I am taking an intro to financial mathematics class and came across the above equation. Unfortunately, I am completely unable to decipher the meaning. I am assuming that Y subscript i is your random variable (could be wrong), however, I can't see how they arrived at the following expansion. Do they successive Y terms represent random values the variable could take at each time step? Also, normally in a Riemann sum (at least at the level I have studied calculus) you end up with a 1/n term, however, here I see (n - 1), (n - 2), etc. and I would like to know how they arrived at this result. Either a qualitative explanation or more complete proof of this explanation would be much appreciated!
If $\{Y_i\}$ represent a stochastic process, where each $Y_i$ is a normally distributed random variable, then what the notes are saying is that $Y_i$'s are not independent, but the random variables $Z_i$, obtained from their 'differences' or increments are independent, i.e. $Z_i=Y_{i+1}-Y_{i}$ are independent. So in general, given the importance of independence, one tries to 'handle' problems regarding $Y_i$'s by somehow `transforming' them to problems regarding the increments.
For example, $Y_2=Y_1+(Y_2-Y_1)$ and $Y_1$ and $Y_2-Y_1$ are independent.
$Y_i $ as you say is the random variable for the process at time $i $.
The sum given in the equation is just a convenient, algebraic identity, and you can verift that it holds.
If you are not familiar with thw definition of the Rieman integral, it may be a good idea to look this up.