$W(t) = (W_{1}(t), \hdots, W_{N}(t))'$, $N \in \mathbb{N}$, is a standard $N$-dimensional Brownian motion. The price of the risk-free asset at time $t$ is denoted by $P_{0}(t)$ and is subject to the equation $$\begin{aligned}
dP_{0}(t) = r P_{0}(t)dt,\ P_{0}(0) = 1,\end{aligned}$$

with constant risk-free interest rate $r > 0$. The remaining $N$ assets in the market are risky assets with price $P_{i}(t)$, $i = 1, \hdots, N$, at time $t$ subject to the stochastic differential equations

$$\begin{aligned}
dP_{i}(t) = {} & P_{i}(t) \left(\mu_{i} dt + \sigma_{i} dW(t)\right) = P_{i}(t)  \left(\mu_{i} dt + \sum_{j = 1}^{N} \sigma_{ij} dW_{j}(t)\right),\ P_{i}(0) = p_{i} > 0,\end{aligned}$$

with constant drift ${\mu = \left(\mu_{1},\hdots,\mu_{N}\right)' \in \mathbb{R}_{+}^{N}}$, ${\mu - r \mathbf{1} > \mathbf{0}}$, and constant volatility vector $\sigma_{i} = (\sigma_{i1}, \hdots, \sigma_{iN}) \in \mathbb{R}_{+}^{1 \times N}$. The volatility matrix is defined by ${\sigma = \left(\sigma_{ij}\right)_{i,j = 1,\hdots,N}}$, the covariance matrix of the log-returns is ${\Sigma = \sigma \sigma'}$ which is assumed to be strongly positive definite, i.e. there exists $K > 0$ such that ${\mathbb{P}}$-a.s. it holds ${x' \Sigma x \ge K x' x}$, ${\forall x \in {\mathbb{R}}^{N}}$. Furthermore, let ${\gamma = \sigma^{-1} (\mu - r \mathbf{1})}$ denote the market price of risk. In this case of 