Consider the SDE $dZ_t = \mu(t,x) d_t + \sigma(t,x) dW_t$. Are there any known (necessary and) sufficient conditions on $\sigma(t,x)$ and on $\mu(t,x)$ guaranteeing that $f(T):=\mathbb{E}[\int_0^T Z_t d<Z>_t]$ is a real-analytic function (in the analysis sense) in $T$?
Some examples to support my question:
-(GBM process) If $\sigma$ and $\mu$ are constants, then $f(T)=Z_0e^{\mu T}$, which is clearly real-analytic.
-(OU-process) If $\mu(t,x):=\theta(\mu-x)$; for some $\theta,\mu\in(0,\infty)$ and $\sigma$ is also constant, then:
$f(T):=Z_0e^{-\theta T} + \mu(1-e^{-\theta T})$.
In both the above cases:$\sigma(t,x)$ and on $\mu(t,x)$ are real-analytic functions (moreover: in fact they are polynomial); will this imply that $f(T)$ will be also?
