**Appendix A. Proofs**

**Proof.** Proof of Proposition 1.

The first step is to ensure the change of measure is well-defined and for this we use Novikov's condition, i.e., generically

$$\mathbb{E}\left[\exp\left(\frac{1}{2}\int\_0^T \lambda^2 \left(\sqrt{\nu(t)} + \frac{b}{\sqrt{\nu(t)}}\right)^2 ds\right)\right] = \epsilon^{\lambda^2 bT} \mathbb{E}\left[\exp\left(\frac{\lambda^2}{2}\int\_0^T \nu(s)ds + \frac{\lambda^2 b^2}{2}\int\_0^T \frac{1}{\nu(s)}ds\right)\right] < \infty.$$

From Grasselli, in order for this expectation to exist, we need two conditions:

$$1 - \frac{\lambda^2}{2} > -\frac{a^2}{2\xi^2} \implies |\lambda| < \frac{a}{\xi} \tag{A1}$$

and

$$-\frac{\lambda^2 b^2}{2} \ge -\frac{(2a\theta - \xi^2)^2}{8\xi^2} \implies |\lambda| \le \frac{2a\theta - \xi^2}{2|b|\xi} \implies \xi^2 \le 2a\theta - 2|\lambda||b|\xi \tag{A2}$$

The latter condition in Equation (A2) implies, in particular, that our volatility processes satisfy Feller's condition under P and Q; in other words, it ensures all our CIR processes stay away from zero under both measures.

Applying Equation (A2) to our setting leads to (*i*, *j* = 1, . . . , *n*):

$$\begin{array}{rcl} \mathbb{Z}\_{j}^{2} & \leq & 2a\_{j}\theta\_{j} - 2\mathbb{Z}\_{j} \max\left\{ |\lambda\_{j}b\_{j}|, \left|\lambda\_{j}^{\perp}b\_{j}\right| \right\} \end{array} \tag{A3}$$

$$
\hat{\xi}\_i^2 \quad \le \quad 2\tilde{\kappa}\_i \tilde{\theta}\_i - 2\tilde{\xi}\_i \max \left\{ \left| \left| \tilde{\lambda}\_i \tilde{\theta}\_i \right| , \left| \tilde{\lambda}\_i^\perp \tilde{\theta}\_i \right| \right\} \right.\tag{A4}
$$

Now, we apply Equation (A1) producing two extra set of conditions (*i*, *j* = 1, . . . , *n*):

$$\max \left\{ \left| \lambda\_j \right|, \left| \lambda\_j^\perp \right| \right\} \quad < \underbrace{\begin{array}{c} a\_j \\ \xi\_j \\ \vdots \end{array} \tag{A5}$$

$$\max \left\{ \left| \tilde{\lambda}\_i \right|, \left| \tilde{\lambda}\_i^\perp \right| \right\} \\ \quad < \begin{array}{ll} \tilde{a}\_i \\ \frac{\tilde{a}\_i}{\tilde{a}\_i} \end{array} \tag{A6}$$

The second step applies to the case *βij* = 0 for *i*, *j* = 1, ... , *n* and it is to ensure the drift of the asset price equal the short rate:

$$L\_i = r,\\ \mathfrak{c}\_i = \sum\_{j=1}^n a\_{ij} \left( \rho\_j \lambda\_j + \sqrt{1 - \rho\_j^2} \lambda\_j^\perp \right),\\ \widetilde{\mathfrak{c}}\_i = \widetilde{\rho}\_i \widetilde{\lambda}\_i + \sqrt{1 - \widetilde{\rho}\_i^2} \widetilde{\lambda}\_i^\perp$$

For the most general case (*βij* = 0 for some *i* or *j*), the second step should be adapted to any particular prescribed drift structure under the Q-measure.

The third step is to ensure the drift-less asset price process is a true Q-martingale and not just a local Q-martingale:

$$\frac{dX\_i(t)}{X\_i(t)} = (.)\,dt + \sum\_{j=1}^n a\_{ij} \left(\sqrt{\upsilon\_j(t)} + \frac{b\_j}{\sqrt{\upsilon\_j(t)}}\right) d\mathcal{W}\_j^Q(t) + \left(\sqrt{\vec{\upsilon}\_i(t)} + \frac{\tilde{b}\_i}{\sqrt{\vec{\upsilon}\_i(t)}}\right) d\tilde{W}\_i^Q(t)$$

Here, we test the martingale property using the Feller nonexplosion test for volatilities, hence considering the following *n*<sup>2</sup> + *n* changes of Brownian motion for the volatility processes and checking the processes do not reach zero under the various measures:

$$d\boldsymbol{B}\_{ij}^{\mathcal{Q}}(\boldsymbol{t}) = \boldsymbol{a}\_{i\bar{j}}\boldsymbol{\rho}\_{\bar{j}}\left(\sqrt{\boldsymbol{v}\_{\bar{j}}(\boldsymbol{t})} + \frac{\boldsymbol{b}\_{\bar{j}}}{\sqrt{\boldsymbol{v}\_{\bar{j}}(\boldsymbol{t})}}\right)\boldsymbol{d}\boldsymbol{t} + d\boldsymbol{B}\_{\bar{j}}^{\mathcal{P}}(\boldsymbol{t}),\\d\boldsymbol{\bar{B}}\_{i}^{\mathcal{Q}}(\boldsymbol{t}) = \boldsymbol{\tilde{\rho}}\_{i}^{\boldsymbol{\r}}\left(\sqrt{\boldsymbol{v}\_{i}(\boldsymbol{t})} + \frac{\boldsymbol{\mathcal{G}}\_{i}}{\sqrt{\boldsymbol{v}\_{i}(\boldsymbol{t})}}\right)\boldsymbol{d}\boldsymbol{t} + d\boldsymbol{\bar{B}}\_{i}^{\mathcal{P}}(\boldsymbol{t})$$

This leads to the following conditions:

$$\mathcal{S}\_{j}^{2} \quad \leq \quad 2a\_{j}\theta\_{j} - 2\left|a\_{i\bar{j}}\rho\_{\bar{j}}b\_{\bar{j}}\right|\mathcal{S}\_{j}, i, j = 1, \dots, n \tag{A7}$$

$$\hat{\xi}\_{i}^{2} \quad \leq \quad 2\tilde{\alpha}\_{i}\hat{\theta}\_{i} - 2\left|\hat{\lambda}\_{i}\hat{\rho}\_{i}\hat{b}\_{i}\right|\hat{\xi}\_{i}, i = 1, \ldots, n \tag{A8}$$

We can combine the first and third steps in Equations (A3), (A7), (A4) and (2) into the final conditions.
