In mathematics, the theory of optimal stopping[1][2] or early stopping[3] 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).
Optimal stopping problems can often be written in the form of a Bellman equation, and are therefore often solved using dynamic programming.
denotes the probability measure where the stochastic process starts at
, the optimal stopping problem is This is sometimes called the MLS (which stand for Mayer, Lagrange, and supremum, respectively) formulation.
[4] There are generally two approaches to solving optimal stopping problems.
[4] 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
is finite, the problem can also be easily solved by dynamic programming.
When the underlying process is determined by a family of (conditional) transition functions leading to a Markov family of transition probabilities, powerful analytical tools provided by the theory of Markov processes can often be utilized and this approach is referred to as the Markov method.
be an open set (the solvency region) and be the bankruptcy time.
The optimal stopping problem is: It turns out that under some regularity conditions,[5] the following verification theorem holds: If a function
These conditions can also be written is a more compact form (the integro-variational inequality): (Example where
converges) You have a fair coin and are repeatedly tossing it.
Each time, before it is tossed, you can choose to stop tossing it and get paid (in dollars, say) the average number of heads observed.
You wish to maximise the amount you get paid by choosing a stopping rule.
If Xi (for i ≥ 1) forms a sequence of independent, identically distributed random variables with Bernoulli distribution and if then the sequences
You wish to maximise the amount you earn by choosing a stopping rule.
You wish to choose a stopping rule which maximises your chance of picking the best object.
An elegant solution to the secretary problem and several modifications of this problem is provided by the more recent odds algorithm of optimal stopping (Bruss algorithm).
Economists have studied a number of optimal stopping problems similar to the 'secretary problem', and typically call this type of analysis 'search theory'.
A special example of an application of search theory is the task of optimal selection of parking space by a driver going to the opera (theater, shopping, etc.).
The goal is clearly visible, so the distance from the target is easily assessed.
The driver's task is to choose a free parking space as close to the destination as possible without turning around so that the distance from this place to the destination is the shortest.
[7] In the trading of options on financial markets, the holder of an American option is allowed to exercise the right to buy (or sell) the underlying asset at a predetermined price at any time before or at the expiry date.
Therefore, the valuation of American options is essentially an optimal stopping problem.
follows geometric Brownian motion under the risk-neutral measure.
When the option is perpetual, the optimal stopping problem is where the payoff function is
The solution is known to be[8] On the other hand, when the expiry date is finite, the problem is associated with a 2-dimensional free-boundary problem with no known closed-form solution.
See Black–Scholes model#American options for various valuation methods here, as well as Fugit for a discrete, tree based, calculation of the optimal time to exercise.