Singular-volatility drift arbitrage (${\sigma }^{T}v=0,v^T\mu >0$)

If a constant-coefficient investment model has a vector $v$ with ${\sigma }^{T}v=0$ and $v^T\mu >0$, investment in direction $v$ produces deterministic positive gains without initial capital. For $A=\sigma {\sigma }^T$, absence of such a direction is equivalent to $\mu \in \mathrm{Range}A$, because $\ker A=\ker {\sigma }^T$ and $\mathrm{Range}A=(\ker A{)}^{\perp }$. Scaling this direction makes any unbounded increasing terminal utility function unbounded on a positive horizon. Thus a finite investment value cannot be inferred from a volatility matrix which is allowed to be singular without this additional condition.