upcrossings and downcrossings

upcrossings and downcrossings

Inequalities

involving the number of times at which a

stochastic process

passes upwards or downwards through a

bounded interval

play an important role in the theory of stochastic processes. Whether or not a

sequence

of

real numbers

converges

to a limit can be expressed in terms of the finiteness of the number of

upcrossings

or

downcrossings

, leading to results such as the

martingale convergence theorem

and

regularity

of

martingale

sample paths

.
As the main applications are to stochastic processes, we suppose that

(

X

t

)

t

∈

𝕋

is a real-valued stochastic process with

totally ordered

time

index set

𝕋

, usually a subset of the real numbers. However, for the definitions here, it is enough to consider

X

t

to be a sequence of real numbers, and the dependence on any underlying

probability space

is suppressed.

For real numbers

a

<

b

, the number of upcrossings of

X

across the

interval

[

a

,

b

]

is the maximum nonnegative integer

n

such that there exists times

s

k

,

t

k

∈

𝕋

satisfying

s

1

<

t

1

<

s

2

<

t

2

<

⋯

<

s

n

<

t

n

(1)

and for which

X

s

k

<

a

<

b

<

X

t

k

. The number of upcrossings is denoted by

U

⁢

[

a

,

b

]

. Note that this is either a nonnegative integer or is

infinite

. Similarly, the number of downcrossings, denoted by

D

⁢

[

a

,

b

]

, is the maximum nonnegative integer

n

such that there are times

s

k

,

t

k

∈

𝕋

satisfying (

1

) and such that

X

s

k

>

b

>

a

>

X

t

k

.

Figure 1:

Process with 3 upcrossings of the interval

[

a

,

b

]

.

Note that between any two upcrossings there is a downcrossing. Given

s

k

,

t

k

satisfying (

1

) and

X

s

k

<

a

<

b

<

X

t

k

, we can set

s

k

′

=

t

k

and

t

k

′

=

s

k

+

1

for

k

=

1

,

…

,

n

-

1

. Then,

X

s

k

′

>

b

>

a

>

X

t

k

′

from which we conclude that

D

⁢

[

a

,

b

]

≥

U

⁢

[

a

,

b

]

-

1

. Similarly, we have

U

⁢

[

a

,

b

]

≥

D

⁢

[

a

,

b

]

-

1

. Hence, the number of upcrossings and the number of downcrossings of an interval differ by at most

1

.

For a finite index set

𝕋

=

{

1

,

2

,

…

,

N

}

, the number of upcrossings can be computed as follows. Set

t

0

=

0

and define

s

1

,

s

2

,

…

and

t

1

,

t

2

,

…

by

t

k

=

inf

⁡

{

t

∈

𝕋

:

t

≥

s

k

,

X

t

>

b

}

∈

𝕋

∪

{

∞

}

.

s

k

=

inf

⁡

{

t

∈

𝕋

:

t

≥

t

k

-

1

,

X

t

<

a

}

∈

𝕋

∪

{

∞

}

.

The number of upcrossings of

[

a

,

b

]

is then equal to the maximum

n

such that

t

n

<

∞

(see Figure

1

).

References

1

David Williams,

Probability

with martingales

,
Cambridge Mathematical Textbooks, Cambridge University Press, 1991.

Title

upcrossings and downcrossings

Canonical name

UpcrossingsAndDowncrossings

Date of creation

2013-03-22 18:49:33

Last modified on

2013-03-22 18:49:33

Owner

gel (22282)

Last modified by

gel (22282)

Numerical id

5

Author

gel (22282)

Entry type

Definition

Classification

msc 40A05

Classification

msc 60G17

Related topic

MartingaleConvergenceTheorem

Related topic

ConvergenceOfASequenceWithFiniteUpcrossings

Defines

upcrossing

Defines

downcrossing

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