stochastic processes

In the realm of mathematical finance there are two relatively disjoint worlds: the “buy-side” and the “sell-side”. Buyers are analyzing the available financial products (e.g. stocks, bonds, derivatives, etc.) and are trying to predict, and profit off of, the future prices of these assets—i.e. they are trying to estimate the real-world probability distribution, commonly denoted $P$, of the future stochastic price movements. On the other side, sellers (aka market-makers) are trying to set a fair price for more illiquid assets, like derivatives, using the prices of more liquid assets, like stocks, which are priced by supply and demand....

The development of mathematical finance, much like the processes it aims to study, has had a particularly jumpy history. A major leap came in 1973, when the Black-Scholes option pricing model was published and mathematically understood. The essential idea is to model the underlying asset $S_t$ of an option as a geometric brownian motion, with a stochastic differential equation (SDE), given by $$ dS_t = \mu S_t \ dt + \sigma S_t \ dW_t, $$...