What I would like to discuss is the following. I don't think that this is a pure duplicate, so I would be happy about comments:
On one hand it is reasonable to model log-returns as Gaussian: $$ \log(S_{t+\Delta}/{S_t}) = \sigma B_{\Delta t} \tag{1} $$ with a Gaussian random variable $B_{\Delta t} \sim N(0,\Delta t)$.
On the other hand as e.g. in the calculations of the equivalent Gaussian volatility for PRIIPS we model $$ S_{t+\Delta} = S_t \exp \left( - \sigma^2/2 \Delta + \sigma \left( B_{t+ \Delta t} - B_{t } \right) \right), $$ and thus $$ \log(S_{t+\Delta}/{S_t}) = - \sigma^2/2 \Delta + \sigma \left( B_{t+ \Delta t} - B_{t } \right), \tag{2} $$ which leads to a non-centered Gaussian.
I know that $(2)$ is the natural model if we want to use the SDE $$ dS_t = \sigma S_t dB_t, $$ whose discretized version is $$ S_{t+ \Delta t} - S_{t } \approx \sigma S_t \left( B_{t+ \Delta t} - B_{t } \right), $$ which can be reformulated as $$ \frac{S_{t+ \Delta t} - S_{t }}{S_t} \approx \sigma \left( B_{t+ \Delta t} - B_{t } \right). $$
So how does all this fit together? In risk management we often assume that log-returns are Gaussian $(1)$ and the regulator of PRIIPS assumes that arithmetic returns are approximately Gaussian? How can we interpret the correction term intuitively in $(2)$?
EDIT: Hopefully doing the right maths:
In setting (A) which gives us equation (1) we have the following stochastic model: $$ S_{t + \Delta t} = S_t \exp \left( \sigma (B_{t + \Delta t} - B_t) \right) $$ then for the log return $R_t$ we have $$ R_t = \log\left(S_{t + \Delta t}/S_t \right) = \sigma (B_{t + \Delta t} - B_t). $$ Then $R_t$ has a Gaussian distribution with expectation $0$ and variance $\sigma^2 \Delta t$.
Setting (B): $$ S_{t + \Delta} = S_t \exp \left( -\frac{\sigma^2}{2} \Delta t + \sigma (B_{t + \Delta t} - B_t) \right) $$ and get for $\log(S_{t + \Delta}/S_t)$ again something Gaussian with expectation $-\frac{\sigma^2}{2} \Delta t$ and variance $\sigma^2 \Delta t$.
For $\Delta t$ small (one or just a couple of days) the difference is negligible but for longer terms (e.g. recommended holding periods) we have to model many $\Delta t$ steps leading to a larger term there. So there is a difference on the long run.