Q: Stochastic equation for deriving return over time Why do you need to do every other day? Why wouldn't you just do two time steps?
I learned this equation, $R_i=\mu dt+\sigma \rho dt$
$R=(S_n-S_{n-1})/S_{n-1} \text{, where $R$ is the return}\\
S_i = \text{stock price at time $i$}\\
\mu = \text{mean return}\\
dt = \text{time step}\\
\sigma = \text{standard deviation of return}\\
\rho = \text{randomness}\\$
This teacher wants to do two time steps, so he says $$\frac{s_{i+2}}{si}-1=\frac{s_{i+2}}{s_i}\frac{s_{i+1}}{s_{i+1}}-1=(R_i+1)(R_{i+1}+1)-1=R_iR_{i+1}+R_i+R_{i+1}+1-1=R_i+R_{i+1}$$ The first term is small and ignored. But he says because $R_i$ is standard normal, $\hat{R_i}=R_i+R_{i+1}$ is a new normal variable with mean $2\mu$ and $\sigma = \sqrt{2}\sigma$
He says this changes the original equation $\rightarrow \hat{R_i}=\mu dt + \sigma \rho (dt)^{1/2}$
I don't get this, why does this equation change like this? Even if he changed $dt=2dt_0$, then $dt^{1/2}=\sqrt{2}\sqrt{dt_0}$
This is not the same. 
Also, doesn't it make more sense to just time step the original equation? 
$
S_2=S_1+S_1(\mu*dt+\sigma*\rho*dt)\\
S_3=S_2+S_2(\mu*dt+\sigma*\rho*dt)$
What is the point of changing R, and adding in the square root of two?
