Doléans-Dade theorem

The Doléans-Dade theorem is a fundamental result in probability theory, particularly in the study of stochastic processes. It states that for a certain class of processes called martingales—models of fair game—there exists an associated measure change (a way to re-weight probabilities) that makes a process with some randomness behave more predictably. This theorem helps in understanding and manipulating complex random processes, especially in finance and other fields where modeling uncertainty accurately is crucial. Essentially, it provides a powerful tool to analyze and transform stochastic processes to better understand their behavior.