Optimal stopping

Overview

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.

Discrete time case

Stopping rule problems are associated with two objects:

A sequence of random variables

X

1

,

X

2

, …

{\displaystyle X_{1},X_{2},\ldots }

, whose joint distribution is something assumed to be known A sequence of 'reward' functions

(

y

i

)

i ≥ 1

{\displaystyle (y_{i})_{i\geq 1}}

which depend on the observed values of the random variables in 1:

y

i

=

Continuous time case

Consider a gain process

G = (

G

t

)

t ≥ 0

{\displaystyle G=(G_{t})_{t\geq 0}}

defined on a filtered probability space

( Ω ,

F

, (

F

t

)

t ≥ 0

,

P

)

{\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t})_{t\geq 0},\mathbb {P} )}

and assume that

Solution methods

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

T

{\displaystyle T}

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 result

Let

Y

t

{\displaystyle Y_{t}}

be a Lévy diffusion in

R

k

{\displaystyle \mathbb {R} ^{k}}

given by the SDE

d

Y

t

= b (

Y

t

) d t + σ (

Y

t

) d

B

Coin tossing

(Example where

E

(

y

i

)

{\displaystyle \mathbb {E} (y_{i})}

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

Bern

(

1 2

)

,

{\displaystyle {\text{Bern}}\left({\frac {1}{2}}\right),}

and if

y

i

=

1 i

∑

House selling

(Example where

E

(

y

i

)

{\displaystyle \mathbb {E} (y_{i})}

does not necessarily converge) You have a house and wish to sell it. Each day you are offered

X

n

{\displaystyle X_{n}}

for your house, and pay

k

{\displaystyle k}

to continue advertising it. If you sell your house on day

n

{\displaystyle n}

, you will earn

y

n

Secretary problem

(Example where

(

X

i

)

{\displaystyle (X_{i})}

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

R

1

, … ,

R

n

{\displaystyle R_{1},\ldots ,R_{n}}

(n is some large number) are the ranks of the objects, and

y

i

{\displaystyle y_{i}}

is the chance you pick the best object if you stop intentionally rejecting objects at step i, then

(

R

Search theory

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.

Parking problem

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.). Approaching the destination, the driver goes down the street along which there are parking spaces – usually, only some places in the parking lot are free. 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.

Option trading

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

r

{\displaystyle r}

be the risk-free interest rate and

δ

{\displaystyle \delta }

and

σ

{\displaystyle \sigma }

be the dividend rate and volatility of the stock. The stock price

S

{\displaystyle S}

follows geometric Brownian motion

S

t

=

S

0

exp ⁡

{

See also

Halting problem Markov decision process Optional stopping theorem Prophet inequality Stochastic control Sequential analysis