I have some financial data i'm trying to fit a random walk too, but the daily change increments have different distributions when studied over the last month, last year, last 5 years etc, along with heteroskedascity. What is the best way to proceed with modelling this? Techniques and interesting areas to look into further?
if you have different distributions for each day, then you'll have different parameters to estimate for each day. if the data is at daily rate then it's impossible to do anything. you have to parameterize the change in distribution then. one way of doing this is GARCH, where you essentially are saying that the variance changes every day but otherwise it's random walk. you also express the functional form of this change. you'll end up estimating a couple more parameters of your random walk.