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

Work in discounted units with a unit bank account and constant asset-gain dynamics $dS=\mu dt+\sigma dW$. Here $\mu$ is the vector of excess dollar drifts, not an unadjusted physical return. If a holdings vector $v$ satisfiesfinance its initial stock cost through the bank. The resulting zero-cost self-financing portfolio has deterministic wealth $t{v}^T\mu >0$ for $t>0$, giving an arbitrage.

For $A=\sigma {\sigma }^T$, absence of such a direction is equivalent to $\mu \in \mathrm{Range}A$. Indeed, $\ker A=\ker {\sigma }^T$ and $\mathrm{Range}A=(\ker A{)}^{\perp }$. If $\mu$ has a nonzero projection onto that kernel, taking $v$ to be that projection gives $v^T\mu =\Vert v{\Vert }^2>0$.

Scaling this deterministic gain makes any unbounded increasing terminal utility function unbounded on a positive horizon. A singular volatility matrix therefore requires this drift-compatibility check before a finite investment value is asserted. With a nonzero bank rate in undiscounted units, the financing return must first be subtracted.