How can I get from the conditional density $f_p(s_t | s_{t-1})$ of stock prices $s_i$ to the density $f_p(a_1,\cdots,a_N)$ of continuous returns a_i?

Question

Let's assume I have determined the continuous returns $a=(a_1,...,a_N)$ of a stock in a certain time interval. Now there are two cases: In one case the price process should follow a Brownian motion. I am looking for the density of $r_1,...,r_N$ under the distribution $f(r_1,...,r_N)$. In the second case the price process is not simply a Brownian motion, so the returns are not simply i.i.d. and normally distributed. The only thing I think I know is the conditional density $f(s_t | s_{t-1})$ of the stock price at time $t$ given the price at time t-1. I would like to know how I can get from there to the distribution f(r_1,...,r_N) of the returns. The final goal is to determine the likelihoods. Any help would be appreciated since my knowledge of this topic is just in its infancy.