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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.

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Well need more information on $f,g$. Here in "Solutions of Stochastic Differential Equations obeying the Law of the Iterated Logarithm, with Applications to Financial Market". They try to cover some cases of having law of iterated logarithm and even more general gauge functions.

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