# Can volatility assume negative values under multi-factor HJM framework?

I could find any reference restricting the sign of the volatilities in the multi-factor HJM framework.

Can someone please confirm if $$\sigma_i(t,T)$$ can assume negative values for some $$i,t$$ and $$T$$?

$$df(t,T) = \left(\sum_i \sigma_i(t,T)\int_t^T \sigma_i(t,u) du \right) dt + \sum_i \sigma_i(t,T) dW_i(t)$$

At first, I also could not find a single source that formally restricts $$\sigma_i(t,T)$$. However, the formally very precise article [1] explicitly states that the $$\sigma_i(t,T)$$ are assumed to be non-negative:
[...] the volatilities $$\sigma_i(\cdot,\cdot,\omega_t)$$ belong to $$\mathcal{F}$$, the set of all functions defined from $${(t, T ) : t \in [0, T ]}\times\Omega$$ onto $$\mathbb{R}$$, that are $$\mathbb{P}$$-almost everywhere non-negative, bounded, square integrable on any finite time horizon, and Lipschitz continuous with respect to the second variable. Further the $$\sigma_i(\cdot,\cdot,\omega_t)$$ are jointly measurable from $$\mathcal{B} {(t, T ) : t \in [0, T]} \times \mathcal{F}_T → \mathcal{B}$$, where $$\mathcal{B}$$ is the Borel $$\sigma$$-algebra restricted to $$[0, T]$$.
Most references quietly assume this by stating that the $$\sigma_i(t,T)$$ are volatility functions.