Suppose I have an SDE of the form:
$$dx_i = x_if(x_1,\cdots,x_n) + \sigma_ix_idW_t $$
By defining $y_i = \log x_i$, I can change the SDE to:
$$dy_i = y_i g(y_1,\cdots,y_n) + \sigma_idW_t $$
Both $f$ and $g$ are smooth and nice functions. The advantage of this is now I've turned a multiplicative noise problem into an additive one. I'm interested in certain properties of each $x_i(t)$, namely the following:
- For all $p>1$, $sup_{t\geq 0} E[x_i(t)^p] \leq K < \infty$ (moment boundedness)
- For any $\epsilon > 0$, there is a constant $H$ so that $\limsup_{t \rightarrow \infty} P\left( |x(t)| \leq H \right) \geq 1- \epsilon$ (stochastic permanence)
- Holder Continuity
My question is: In general, for my equation, I have been unable to prove these properties for $x_i$. However, I can prove them for $y_i$. Since $y$ is just a log-transformation of $x$, can one conclude that the properties are true for $x$ if they are true for $y$? There's probably a counter-example somewhere but I can't really think of a reason for why it wouldn't be true