n}(0) +\int_0^{T_\varepsilon \wedge t}n\kappa(v(s) - \gamma)v^{-(n+1)} (s)ds -\int_0^{T_\varepsilon \wedge t}n\eta \sqrt{v(s)}v^{-(n+1)}(s)dW_s\\ + \frac{1}{2}\int_0^{T_\varepsilon \wedge t}n(n+1)\eta^2v(s)v^{-(n+2)} (s)d s$$ $$=v^{-n}(0) + n\kappa \int_0^t v^{-n}(T_\varepsilon \wedge s)ds -\int_0^t n\eta v^{-(n+1/2)} (T_\varepsilon \wedge s)dW_s +\int_0^t[\frac{n(n+1)}{2}\eta^2-n\kappa \gamma]v^{-(n+1)}(T_\varepsilon \wedge s)ds$$.
Then he end up by saying taking expectation in the latter equality, we arrive at the estimate
$$\mathbb{E}[v^{-n} (T_\varepsilon \wedge t)] \leq v^{-n}(0) + n\kappa \int_0^t \mathbb{E}[v^{-n}(T_\varepsilon \wedge s)]ds.$$
so my question here where the stochastic term disappears, it seems he deal with it as a positive term which yields the last estimate but I don't understand why or if there is any other explanation to it?!
any advice will be appreciated.