hitting times are stopping times

hitting times are stopping times

Let

(

ℱ

t

)

t

∈

𝕋

be a

filtration

(

http://planetmath.org/FiltrationOfSigmaAlgebras

) on a

measurable space

(

Ω

,

ℱ

)

. If

X

is an

adapted

stochastic process

taking values in a measurable space

(

E

,

𝒜

)

then the

hitting time

of a set

S

∈

𝒜

is defined as

τ

:

Ω

→

𝕋

∪

{

±

∞

}

,

τ

⁢

(

ω

)

=

inf

⁡

{

t

∈

𝕋

:

X

t

⁢

(

ω

)

∈

S

}

.

We suppose that

𝕋

is a

closed subset

of

ℝ

, so the hitting time

τ

will indeed lie in

𝕋

whenever it is finite. The main cases are discrete-time when

𝕋

=

ℤ

+

and continuous-time where

𝕋

=

ℝ

+

. An important property of hitting times is that they are

stopping times

, as stated below for the different cases.

Discrete-time processes

For discrete-time processes, hitting times are easily shown to be stopping times.

Theorem

.

If the

index set

T

is

discrete

, then the hitting time

τ

is a stopping time.

Proof.

For any

s

≤

t

∈

𝕋

then

X

s

will be

ℱ

t

/

𝒜

-

measurable

, as it is adapted. So, by the fact that the

σ

-

algebra

ℱ

t

is

closed under

taking

countable

unions,

{

τ

≤

t

}

=

⋃

s

∈

𝕋

,

s

≤

t

X

s

-

1

(

S

)

∈

ℱ

t

as required.
∎

Continuous processes

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.

Theorem

.

Suppose that

X

is a continuous and adapted process taking values in a

metric space

E

. Then, the hitting time

τ

of any closed subset

S

⊆

E

is a stopping time.

Proof.

We may suppose that

S

is nonempty, and define the

continuous function

d

S

⁢

(

x

)

≡

inf

⁡

{

d

⁢

(

x

,

y

)

:

y

∈

S

}

on

E

. Then,

τ

is the first time at which

Y

t

≡

d

S

⁢

(

X

t

)

hits

0

. Letting

U

be any countable and

dense subset

of

𝕋

∩

[

0

,

t

]

then the

continuity

of the sample paths of

Y

gives,

{

τ

≤

t

}

=

{

inf

u

∈

U

Y

u

=

0

}

.

As the

infimum

of a

countable set

of

measurable functions

is measurable, this shows that

{

τ

≤

t

}

is in

ℱ

t

.
∎

Right-continuous processes

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.

Theorem

.

Suppose that

X

is a right-continuous and adapted process taking values in a metric space

E

, and that the filtration

(

F

t

)

is universally complete. Then, the hitting time

τ

of any closed subset

S

⊆

E

is a stopping time.

In particular, the hitting time of any

closed set

S

⊆

ℝ

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

ℱ

t

is universally complete is necessary.

Progressively measurable processes

The début

D

⁢

(

A

)

of a set

A

⊆

𝕋

×

Ω

is defined to be the hitting time of

{

1

}

for the process

1

A

,

D

⁢

(

A

)

⁢

(

ω

)

=

inf

⁡

{

t

∈

𝕋

:

(

t

,

ω

)

∈

A

}

.

An important result for continuous-time stochastic processes is the début

theorem

.

Theorem

(Début theorem).

Suppose that the filtration

(

F

t

)

is right-continuous and universally complete. Then, the début

D

⁢

(

A

)

of a

progressively measurable

A

⊆

T

×

Ω

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

X

taking values in a measurable space

(

E

,

𝒜

)

is said to be progressive if the set

X

-

1

⁢

(

S

)

is progressively measurable for every

S

∈

𝒜

. In particular, the hitting time of

S

is equal to the début of

X

-

1

⁢

(

S

)

and the début theorem has the following immediate corollary.

Theorem

.

Suppose that the filtration

(

F

t

)

is right-continuous and universally complete, and that

X

is a progressive process taking values in a measurable space

(

E

,

A

)

. Then, the hitting time

τ

of any set

S

∈

A

is a stopping time.

Title

hitting times are stopping times

Canonical name

HittingTimesAreStoppingTimes

Date of creation

2013-03-22 18:39:06

Last modified on

2013-03-22 18:39:06

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

7

Author

gel (22282)

Entry type

Theorem

Classification

msc 60G40

Classification

msc 60G05

Defines

hitting time

Defines

début

Defines

debut

Defines

début theorem

Defines

debut theorem

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