Suppose I have a stochastic differential equation ($X_t$ is a vector) $dX_t = f(X_t) dt + \sigma g(X_t) d\eta(t)$ and define $V = \sum_{i=1}^{n} x_i$. Here, $\eta(t)$ is an Ornstein-Uhlenbeck process.
So now what I want to do is something similar to the law of iterated logarithms, and that is to find a function $\phi(t)$ so that for some $\lambda > 0$:
$$ \limsup_{t \rightarrow \infty} \frac{\int_{0}^{t}\frac{e^s}{V}\lambda \eta(s)x^T\sigma x ds}{\phi(t)} \leq K < \infty$$
I really just need an upper bound. I'm trying to model it after the proof of the normal Law of Iterated Logarithms for a standard Ornstein-Uhlenbeck process (or Brownian Motion), but in the proof, they just say "Let $\phi(t)$ be this and look at all the good stuff that happens" so it's not quite clear how to modify $\phi(t)$. Any advice on how to get started would be really appreciated.