Measurement changes and martingale representation
Brownian motion and stochastic calculus

The foundations of the theory of financial mathematics consist of two theoretical results. The first gives a necessary and sufficient condition for having no arbitrage opportunities (the game is balanced, one cannot win strictly more than one's stake with a strictly positive probability), and this condition translates into the fact that there exists a probability, equivalent to the initial probability, under which the prices of all assets (stocks and bonds), as time-indexed processes, are martingales. The second gives a necessary and sufficient condition for having a "complete market", meaning that every option payoff can be obtained by a suitable strategy, and this condition translates into the fact that the above equivalent probability, under which all asset prices are martingales, is unique.

The purpose of this paragraph is to provide the mathematical basis for...