Research partially supported by the U.S. Department of Navy, Office of
Naval Research, Contract Nonr 855(09).
STOPPING TIMES ON BROWNIAN MOTION:
SOME PROPERTIES OF ROOT'S CONSTRUCTION
by
R. M. LOYNES 1
Department of Statistics
University of North Carolina at ~hapel Hill
Institute of Statistics Mimeo Series No. 637
JULY 1969
This work was done while the author was visiting from the University of
Cambridge.
i
Stopping Times on Brownian Motion: Some Properties of Root's Construction
by
2
R. M. Loynes
University of Cambridge
1.
Introduction and Summary.
The problem of finding a stopping-time
for Brownian motion in such a way that the stopped value will have a
given distribution has received considerable attention since the origina1 work of Skorokhod [6]: see Breiman [1], Dubins [2], and Root [5].
(A further method has also been given by W. J. Hall, but has not been
published.)
Although all approaches coincide when the given distri-
bution is concentrated on just two points, they otherwise give
ferent results, and have different characteristics.
dif-
Skorokhod's
method requires external randomization, for example, as does Breiman's
modification; Dubin's method involves, in an essential way, a limiting
process.
As these features could be disadvantages in certain circum-
stances, it seems worthwhile to consider Root's method carefully, since
it is at least in principle very straightforward, defining the
stopping-time as the hitting-time of a certain barrier.
(It is not,
however, easy to find the barrier corresponding to any particular distribution of the stopped process.)
Here we shall investigate the relationship between the barrier,
the stopping-time and the distribution of the stopped process, obtaining uniqueness and continuity properties.
This work was begun in
an attempt to solve the embedding problem for reverse martingales in
the following way.
If
{X: n
n
~
1}
is the martingale, then fbr every
1 Research partially supported by the Office of Naval Research, Contract Nonr 855(09).
2Tl1is work was done while the author was at the Department of Statistics,
University of North Carolina at Chapel Hill
I
1
2
n
the set
{X.: n
J
~ j
with stopping times
~
I}
may be embedded as a forward martingale,
{T~: n ~ j ~ I}
J
(Strassen [7], Theorem 4.3):
there seems a possibility that, for fixed
T.
J
and solve the problem.
j,
T;
may converge to some
The attempt was unsuccessful, but the re-
suIts, nevertheless, seemed worth presenting.
Certain general results are given in Section 2.
The main results,
which require certain restrictions,form Section 3.
2.
General Results.
The notation and terminology will be as in [5].
In particular, the Brownian motion sample path will be denoted by
w,
and a barrier will be defined as follows.
Definition.
A barrier is a subset
B of
[0, + 00]
x
[-
00, - 00 ]
satisfying
(1)
B is closed;
(2)
(+ 00, x) e: B
(3)
(t, ±oo) e: B
(4)
if
for all
for all
(t, x) e: B then
x·,
t e: [0, +00];
(s, x) e: B whenever
The set of all barriers is denoted by
sponding to
B,
s
>
t.
and the stopping time corre-
B e: B will be denoted by
Here and elsewhere we shall omit labels, such as
can be caused.
B,
when no confusion
3
Proposition
(i)
or (ii)
1:
If
B
B,
E
then either
B
contains at least one point
x
finite and
P['B < 00]
(t, x)
ph B = 00] =
s
~
and
only,
00
l.
The proof is trivial since in case (i)
{(s,x):
t
=1
B is the barrier consisting of all points at
and
line
with both
B
contains a horizontal
t}.
It will be recalled that Root defined a metric
r
on the space of
barriers, by
r(C, D)
where the metric
max{sup
=
r
XE
sup
yE
D r(y, C)} ,
on the points of the closed half plane
induced by the homeomorphism
the rectangle
C r(x, D),
(t,x) + (t/(l+t), x/(1+lxl))
[O,lJ x [-1, +1]
H is that
of
H onto
endowed with the Euclidean metric
p.
B is then compact and separable.
The space
The following lemma is a strengthened version of Lemma 2.4 of [5].
Lemma 1:
Let
peT < ooJ
= 1,
r(B,B)
0
<
B be a barrier with stopping time
for any
o
and that
B
°
there exists
P[I,-TI > E]
implies
corresponding to
E >
E
B.
r(B,B) < 0
If
PI,
implies
<
E,
0
where
= ooJ =
1,
PIT < AJ <
>
,
,.
0
Then if
and that
is the stopping time
then given
A
there exists
E.
Clearly both parts of the lemma can be combined into one statement
i f desired.
Proof:
details.
As this is quite similar to Root's proof, we omit some of the
Since, moreover, the second part is trivial, we confine our
attention to the first part.
Choose
n as in [5], and T so that
4
T > E
and
P[T
~
T] < E/3.
pro ~s~p~ 2T !w(t) I ~
(t,x)
with
B
E
p«t,x),
B)
<
there, that
Now
W E
then
r(B,B) < 0
0
so that if
implies
A is defined as in [5], it follows, as
P[T-T > E] < E.
P[T
we define
so that
1'1
and after that choose
Ixl ~ 1'1,
If the set
~
2T]
~
t ~ 2T, jx] ~ 1'1
with
1'1] < E/3,
t ~ 2T,
n.
Next choose
A = {w: w
A implies
T
<
P[T
~
then
T] + P[T
T+E]
>
r(B,B) < 0
~
4E/3,
and if
(t,x)
p«t,x), B) < n.
implies
B
E
If
satisfies (i)', (ii)', (iii)' below}, we find
T+E,
while
PiA]
the lemma then follows except that
>
3E
l-2E.
The first statement of
appears instead of
E,
while
the proof of the second part is trivial.
(i) ,
Same as (i) of [5], with
(ii) ,
T(W)
(iii)'
<
,
TO
2T;
Same as (iii) of [5], with
B with stopping time
T replacing
L(X)
Let us call a probability law
barrier
T replacing
T.
achievable if there exists a
T such that
L(w(T»
=
L(X):
then the
question naturally arises as to how large the class of achievable laws
is.
Certainly, every law with finite variance and zero mean is achiev-
able, for this is Theorem 2.1 of [5].
Possibly every law is achiev-
able, though we have not succeeded in proving this; the following is a
partial result.
Proposition
1:
If
L(X)
is concentrated on a half line, it is achiev-
able.
Proof:
First consider the case in which
finite interval
n
[a, b],
construct a law
where
a
<
b,
L(X)
is concentrated on a
and suppose
EX
>
O.
For every
L,
with zero mean and finite variance, by writing
n
5
P
n
to
1
=
1
(1 - -)p + -PI '
n
n ,n
L , L (X) ,
to
L,
n
and let
B,
B of the
the barrier at
L
n
for which
point
P ,P,
n
respectively, and
n
- (n-l)EX,
where
n
P
l,n
with stopping time
L(X).
B generates
and provided
L,
(O,x)
E
B
n
b
L(X),
is in the support of
for
~
x
b
for all
and this implies that
B
n
(-00, b)
or
is not
If we suppose, as we
then we may also suppose that
because of the finiteness of
E(L),
contains finite points.
If now we return to the general case, in which
trated on
B
(It is this provision that makes
it difficult to apply the argument more widely.)
may, that
at
1
Then, as in [5], there exists a limit
n
00,
places probability
be barriers and stopping times corresponding
,
E(L) < 00.
B ,
n
are the measures corresponding
[a,oo)
for some
a
or
b,
L(x)
is concen-
then truncation
to a finite interval together with an argument of the above type will
complete the proof of the proposition.
It will be convenient to define, for a given barrier
barrier function
f
(1)
fB(x)
Proposition
1:
(i)
(ii)
(iii)
B
=
as follows:
inf{t: (t,x)
::; 00
(f (x) , x)
B
B
E
(-00 ::; x ::; 00).
B}
E
for all
B
for all
x·,
x;
is lower semi-continuous;
(iv)
B = { (t ,x) :
(v)
f (±oo) = 0.
B
t ~ fB(x)};
Moreover any lower semi-continuous function
[0,00]
the
The barrier function has the following properties;
°::; fB(x)
f
B,
and vanishing at
too
f
taking values in
is the barrier function of the barrier
n
6
(Z)
{(t,x):
=
t
f(x)}.
:?:
Properties (i), (iv) and (v) are obvious, and properties (ii) and
(iii) follow from the fact that barriers
observed by Root [4].
that given
implies
way if
E
The usual definition of lower semi-continuity,
and
> 0
E,
f(x O)
=
or
o>
there exists
f(x) > f(x ) O
Ixol
are closed sets; (iii) was
such that
0
has of course to be modified in the obvious
00.
As above, we shall refer to the law or distribution of
being generated by
Proposition
!t.:
B
If
(or
B
l
and
generate the same law, then
Corollary:
B ,
2
with stopping times
B n B
2
l
B u B
and
Z
l
'1
II
l.
'z'
also generate
and
this
v,
'1
Z
=
respectively.
If
B ,
i
with stopping times
inf ,.,
spective stopping times
nB.
and
'1
'z = min('l' 'Z) ,
'i'
the same law, then this is also generated by
that
as
'B)'
same law and have stopping times
max('l' 'Z)
w('B)
uB.l.
and
nB.,
generate
with re-
l.
provided in the latter case
sup ' i '
l.
= 1, 2, .•• ),
(i
contains finite points, or equivalently
sup '.
l.
>
00
with
prob abili ty 1.
Proof:
which
It will be convenient to denote that part of a barrier
x
lies in a set
Let us write
K
= {x:
given sample function
before
c
BZ(K),
K by
E
E
c
K,
K.
w(,Z)
E
K]
= O.
Now suppose that for a
This means that
which implies that it hits
this in turn implies that it hits
P[w('l)
for
B(K).
flex) < fz(x)}.
w, w(,Z)
B
Bl(K)
B (K)
l
before
w hits
before
c
Bl(K):
From this and the fact that
B (K)
Z
c
B (K ),
2
thus
and
7
=
it follows that if we write
(3)
then
(4)
But it is now clear that if
while if
'B
= '1
w('2)
A
'2'
That
W
w( 'B)
wEAl
EA
2
we must have
'B
= '1
~
'2'
where
on the other hand
thus
has the same distribution as
is also easily seen, and completes the proof
(D(, 1)
and
of one half of the
proposition; the other half follows similarly.
In the part of the corollary dealing with unions, we may suppose
increasing, since if necessary, we may replace the original
B.
1.
U ••
J=l.
B .•
J
Lemma 1
B
=
UBi'
Now a subsequence of
'B
= lim
since
'i:
{B.}
1.
converges, to
B.
1.
by
B say, and by
thus it is only necessary to show that
L(w('B) = lim L(w('i»'
This is straightforward, and
as the other part may be dealt with similarly, the proof is complete.
Whether it is possible to have two regular barriers (see Section 3
for the definition) generating the same distribution, neither of which
includes the other, is unknown, though it seems a little unlikely.
8
3.
Uniqueness and Continuity Results.
to a well defined stopping time
T
B
,
A given barrier
B
gives rise
and consequently (provided it has
L(W(T )).
some finite points) generates exactly one distribution,
B
It
would be useful to be able to reverse this statement, and conclude that
each achievable law arises from exactly one barrier, but unfortunately
this is not true without certain restrictions.
B fB(x )
O
for a barrier
X
o
(i.e., with
x >
X
o
=
° for some
if
o~
inaccessible, and consequently
changing
L(W(T B));
0,
X
f
B
x
o'
In the first place, if
then the barrier beyond
and with
x <
X
o
if
o~
X
0)
is
can be changed arbitrarily without
and in the second place essentially different bar-
riers can generate the same law, as the example of the barriers
B = {(t,O):
c
t
~
c}
for arbitrary
c
~
0
shows.
The first difficulty
is easily overcome, but the second is more troublesome.
A possibility
of resolving it arises from the corollary to Proposition 4, since we
might agree to use only barriers with minimal stopping times, but apart
from the fact that uncountable collections might occur, there is also
the drawback that it would be difficult to recognise such barriers.
As
it happens, however, in the practically important case of zero mean and
finite variance laws, these difficulties disappear, and allow an attractive solution.
Definition:
The first positive zero of a non-negative lower semi
continuous function
for
0
~
x < x .
O
f
is at
X
o
if
X
o
~
0,
f(x )
O
=0
and
f(x)
>
The first negative zero is similarly defined.
Note that although barrier functions have first zeroes, they may be at
0
9
Definition:
Two barriers
Band
C or barrier functions
are equivalent if the latter agree on the interval
x+
f
B
[x_, x+]
and
f
where
C
x_,
are the first negative and positive zeroes of either.
Definition:
A barrier
B or barrier function
vanishes outside the interval
f
B
is regular if
f
B
[x_, x+].
Clearly any barrier is equivalent to exactly one regular barrier.
Lemma 2:
If
Band
if and only if
Band
Corollary:
B
If
C are two barriers,
and
T
B
= T
C
with probability 1
C are equivalent.
are two regular barriers,
C
probability 1 if and only if
Band
T
B
= T C with
C coincide.
The qualification 'with probability l' may of course be omitted.
That the corollary follows from the lemma is obvious, and the 'if' part
of the lemma needs no proof.
Suppose, therefore, that
two regular barriers which are not identical.
fC(x )
O
# 0,
and because
Care
Then without loss of
generality, we may suppose that for some finite
Thus
Band
X
o
> 0
fB(x ) < fC(x ).
O
O
C is regular and a lower semi-contin-
uous function achieves its minimum on a compact set, there exists an
s
>
0
such that
chosen so that
fC(x)
>
s
for
o
:0;;
x
2s < fC(x O) - fB(x O),
:0;;
x .
O
Moreover,
s
and then there exists
may be
0 > 0
f (x) > fC(x ) - s > fB(x ) + s .
O
O
C
Clearly there is a non-zero probability that w will hit the line
such that
x
Jx-xol < 0
implies
= Xo in the time-interval [fB(xO)' fB(x O) + s] while not hitting
until after time
fB(x ) + s,
O
and thus
P[T
B
# T ] # O.
C
C
10
Theorem 1:
L(X)
If
has zero mean and finite variance, it is generated
by exactly one regular barrier
E(T ).
B
B with finite
That there is at least one such barrier is Root's Theorem 2.1, so
we need only prove uniqueness.
ular barriers,
Band
TB = TB
C
1\
Thus
generating
L(X),
B u C also generates
finite mean.
C,
Suppose however that there are two reg-
L(x):
then by Proposition 4
and has stopping time
2
E(T ) = E (X ) = E(T
B
B
T with probability 1:
C
1\
T
B
1\
also with
TC'
which shows that
TC) ,
by the corollary to Lemma 2
B and
must coincide.
Tile essentials of the proof are contained in the following result.
Proposition
2:
If
L(X) ,
erated oy two barriers
with zero mean and finite variance, is gen-
Band
C,
and
E(T )
B
<
00,
then
T
B
~
T '
C
Under these conditions, there is therefore a minimum stopping time
and a maximum boundary.
In the first paragraph of this section, it was shown by example
that non-equivalent barriers can generate the same law.
Whether this
is true for every law is not known, though it seems quite likely.
One
can discover some properties of the situation without much difficulty.
Consider, for example, the uniform distribution on
[-1, +1]:
this is
generated by a regular barrier with finite stopping time, and for this
= 0,
barrier
fB(±l)
and with this latter condition the barrier is
unique.
If we relax this, at
+1
say, we must have
fB(+l)
otherwise there would be positive probability attached to
Our final results concern continuity properties.
the following notation will be convenient.
= 00,
since
[+1, 00).
To state them,
11
B
reg
=
is a regular barrier};
{B: B
T
= h:
L
=
L
for some barrier
TO
= h:
L
E
T,
L < 00
f
T
= h:
L
E
T,
E(L) < oo};
K
T
h:
L
E
T,
E(L)
Va =
{F: F
is an achievable probability distribution} ;
Vf
{F: F
is a probability distribution with mean
B
B} ;
with probability 1
K} ,
::::;
where
°
and variance
VK
{F:
F
E
Vf ,
v(F)
::::;
r;
on
T
in probability, denoted by
K < 00 ,•
lJ
(F)
0
v(F) < 00);
K} .
On these spaces, we shall put a topology:
corresponding to
::::;
Breg
on
the metric topology
and its subsets, the topology of convergence
cp;
and on
ogy of weak convergence, denoted by
w.
Va
and its subsets, the topo1-
The natural maps from barrier
to stopping time and from stopping time to distribution are the obvious
ones.
Theorem 2:
K ;;;: 0,
The following spaces are compact:
K
(T , cp)
and
(vK,
(T, cp),
and, for all
w).
The following natural maps are onto, and also have the properties
indicated:
, r) + (T, cp)
(a)
(B
(b)
(TO, cp) +
(Va,
w)
is continuous;
(c)
f
(T , cp) +
(V f ,
w)
is one-one and continuous;
(d)
K
(T , cp) +
(VK,
w)
is a homeomorphism.
Proof:
reg
is one-one and continuous;
That the maps are onto is a consequence of definition, and Root's
main theorem.
Moreover, (a) follows from Lemmas 1 and 2, and since
12
(8, r)
+
(T, cp)
is also continuous (though not one-one)
compact, it follows that
that
T
is compact:
K
T
is closed, and thus compact.
that i f
T
n
then
+ T,
F
=l>
n
F
from Fatou's lemma, it follows
For (b), it is necessary to show
where
'
F, F
correspond to
n
For this, it is sufficient to show that every subsequence of
tains a subsubsequence converging to
then
T ,
+
n
T
B is
and
F:
if
F,
F
n
con-
is the subsequence,
n
in probability, so that there exists a subsubsequence
Tn" + T with probability 1, and this implies
From Theorem 1,
F ,,=>F.
n
(c) now follows, and (d) is then obvious in view of the compactness of
K
T ;
K
V
the compactness of
is a consequence of (d) and the proof is
complete.
None of these results except possibly (c) can be improved in any
direct and obvious way.
Clearly
B
reg
B , and
is not closed in
therefore not compact (and none of the other spaces is compact):
(a) can not be a homeomorphism.
thus
The map in (b) is not one-one.
Whether (c) can be improved to state that the map is a homeomorphism has
not been settled:
T
impossibility of
corresponds to
an affirmative answer is equivalent to proving the
n
F
n
+
00
and
with probability 1 while
E(T),
n
v(F ),
n
v(F) <
F
n
+
F,
where
T
n
00.
The impossibility of improving (a) is merely another way of saying
that
r
is not the appropriate metric on
B
reg
for the present purposes.
It appears to be possible to modify the metric in a natural way to
in such a way that
follows that
under
T*.
(B
reg
,r*)
+
(T, cp)
r*
is a homeomorphism; it then
K
K
, the inverse image of T ) is compact
(and B
B
reg
reg
As, however, the details are rather tedious, and for many
purposes it is the stopping times rather than the barriers which are
important, they will not be given here.
13
A probably
more useful result is obtained by strengthening the
f
T
topology on
and
f
V •
f
T ,
On
On
F -+ F
F => F
if
n
n
Proposition 6:
(i)
F
n
and
we introduce the usual
v(F) -+ v(F).
n
and
This we shall denote by
= IA
Q (A)
n
f
V :
2
x dF (x),
=IA
Q(A)
n
It will not usually be the case that
2
x dF(x).
Qn' Q are probability meas-
ures; however, we define
Q ==> Q in the usual way by requiring
Ih(x)dQ (x) -+ Ih(x)dQ(x)
for every bounded continuous
n
Proof:
x
2
n
h.
Assume (i): then from Theorem A of [3], p. 183, it follows that
F ,
is uniformly integrable in
that
h(x) x
2
and since
n
F .
is uniformly integrable in
n
is bounded it follows
h
Now since
2
Ih(x) dQ (x) = Ih (x) x dF (x)
n
n
the truth of (ii) is clear.
suppose (ii) holds: then i f
±e;
choose in effect
find
h(x) = x
-2
Conversely,
are continuity points of
Ixl > e;, =
if
F (Ixl > e;) -+ F( Ixl > e;),
n
so that
a
if
Q,
::; e;,
Ixl
we can
and we
F (Ixl ::; e;) -+ F( Ixl ::; e;).
n
Because of the continuity of the admissible
h
it is therefore suffi2
cient to show that
-+
q.
v(F)
n -+ v(F)
.
where
(ii)
metric,
1
we introduce the topology defined by
The following are equivalent for
=> F
1
II x I >e; h(x)dF n (x) = II x I >e; h(x)x- dQ n (x)-+
-2
II x I>e; h(x)x dQ(x) = II x I>e; h(x)dF(x)
for every
e;
such that
are continuity points, and this is straightforward.
Theorem 3:
Proof:
then
n
-+ F
is a homeomorphism.
The map is known to be one-one.
E(T) -+ E(T),
so that
F
The natural map
n
F
n
~
F
so that
v(F) -+ v(F),
n
from Theorem 2.
in q-topology: then
Suppose that
F
n
,
and
T
n
-+
T
T -+ T is probability,
n
Suppose on the other hand that
F
E
K for some
V
K,
and from Theorem
14
2
•
n
+.
in probability, and in addition
the result at the top of p. 163 of [3],
E(.)
+
n
E!. n-.1
+
E(.),
and from
o.
References:
[1]
Breiman, L.
Probability Theory.
Reading: Addison-Wesley, 1968.
[2J
Dubins, L. On a theorem of Skorokhod.
(1968), 2094-2097.
[3J
Loeve, Michel Probability Theory (3rd edition).
Van Nostrand, 1963.
[4]
Root, D.h. The existence of certain stopping times on Brownian
motion. Technical report no. 15, Dept. of Mathematics,
University of Washington.
[5J
Root, D.H. The existence of certain stopping times on Brownian
motion. Ann. Math. Statist. 40 (1969), 715-718.
[6J
Skorokhod, A. Studies in the theory of random processes.
Reading: Addison-Wesley, 1965.
[7]
Strassen, Volker. Almost sure behavior of sums of independent random variables and martingales. Proc. 5th Berkeley Symposium on Mathematical Statistics and Probability 1 (1),
315-343.
Ann. Math. Statist. 39
Princeton:
IINQ,ASSTFTED
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.
CI aSSl'flcatlOn
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l. ORIGINATING ACTIVITY
(Corporate author)
28. REPORT SECURITY CLASSIFICATION
UNCLASSIFIED
Department of Statistics
University of North Carolina
Chapel Hill, North Carolina 27514
2b.
GROUP
3. REPORT TITLE
!
STOPPING TIMES ON BROWNIAN MOTION:
4. OESCRIPTIVE NOTES
SOME PROPERTIES OF ROOT'S CONSTRUCTION
(Type of report and inclusive dates)
TECHNICAL REPORT
5· AU THORIS)
(First name, middle initial. last name)
Robert M. Loynes
,
\
i
6· REPORT DATE
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July 1969
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Nonr 855(09)
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-07-
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12. SPONSORING MILITARY ACTIVITY
L.ogistics and Mathematical Statistics
Branch Office of Naval Research
fJashington, D.C. 20360
13. ABSTRACT
I
'J
\)
fi
Ii
~
I
Root recently IAnn. Math. Statist. 40] described a method of constructing an
J
i
absorbing barrier for Brownian motion with the property that the value of the process i
at the moment i t stops has any desired distribution with zero mean and finite variance
I
Further properties of the construction are investigated here, concerning uniqueness
.;
:1
:!''I:
and continuity.
f,I
"
I
I
Security Classification