that we denote by ${\mathbf{V}}=\left\{{\mathbf{V}}(t)= (V_1(t),...,V_d(t)), 0\leq t \leq \bar T\right\}$ . The $k$-th component $V_k$ of the vector process ${\mathbf{V}}$ is assumed to solve the SDE $$\begin{aligned}
\begin{aligned}
{\mathrm{d}}V_k(t)=& \kappa_k ( \theta_k - V_k(t)){\mathrm{d}}t + \sigma_k V_k(t)^{1/2} {\mathrm{d}}W_k(t),\\
V_k(0) &= v_k>0,
\label{CIRs}
\end{aligned}\end{aligned}$$ for $t\in[0, \bar T]$, where the parameters $\kappa_k>0,\theta_k>0,\sigma_k>0, $ are admissible in the sense of [@article_DFS], $k=1,...,d$. In addition, to avoid zero volatility factors, we impose the following assumption.

For every $k=1,\ldots , d,$ the parameters in satisfy the relation $$\begin{aligned}
2\kappa_k\theta_k-\sigma^2_k\geq 0.\end{aligned}$$

We also allow for non-zero correlation between assets and their volatilities via the following condition:

\[assumptionCorrelation\] The Brownian motions ${\mathbf{Z}}$ and ${\mathbf{W}}$ have a covariation satisfying $$\begin{aligned}
\frac{{\mathrm{d}}\langle W_k,Z_l\rangle(t)}{{\mathrm{d}}t}=\delta_{kl}\rho_k, \ k,l=1,..., d,\end{aligned}$$ where $\delta_{kl}$ denotes the Dirac delta function for the indices $k$ and $l$.

We then proceed to provide a general specification for the family of market prices of risk.

\[piandr\] We assume that the $i$-th market price of risk vector $\boldsymbol{\pi}^i(t)$ is a projection of the common volatility factor ${\mathbf{V}}$, along a direction parametrized by a constant vector ${\mathbf{a}}^i \in {\mathbb{R}}^d$ and a projection of the inverted elements of ${\mathbf{V}}$ along another direction parametrized by ${\mathbf{b}}^i\in {\mathbb{R}}^d$, according to the following 