I'm trying to derive a property, for which I need to evaluate the following.
$W_t$ is a Wiener process, and I'm interested in finding: $$\mathbb{E}(X) = \mathbb{E} \left( (1+W_t)\frac{C}{C+KW_t}\right)$$ where $C$ and $K$ are constants.
I know that both $\mathbb{E}(W_t)=0$ and that processes are independent with time. That is, $W_{t, t-s}$ is independent of $W_{t-s, 0}$ and hence the expectation can be computed as a product of expectations.
In this case, the processes are not independent. I assume $W_t$ has normal increments $N(0, \sigma^2)$. Let $Z(x)=\frac{1}{\sqrt{2 \pi \sigma^2}} e^{-x^2/(2\sigma^2)}$
\begin{equation} \begin{split} \mathbb{E}(X) & = \int^\infty_{-\infty} x f(x) dx\\ & = \int^\infty_{-\infty} \frac{CZ(x)}{C+KZ(x)} xdx + \int^\infty_{-\infty} \frac{C}{C+KZ(x)}x dx \end{split} \end{equation}
Is everything correct up-to now? From here, I don't really know how to proceed?