Doob

Doob is a surname, most notably associated with Joseph L. Doob, an American mathematician whose work helped establish modern probability theory. Doob's research laid the groundwork for the systematic study of stochastic processes, with particular emphasis on martingales and their applications. He authored the influential monograph Stochastic Processes, which helped popularize measure-theoretic approaches to probability.

Among concepts named after him are Doob's inequality and the Doob maximal inequality for submartingales, as well as the Doob–Meyer decomposition, which states that every submartingale can be decomposed into a martingale and a predictable increasing process. He also introduced or helped develop the Doob transform (h-transform) for conditioning Markov processes on events such as hitting or avoiding a state, and is associated with the Doob optional stopping theorem, which gives conditions under which stopping times preserve expectations. His work characterized many fundamental tools in stochastic analysis and probabilistic potential theory.

Doob's contributions influenced subsequent generations of probabilists and provided techniques widely used in finance, physics, and applied probability. His role helped shape the modern study of martingales, stochastic calculus, and the rigorous foundation of stochastic processes.