Stopped Process
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a stopped process is a
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the family often has the interpretation of time. Sto ...
that is forced to assume the same value after a prescribed (possibly random) time.


Definition

Let * (\Omega, \mathcal, \mathbb) be a
probability space In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models ...
; * (\mathbb, \mathcal) be a
measurable space In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured. It captures and generalises intuitive notions such as length, area, an ...
; * X : , + \infty) \times \Omega \to \mathbb be a stochastic process; * \tau : \Omega \to
stopping time In probability theory, in particular in the study of stochastic processes, a stopping time (also Markov time, Markov moment, optional stopping time or optional time ) is a specific type of "random time": a random variable whose value is interpre ...
with respect to some
filtration Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filte ...
\ of \mathcal. Then the stopped process X^ is defined for t \geq 0 and \omega \in \Omega by :X_^ (\omega) := X_ (\omega).


Examples


Gambling

Consider a Gambling">gambler Gambling (also known as betting or gaming) is the wagering of something of value ("the stakes") on a random event with the intent of winning something else of value, where instances of strategy are discounted. Gambling thus requires three ele ...
playing roulette. ''X''''t'' denotes the gambler's total holdings in the casino at time ''t'' ≥ 0, which may or may not be allowed to be negative, depending on whether or not the casino offers credit. Let ''Y''''t'' denote what the gambler's holdings would be if he/she could obtain unlimited credit (so ''Y'' can attain negative values). * Stopping at a deterministic time: suppose that the casino is prepared to lend the gambler unlimited credit, and that the gambler resolves to leave the game at a predetermined time ''T'', regardless of the state of play. Then ''X'' is really the stopped process ''Y''''T'', since the gambler's account remains in the same state after leaving the game as it was in at the moment that the gambler left the game. * Stopping at a random time: suppose that the gambler has no other sources of revenue, and that the casino will not extend its customers credit. The gambler resolves to play until and unless he/she goes broke. Then the random time \tau (\omega) := \inf \ is a stopping time for ''Y'', and, since the gambler cannot continue to play after he/she has exhausted his/her resources, ''X'' is the stopped process ''Y''''τ''.


Brownian motion

Let B : , + \infty) \times \Omega \to \mathbb be one-dimensional standard Brownian motion starting at zero. * Stopping at a deterministic time T > 0: if \tau (\omega) \equiv T, then the stopped Brownian motion B^ will evolve as per usual up until time T, and thereafter will stay constant: i.e., B_^ (\omega) \equiv B_ (\omega) for all t \geq T. * Stopping at a random time: define a random stopping time \tau by the first hitting time for the region \: \tau (\omega) := \inf \. Then the stopped Brownian motion B^ will evolve as per usual up until the random time \tau, and will thereafter be constant with value a: i.e., B_^ (\omega) \equiv a for all t \geq \tau (\omega).


See also

* Killed process


References

* {{DEFAULTSORT:Stopped Process Stochastic processes