Given a "hybrid" Black-Scholes Hull White (BSHW) model. That is, the stock price is modelled by a Black Scholes SDE:
\begin{equation} dS(t) = \mu(t)S(t)dt + \sigma\_{S}(t)S(t)dW^{\mathbb{P}}\_{S}(t)
\end{equation}


and short rate evolution is given by a Hull-White model
\begin{equation} dr(t) = (\theta(t) - ar(t))dt + \sigma\_{r}dW^{\mathbb{P}}\_{r}(t),
\end{equation}
where as $W^{\mathbb{P}}\_{S}$ and $W^{\mathbb{P}}\_{r}$ are Wiener processes on a filtered probability space $(\Omega,\mathcal{F},\mathbb{F},\mathbb{P})$ with correlation $\rho$.


My question is now:


1. Is this model in general abritrage free?
2. Is it generally complete?
3. Respectively, is there a unique risk neutral measure $\mathbb{Q}$?
4. If not, what are sufficient conditions for this to hold? For example $\theta(t)=f(0,t)+\frac{\sigma\_{r}^{2}}{2a^{2}}(1-e^{-at})^{2}$?



Is the "hybrid" Black-Scholes Hull-White model arbitrage free?
The discounted stock price satisfies 
$$
dX(t) = \big(\mu(t) - r(t)\big)X(t) dt + \sigma\_S(t) X(t) dW\_S^{\mathbb P}(t).
$$
The Girsanov density for $X$ is 
$$
Z(T) = \exp\left\{\int\_0^T \nu(t)dW^{\mathbb P}\_S(t) - \frac12\int\_0^T \nu(t)^2 dt \right\}
$$
with $\nu(t) = (r(t)-\mu(t))/\sigma\_S(t)$. The problem, as usual, is to establish that $\mathbb{E}[Z(t)]=1$. Since $r$ has Gaussian distribution, it is not possible to proceed with Novikov or Kazamaki conditions, as usual. However, when $\mu$, $\theta$ and $\sigma\_S$ are reasonable enough, it is possible to use the less known Beneŝ condition as follows. The solution to the equation for interest rate is
$$
r(t) = r(0)e^{-at} + \int\_0^t e^{a(s-t)}\big(\theta(s) ds + \sigma\_r dW^{\mathbb{P}}\_r(s)\big) \\
 = r(0)e^{-at} + \int\_0^t e^{a(s-t)}\theta(s) ds + \sigma\_r \int\_0^t e^{a(s-t)} d\big(\rho W^{\mathbb{P}}\_S(s) +\gamma(s)\big) \\
= \Gamma(t) + \Lambda(t), 
$$
where $$\Gamma(t) = r(0)e^{-at} +\int\_0^t e^{a(s-t)}\theta(s) ds + \int\_0^t e^{a(t-s)}d\gamma(s),\\
\Lambda(t) = \sigma\_r \rho\left( W^{\mathbb{P}}\_S(t) + a \int\_0^t e^{a(t-s)}W^{\mathbb{P}}\_S(s)ds\right),
$$ and $\gamma$ is, up to a constant, a Wiener process independent of $W^{\mathbb{P}}\_S$. Then, assuming $\mu$, $\theta$ and $\sigma\_S$ to be nice (say, all bounded, and $\inf|\sigma\_S|>0$), we have $|\nu(t)|\le C(\gamma)+ C\max\_{s\in [0,t]} |W\_S^{\mathbb P}(s)|$, where $C(\gamma)$ is a finite random variable depending on $\gamma$. Then we have, see e.g. [here](https://arxiv.org/pdf/1112.0430.pdf), 
$$
\mathbb{E}\left[Z(T) \mid \gamma\right]=1,\tag{1}
$$
whence $\mathbb{E}\left[Z(T)\right]=1$, as required. 


As @michael wrote, there is no uniqueness (unless $|\rho| = 1$ or $\sigma\_r = 0$). Financially, the explanation is that the interest rate process is non-traded, thus there is no completeness. Mathematically, in view of $(1)$, for any $\gamma$-measurable positive random variable $L(\gamma)$, which integrates to $1$, $Z'(T) = L(\gamma) Z(T)$ defines a density of equivalent martingale measure.


