Optional stopping theorem | Wikipedia audio article

In probability theory, the optional stopping
theorem (or Doob's optional sampling theorem) says that, under certain conditions, the expected
value of a martingale at a stopping time is equal to its initial expected value. Since
martingales can be used to model the wealth of a gambler participating in a fair game,
the optional stopping theorem says that, on average, nothing can be gained by stopping
play based on the information obtainable so far (i.e., without looking into the future).
Certain conditions are necessary for this result to hold true. In particular, the theorem
applies to doubling strategies. The optional stopping theorem is an important
tool of mathematical finance in the context of the fundamental theorem of asset pricing. == Statement of theorem ==
A discrete-time version of the theorem is given below:
Let X = (Xt)tβˆˆβ„•0 be a discrete-time martingale and Ο„ a stopping time with values in β„•0
βˆͺ {∞}, both with respect to a filtration (Ft)tβˆˆβ„•0. Assume that one of the following
three conditions holds: (a) The stopping time Ο„ is almost surely
bounded, i.e., there exists a constant c ∈ β„• such that Ο„ ≀ c a.s.
(b) The stopping time Ο„ has finite expectation and the conditional expectations of the absolute
value of the martingale increments are almost surely bounded, more precisely, E [
Ο„ ]
< ∞ {\displaystyle \mathbb {E} [\tau ]<\infty
} and there exists a constant c such that E [ | X t
+ 1 βˆ’ X t | | F t ] ≀
c {\displaystyle \mathbb {E} {\bigl [}|X_{t+1}-X_{t}|\,{\big
\vert }\,{\mathcal {F}}_{t}{\bigr ]}\leq c} almost surely on the event {Ο„ > t} for all
t ∈ β„•0.

(c) There exists a constant c such that |Xtβˆ§Ο„|
≀ c a.s. for all t ∈ β„•0 where ∧ denotes the minimum operator.Then XΟ„ is an almost
surely well defined random variable and E [ X Ο„ ]
= E [ X 0 ]
. {\displaystyle \mathbb {E} [X_{\tau }]=\mathbb
{E} [X_{0}].} Similarly, if the stochastic process X is
a submartingale or a supermartingale and one of the above conditions holds, then E [ X Ο„ ]
β‰₯ E [ X 0 ]
, {\displaystyle \mathbb {E} [X_{\tau }]\geq
\mathbb {E} [X_{0}],} for a submartingale, and E [ X Ο„ ]
≀ E [ X 0 ]
, {\displaystyle \mathbb {E} [X_{\tau }]\leq
\mathbb {E} [X_{0}],} for a supermartingale. === Remark ===
Under condition (c) it is possible that Ο„ = ∞ happens with positive probability. On
this event XΟ„ is defined as the almost surely existing pointwise limit of X, see the proof
below for details. == Applications ==
The optional stopping theorem can be used to prove the impossibility of successful betting
strategies for a gambler with a finite lifetime (which gives condition (a)) and a house limit
on bets (condition (b)).

Suppose that the gambler can wager up to c dollars on a fair
coin flip at times 1, 2, 3, etc., winning his wager if the coin comes up heads and losing
it if the coin comes up tails. Suppose further that he can quit whenever he likes, but cannot
predict the outcome of gambles that haven't happened yet. Then the gambler's fortune over
time is a martingale, and the time Ο„ at which he decides to quit (or goes broke and is forced
to quit) is a stopping time. So the theorem says that E[XΟ„] = E[X0].

In other words,
the gambler leaves with the same amount of money on average as when he started. (The
same result holds if the gambler, instead of having a house limit on individual bets,
has a finite limit on his line of credit or how far in debt he may go, though this is
easier to show with another version of the theorem.)
Suppose a random walk starting at a β‰₯ 0 that goes up or down by one with equal probability
on each step.

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Suppose further that the walk stops if it reaches 0 or m β‰₯ a; the time
at which this first occurs is a stopping time. If it is known that the expected time at which
the walk ends is finite (say, from Markov chain theory), the optional stopping theorem
predicts that the expected stop position is equal to the initial position a. Solving a
= pm + (1 – p)0 for the probability p that the walk reaches m before 0 gives p = a/m.
Now consider a random walk X that starts at 0 and stops if it reaches –m or +m, and
use the Yn = Xn2 – n martingale from the examples section. If Ο„ is the time at which
X first reaches Β±m, then 0 = E[Y0] = E[YΟ„] = m2 – E[Ο„]. This gives E[Ο„] = m2.
Care must be taken, however, to ensure that one of the conditions of the theorem hold.
For example, suppose the last example had instead used a 'one-sided' stopping time,
so that stopping only occurred at +m, not at βˆ’m. The value of X at this stopping time
would therefore be m. Therefore, the expectation value E[XΟ„] must also be m, seemingly in
violation of the theorem which would give E[XΟ„] = 0.

The failure of the optional stopping
theorem shows that all three of the conditions fail. == Proof ==
Let X τ denote the stopped process, it is also a martingale (or a submartingale or
supermartingale, respectively). Under condition (a) or (b), the random variable XΟ„ is well
defined. Under condition (c) the stopped process X τ is bounded, hence by Doob's martingale
convergence theorem it converges a.s. pointwise to a random variable which we call XΟ„.
If condition (c) holds, then the stopped process X τ is bounded by the constant random variable
M := c. Otherwise, writing the stopped process as X t Ο„ = X 0 + βˆ‘ s
= 0 Ο„
∧ t
βˆ’ 1 ( X s
+ 1 βˆ’ X s )
, t
∈ N 0 , {\displaystyle X_{t}^{\tau }=X_{0}+\sum _{s=0}^{\tau
\land t-1}(X_{s+1}-X_{s}),\quad t\in {\mathbb {N} }_{0},}
gives |XtΟ„| ≀ M for all t ∈ β„•0, where M
:= | X 0 | + βˆ‘ s
= 0 Ο„
βˆ’ 1 | X s
+ 1 βˆ’ X s | = | X 0 | + βˆ‘ s
= 0 ∞ | X s
+ 1 βˆ’ X s | β‹… 1 {
Ο„ >
s } {\displaystyle M:=|X_{0}|+\sum _{s=0}^{\tau
-1}|X_{s+1}-X_{s}|=|X_{0}|+\sum _{s=0}^{\infty }|X_{s+1}-X_{s}|\cdot \mathbf {1} _{\{\tau
>s\}}} .By the monotone convergence theorem E [
M ]
= E [ | X 0 | ]
+ βˆ‘ s
= 0 ∞ E [ | X s
+ 1 βˆ’ X s | β‹… 1 {
Ο„ >
s } ] {\displaystyle \mathbb {E} [M]=\mathbb {E}
[|X_{0}|]+\sum _{s=0}^{\infty }\mathbb {E} {\bigl [}|X_{s+1}-X_{s}|\cdot \mathbf {1}
_{\{\tau >s\}}{\bigr ]}} .If condition (a) holds, then this series
only has a finite number of non-zero terms, hence M is integrable.
If condition (b) holds, then we continue by inserting a conditional expectation and using
that the event {Ο„ > s} is known at time s (note that Ο„ is assumed to be a stopping
time with respect to the filtration), hence E [
M ] = E [ | X 0 | ]
+ βˆ‘ s
= 0 ∞ E [ E [ | X s
+ 1 βˆ’ X s | | F s ] β‹… 1 {
Ο„ >
s } ⏟ ≀ c 1 {
Ο„ >
s } a.s.

By (b) ] ≀ E [ | X 0 | ]
+ c βˆ‘ s
= 0 ∞ P (
Ο„ >
s ) = E [ | X 0 | ]
+ c E [
Ο„ ]
< ∞
, {\displaystyle {\begin{aligned}\mathbb {E}
[M]&=\mathbb {E} [|X_{0}|]+\sum _{s=0}^{\infty }\mathbb {E} {\bigl [}\underbrace {\mathbb
{E} {\bigl [}|X_{s+1}-X_{s}|{\big |}{\mathcal {F}}_{s}{\bigr ]}\cdot \mathbf {1} _{\{\tau
>s\}}} _{\leq \,c\,\mathbf {1} _{\{\tau >s\}}{\text{ a.s. by (b)}}}{\bigr ]}\\&\leq \mathbb {E}
[|X_{0}|]+c\sum _{s=0}^{\infty }\mathbb {P} (\tau >s)\\&=\mathbb {E} [|X_{0}|]+c\,\mathbb
{E} [\tau ]<\infty ,\\\end{aligned}}} where a representation of the expected value
of non-negative integer-valued random variables is used for the last equality.
Therefore, under any one of the three conditions in the theorem, the stopped process is dominated
by an integrable random variable M. Since the stopped process X τ converges almost
surely to Xτ , the dominated convergence theorem implies E [ X Ο„ ]
= lim t
β†’ ∞ E [ X t Ο„ ]
.

{\displaystyle \mathbb {E} [X_{\tau }]=\lim
_{t\to \infty }\mathbb {E} [X_{t}^{\tau }].} By the martingale property of the stopped
process, E [ X t Ο„ ]
= E [ X 0 ]
, t
∈ N 0 , {\displaystyle \mathbb {E} [X_{t}^{\tau }]=\mathbb
{E} [X_{0}],\quad t\in {\mathbb {N} }_{0},} hence E [ X Ο„ ]
= E [ X 0 ]
. {\displaystyle \mathbb {E} [X_{\tau }]=\mathbb
{E} [X_{0}].} Similarly, if X is a submartingale or supermartingale,
respectively, change the equality in the last two formulas to the appropriate inequality.

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