I am reading a finance article and they made an assumption about underlying price S:
" We assume that the underlying S satisfies the following stochastic differential equation over a period ${\delta{t}}$ :
$$\frac{S_{j+1}-S_{j}}{S_{j}} = {\sigma_{s} \sqrt{\delta{t}} X_{j}}$$
where $X_{j}$ is independent standard normal distribution variables, ${\sigma_{s}}$is the realised volatility. "
Is there a mathematical or financial explanation behind this assumption ? Thank you.
It says that the returns on the stock over a time period of length $\delta t$ are normally distributed with mean $0$ and variance $\sigma_s^2 \delta t$. The fact that the $X_j$ are independent says that the returns over this time period are independent of the returns over any earlier time period, i.e. you cannot learn anything about the stock's future trajectory by looking at the past.