Optimal stopping theory has its roots in classical calculus of variations. The three problem formulations due to Lagrange (18th century), Mayer (19th century) and Bolza (in 1913) were inherited by (stochastic) optimal control theory in the 1940's and 50's (Bellman). At about the same time optimal stopping problems emerged from work by Wald in relation to problems of sequential testing. Key principles of optimal stopping were established by Snell in 1952 (Snell's envelope) and Dynkin in 1963 (superharmonic characterization). In the 1950's and 1960's a fundamental connection between optimal stopping and free boundary problems was discovered by a number of researchers (Mikhalevich, Chernoff, Lindley, McKean, Shiryaev). This equivalence penetrates deeply into the fundamentals of modern probability theory (Einstein, Wiener, Kolmogorov) via exit problems (Kakutani) and their connection with analysis through boundary value problems (Dirichlet, Poisson, Neumann, Cameron, Martin, Girsanov, Feynman, Kac). This connection is one of the most fascinating accomplishments of modern probability theory (or even of modern mathematics). Studies of martingales and Markov processes are central to optimal stopping problems and the concepts of filtration (information) and stopping times (non-anticipation) are key to real world applications.
A huge stimulus to the development
of optimal stopping theory was provided by option pricing theory (financial
mathematics) developed in the late 1960's and 1970's. According to the
modern theory of finance, pricing an American option in a complete market
is equivalent to solving an optimal stopping problem (with a corresponding
generalisation in incomplete markets), the optimal stopping time being
the rational time for the option to be exercised. Due to the enormous importance
of the American price mechanism in finance, this line of research has been
intensively pursued in recent times. The most recent breakthrough involves
extensions of the classic Itô formula to account for local time on
curves and surfaces leading to the so-called 'local time-space calculus'.
In parallel to the classic problem formulations of Lagrange, Mayer and
Bolza mentioned above, in the 1990's several researchers have independently
started to study problem formulations based on the maximum process (Jacka,
Dubins, Shepp, Shiryaev, Graversen, Pedersen, Peskir). Solutions of optimal
stopping problems of this type are intimately related to the fundamental
inequalities of probability theory (Doob, Hardy-Littlewood, Burkholder,
Davis, Gundy) and several deep and intriguing problems are still open.
Further studies of optimal stopping conducted presently involve optimal
prediction, sequential testing, quickest detection, and free-boundary problems
for PIDEs (instead of PDEs).