In mathematics, the theory of optimal stopping or early stopping is concerned with the problem of choosing a time to take a particular action, in order to maximise an expected reward or minimise an expected cost. Optimal stopping problems can be found in areas of statistics, economics, and mathematical finance (related to the pricing of American options). A key example of an optimal stopping problem is the secretary problem. Optimal stopping problems can often be written in the form of a Bellman equation, and are therefore often solved using dynamic programming.

Stopping rule problems are associated with two objects:

Given those objects, the problem is as follows:

Consider a gain process ${\displaystyle G=(G_{t})_{t\geq 0}}$ defined on a filtered probability space ${\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\geq 0},\mathbb {P} )}$ and assume that ${\displaystyle G}$ is adapted to the filtration. The optimal stopping problem is to find the stopping time ${\displaystyle \tau ^{*}}$ which maximizes the expected gain

where ${\displaystyle V_{t}^{T}}$ is called the value function. Here ${\displaystyle T}$ can take value ${\displaystyle \infty }$.

A more specific formulation is as follows. We consider an adapted strong Markov process ${\displaystyle X=(X_{t})_{t\geq 0}}$ defined on a filtered probability space ${\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\geq 0},\mathbb {P} _{x})}$ where ${\displaystyle \mathbb {P} _{x}}$ denotes the probability measure where the stochastic process starts at ${\displaystyle x}$. Given continuous functions ${\displaystyle M,L}$, and ${\displaystyle K}$, the optimal stopping problem is

This is sometimes called the MLS (which stand for Mayer, Lagrange, and supremum, respectively) formulation.

There are generally two approaches to solving optimal stopping problems. When the underlying process (or the gain process) is described by its unconditional finite-dimensional distributions, the appropriate solution technique is the martingale approach, so called because it uses martingale theory, the most important concept being the Snell envelope. In the discrete time case, if the planning horizon ${\displaystyle T}$ is finite, the problem can also be easily solved by dynamic programming.