Stopped process

In probability theory, a stopped process is a special stochastic process that is stopped at a certain random point in time. Formally, this is done using a stop time . Stopped processes are used, for example, when examining game abandonment strategies. There, stopping the process corresponds to canceling the game. Stopped processes find a more theoretical application in the localization of process classes through which, for example, the martingales are expanded to include the local martingales .

definition

Given is a stochastic process with at most a countable index set and a stop time with values in . Then the process is called ${\ displaystyle X = (X_ {t}) _ {t \ in T}}$ ${\ displaystyle T}$ ${\ displaystyle \ tau}$ ${\ displaystyle T}$

{\ displaystyle X ^ {\ tau}: = (X_ {t \ wedge \ tau}) _ {t \ in T} = (X _ {\ min (t, \ tau)}) _ {t \ in T}}

the stopped process regarding . It is ${\ displaystyle \ tau}$

{\ displaystyle X _ {\ min (t, \ tau)} \ colon \ omega \ to X _ {\ min (t, \ tau (\ omega))} (\ omega)}

In purely formal terms, the process is not halted, but rather no longer changes its value after the point in time . ${\ displaystyle \ tau}$

Explanation

If there is a stochastic process , the stopped process arises as follows: ${\ displaystyle (X_ {n}) _ {n \ in \ mathbb {N}}}$

It is because in the zeroth time step stopping the process makes no difference. ${\ displaystyle X_ {0} = X_ {0} ^ {\ tau}}$
In the first time step, the process remains stopped on the set , but otherwise behaves like the original process, so it is ${\ displaystyle \ {\ tau = 0 \}}$

{\ displaystyle X_ {1} ^ {\ tau} = X_ {1} \ mathbf {1} _ {\ {\ tau \ geq 1 \}} + X_ {0} \ mathbf {1} _ {\ {\ tau = 0 \}}}

.

In the second time step, the stopped process on the set remains unchanged, but is also stopped on the set . So is ${\ displaystyle \ {\ tau = 0 \}}$ ${\ displaystyle \ {\ tau = 1 \}}$

{\ displaystyle X_ {2} ^ {\ tau} = X_ {2} \ mathbf {1} _ {\ {\ tau \ geq 2 \}} + X_ {1} \ mathbf {1} _ {\ {\ tau = 1 \}} + X_ {0} \ mathbf {1} _ {\ {\ tau = 0 \}}}

.

Thus, the nth random variable in the stopped process is given by

{\ displaystyle X_ {n} ^ {\ tau} = X_ {n} \ mathbf {1} _ {\ {\ tau \ geq n \}} + \ sum _ {i = 0} ^ {n-1} X_ {i} \ mathbf {1} _ {\ {\ tau = i \}}}

.

If you consider a stopped process only on the set for one , it behaves on this set up to the kth step like the actual process and does not change its values afterwards. ${\ displaystyle \ {\ tau = k \}}$ ${\ displaystyle k \ in \ mathbb {N}}$

comment

The stopped process should not be with the "sampled" random variable ${\ displaystyle X ^ {\ tau}}$

{\ displaystyle X _ {\ tau} = \ sum _ {n = 0} ^ {\ infty} \ mathbf {1} _ {\ {\ tau = n \}} X_ {n}}

of a stochastic process , especially since the notation in the literature is not clear. ${\ displaystyle (X_ {n}) _ {n \ in \ mathbb {N}}}$

Statements about stopped processes

The most important statements about stopped processes include the optional stopping theorem and the optional sampling theorem . They examine how stopped (sub- / super) martingales behave and what statements can be made about the expected values of the stopped processes.

literature

Achim Klenke: Probability Theory . 3. Edition. Springer-Verlag, Berlin Heidelberg 2013, ISBN 978-3-642-36017-6 , doi : 10.1007 / 978-3-642-36018-3 .
David Meintrup, Stefan Schäffler: Stochastics . Theory and applications. Springer-Verlag, Berlin Heidelberg New York 2005, ISBN 978-3-540-21676-6 , doi : 10.1007 / b137972 .
Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , doi : 10.1007 / 978-3-642-45387-8 .