Black-Scholes market dynamics, according to [@KaratzasShreve1998], there exists a unique risk-neutral probability measure ${{\mathbb{Q}}\sim {\mathbb{P}}}$ defined by ${\frac{d{\mathbb{Q}}}{d{\mathbb{P}}} | _{\CMcal{F}_{t}} := e^{-\frac{1}{2} \|\gamma\|^{2} t - \gamma' W(t)}}$ and the market is complete (that allows to value payment streams under the measure ${\mathbb{Q}}$ as expected discounted values, meaning that the cost of a portfolio replicating the contract is given by its expected discounted value under ${\mathbb{Q}}$). The corresponding pricing kernel or state price deflator, denoted by $\tilde{Z}(t)$, is defined as $$\begin{aligned}
 \label{eq:PricingKernel}
\tilde{Z}(t) := e^{- \left(r + \frac{1}{2}\|\gamma\|^{2}\right) t - \gamma' W(t)}\end{aligned}$$ and can be used for the valuation of payment streams under the real-world probability measure. Its dynamics are subject to the stochastic differential equation $$\begin{aligned}
d \tilde{Z}(t) = - \tilde{Z}(t) \left(r dt + \gamma' dW(t)\right),\ \tilde{Z}(0) = 1.\end{aligned}$$ We consider $\CMcal{F}_{t}$-progressively measurable trading strategies $\varphi = (\varphi_{0}, \hat{\varphi})'$, ${\hat{\varphi} = (\varphi_{1}, \hdots, \varphi_{N})'}$, such that ${\mathbb{P}}$-a.s. it holds $\int_{0}^{T} |\varphi_{0}(t)| dt < \infty$ and $\int_{0}^{T} \varphi_{i}(t)^{2} dt < \infty$. $\varphi_{i}(t)$ represents the number of individual shares of asset $i$ held by the investor at time $t$. The corresponding relative portfolio process is denoted by ${\pi = (\pi_{0}, \hat{\pi}')'}$ with risky relative investment ${\hat{\pi} = (\pi_{1}, \hdots, \pi_{N})'}$ and risk-free 