= Singular-volatility drift arbitrage
{title2=$\sigma^T v=0,\quad v^T\mu>0$}
If a constant-coefficient investment model has a vector $v$ with $\sigma^Tv=0$ 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 $\mu\in\operatorname{Range}A$, because $\ker A=\ker\sigma^T$ and $\operatorname{Range}A=(\ker A)^\perp$. 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.
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