hitting times are stopping times
Let be a filtration (http://planetmath.org/FiltrationOfSigmaAlgebras) on a measurable space . If is an adapted stochastic process taking values in a measurable space then the hitting time of a set is defined as
We suppose that is a closed subset of , so the hitting time will indeed lie in whenever it is finite. The main cases are discrete-time when and continuous-time where . An important property of hitting times is that they are stopping times, as stated below for the different cases.
For discrete-time processes, hitting times are easily shown to be stopping times.
For continuous-time processes it is not necessarily true that a hitting time is even measurable, unless further conditions are imposed. Processes with continuous sample paths can be dealt with easily.
Suppose that is a continuous and adapted process taking values in a metric space . Then, the hitting time of any closed subset is a stopping time.
We may suppose that is nonempty, and define the continuous function on . Then, is the first time at which hits . Letting be any countable and dense subset of then the continuity of the sample paths of gives,
Right-continuous processes are more difficult to handle than either the discrete-time and continuous sample path situations. The first time at which a right-continuous process hits a given value need not be measurable. However, it can be shown to be universally measurable, and the following result holds.
Suppose that is a right-continuous and adapted process taking values in a metric space , and that the filtration is universally complete. Then, the hitting time of any closed subset is a stopping time.
In particular, the hitting time of any closed set for an adapted right-continuous and real-valued process is a stopping time.
The proof of this result is rather more involved than the case for continuous processes, and the condition that is universally complete is necessary.
Progressively measurable processes
The début of a set is defined to be the hitting time of for the process ,
An important result for continuous-time stochastic processes is the début theorem.
Theorem (Début theorem).
Suppose that the filtration is right-continuous and universally complete. Then, the début of a progressively measurable is a stopping time.
Proofs of this typically rely upon properties of analytic sets, and are therefore much more complicated than the result above for right-continuous processes.
A process taking values in a measurable space is said to be progressive if the set is progressively measurable for every . In particular, the hitting time of is equal to the début of and the début theorem has the following immediate corollary.
Suppose that the filtration is right-continuous and universally complete, and that is a progressive process taking values in a measurable space . Then, the hitting time of any set is a stopping time.
|Title||hitting times are stopping times|
|Date of creation||2013-03-22 18:39:06|
|Last modified on||2013-03-22 18:39:06|
|Last modified by||gel (22282)|