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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.

Contents

DefinitionEdit

Discrete time caseEdit

Stopping rule problems are associated with two objects:

  1. A sequence of random variables  , whose joint distribution is something assumed to be known
  2. A sequence of 'reward' functions   which depend on the observed values of the random variables in 1.:
     

Given those objects, the problem is as follows:

  • You are observing the sequence of random variables, and at each step  , you can choose to either stop observing or continue
  • If you stop observing at step  , you will receive reward  
  • You want to choose a stopping rule to maximise your expected reward (or equivalently, minimise your expected loss)

Continuous time caseEdit

Consider a gain processes   defined on a filtered probability space   and assume that   is adapted to the filtration. The optimal stopping problem is to find the stopping time   which maximizes the expected gain

 

where   is called the value function. Here   can take value  .

A more specific formulation is as follows. We consider an adapted strong Markov process   defined on a filtered probability space   where   denotes the probability measure where the stochastic process starts at  . Given continuous functions  , and  , the optimal stopping problem is

 

This is sometimes called the MLS (which stand for Mayer, Lagrange, and supremum, respectively) formulation.[1]

Solution methodsEdit

There are generally two approaches of solving optimal stopping problems.[1] 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. The solution is usually obtained by solving the associated free-boundary problems (Stefan problems).

A jump diffusion resultEdit

Let   be a Lévy diffusion in   given by the SDE

 

where   is an  -dimensional Brownian motion,   is an  -dimensional compensated Poisson random measure,  ,  , and   are given functions such that a unique solution   exists. Let   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,[2] the following verification theorem holds:

If a function   satisfies

  •   where the continuation region is  ,
  •   on  , and
  •   on  , where   is the infinitesimal generator of  

then   for all  . Moreover, if

  •   on  

Then   for all   and   is an optimal stopping time.

These conditions can also be written is a more compact form (the integro-variational inequality):

  •   on  

ExamplesEdit

Coin tossingEdit

(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  , and   are the objects associated with this problem.

House sellingEdit

(Example where   does not necessarily converge)

You have a house and wish to sell it. Each day you are offered   for your house, and pay   to continue advertising it. If you sell your house on day  , you will earn  , where  .

You wish to maximise the amount you earn by choosing a stopping rule.

In this example, the sequence ( ) is the sequence of offers for your house, and the sequence of reward functions is how much you will earn.

Secretary problemEdit

(Example where   is a finite sequence)

You are observing a sequence of objects which can be ranked from best to worst. You wish to choose a stopping rule which maximises your chance of picking the best object.

Here, if   (n is some large number, perhaps) are the ranks of the objects, and   is the chance you pick the best object if you stop intentionally rejecting objects at step i, then   and   are the sequences associated with this problem. This problem was solved in the early 1960s by several people. 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).

Search theoryEdit

Economists have studied a number of optimal stopping problems similar to the 'secretary problem', and typically call this type of analysis 'search theory'. Search theory has especially focused on a worker's search for a high-wage job, or a consumer's search for a low-priced good.

Option tradingEdit

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. Consider a classical Black-Scholes set-up and let   be the risk-free interest rate and   and   be the dividend rate and volatility of the stock. The stock price   follows geometric Brownian motion

 

under the risk-neutral measure.

When the option is perpetual, the optimal stopping problem is

 

where the payoff function is   for a call option and   for a put option. The variational inequality is

 

for all   where   is the exercise boundary. The solution is known to be[3]

  • (Perpetual call)   where   and  
  • (Perpetual put)   where   and  

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. Various numerical methods can however be used. 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.

See alsoEdit

ReferencesEdit

  1. ^ a b Peskir, Goran; Shiryaev, Albert (2006). "Optimal Stopping and Free-Boundary Problems". Lectures in Mathematics. ETH Zürich. doi:10.1007/978-3-7643-7390-0. ISBN 978-3-7643-2419-3. 
  2. ^ Øksendal, B.; Sulem, A. S. (2007). "Applied Stochastic Control of Jump Diffusions". doi:10.1007/978-3-540-69826-5. ISBN 978-3-540-69825-8. 
  3. ^ Karatzas, Ioannis; Shreve, Steven E. (1998). "Methods of Mathematical Finance". Stochastic Modelling and Applied Probability. 39. doi:10.1007/b98840. ISBN 978-0-387-94839-3. 

External linksEdit