3 Sample paths of the Brownian motion

3
Sample paths of the Brownian motion
In this lecture we discuss some properties which are shared, with probability 1, by sample paths of
Brownian motion. We have already seen that there are positive results (paths are continuous, and
H¨
older continuous with arbitrary parameter γ less than 12 on bounded intervals, with probability
1). We can ask about differentiability. The answer is surprising: a theorem of Dvoretzky, Erd¨
os
and Kakutani says that sample paths are (almost surely) nowhere differentiable on R + . Then, if
continuous, nowhere differentiable functions are considered as pathological in mathematical analysis,
they seem to be the norm for trajectories of Brownian motion.
We will not prove the previous result, but the following simple reasoning can give a justification
for the theorem, by considering differentiability at t = 0. Remember that, by Proposition 2.18, the
process Yt = tB1/t is a Brownian motion, with Y0 = 0. Since
Yt
= B1/t
t
the derivative of Yt exists for t = 0 if and only if Bt has a limit as t → ∞. But, in fact, we have
sup Bn = +∞,
n∈N
inf Bn = −∞
n∈N
(3.1)
with probability 1; hence, almost surely, paths of the Brownian motion Yt are not differentiable at 0.
Of course, the non-differentiability of Brownian paths can be phrased by saying that Brownian
particles show at no point a finite velocity, thus they can only be regarded as an approximating model
of the physical reality.
Maxima for Brownian paths
The material contained in this lecture can be found in many classical textbooks, as Doob [Do53],
Freedman [Fr71] or L´
evy [L´e54, L´e65].
We consider a real Brownian motion {Bt , t ≥ 0} on a stochastic basis (Ω, F, {Ft }, P). We will
often write B(t) instead of Bt .
Proposition 3.1. For almost every ω, the function B(·, ω) is monotone in no interval.
34
3 Sample paths of the Brownian motion
Proof. We can simplify the situation. First, it suffices to prove the result for an interval with rational extremes; then, thanks to Exercise 2.7, we can reduce to prove that, a.s., no path is monotone
nondecreasing on [0, 1].
Define, for every n, the events
\
n−1
i
i
An = ω : B( i+1
B( i+1
)
−
B(
)
≥
0
for
i
=
0,
.
.
.
,
n
−
1
=
n
n
n ) − B( n ) ≥ 0 .
i=1
Note that the set of nondecreasing paths on [0, 1] is contained in ∩ An . But
n
P(An ) =
n−1
Y
1
i
P( B( i+1
n ) − B( n ) ≥ 0 ) = 2n → 0
i=0
(3.2)
and the proof is complete.
2
Problem 3.1. Prove claim (3.2).
Let us recall that a continuous function f has a local maximum at t if there exists ε > 0 such that
f (s) ≤ f (t) for all s ∈ (t − ε, t + ε). The maximum is strict if the inequality is strict, i.e., f (s) < f (t)
for all s ∈ (t − ε, t + ε).
We shall need the following properties of continuous functions.
Lemma 3.2. Let f be a continuous function on [0, 1], monotone in no interval. Then, the set of local
maxima of f is dense in [0, 1].
Proof. Using time scaling invariance, it is sufficient to prove the existence of just one local maximum
in [0, 1].
At first, let us assume that f (0) < f (1), and denote z the last point in [0, 1] such that f (z) = f (0).
Then, since f (x) > f (z) = f (0) for all x ∈ (z, 1], but the function is not monotone increasing in [z, 1],
hence there must be z < a < b < 1 with f (a) > f (b) > f (z). In the interval [z, b], the function is
initially increasing (since f (a) > f (z)) and then decreasing (having f (b) < f (a)), therefore there must
necessarily be a local maximum.
The converse case f (0) > f (1) is treated similarly, using z the first point with f (z) = f (1), and
proving the existence of a local maximum in [0, z].
Finally, in case f (0) = f (1), since f is not constant, there exists c such that f (c) 6= f (0). Then
one can proceed with the same construction as above on either [0, c] or [c, 1].
2
We are ready to state the following result.
Theorem 3.3. For almost every ω, all local maxima of {Bt (ω), t ∈ [0, 1]} are strict and constitute a
dense set in [0, 1].
Proof. Let us denote M [a, b] the maximum value taken by the Browian motion Bt as t varies in [a, b].
We claim that for consecutive intervals a < b < c < d,
3 Sample paths of the Brownian motion
P(M [a, b] 6= M [c, d]) = 1.
35
(3.3)
This clearly implies the thesis. Let us prove (3.3). Setting
X = B(c) − B(b),
Y = max{B(c + t) − B(c), t ∈ [0, d − c]},
we have
{M [a, b] 6= M [c, d]} = {B(c) + Y 6= M [a, b]} = {X 6= M [a, b] − B(b) − Y }.
Notice that Fb , X and Y are independent, M [a, b] and B(b) are Fb -measurable; therefore, denoting
by µX the law of X,
P(X 6= M [a, b] − B(b) − Y ) = P X + (M [a, b] − B(b) − Y ) = 0
Z
=
µX ⊗ µM [a,b]−B(b)−Y (dx dz) = 0
x+z=0
since X has a gaussian distribution and P(X 6= x) = 1.
2
The zero set of Brownian motion
We have seen the link between behaviour of the Brownian motion for t near zero and the asymptotic
behaviour as t → ∞. Before we proceed with the study of the set of zeroes for the Brownian motion,
let us exploit some more this link. First, we propose the following exercise.
Problem 3.2.
1. Prove claim (3.1):
sup Bn = +∞,
n∈N
2. Prove that
lim
t→+∞
inf Bn = −∞.
n∈N
Bt
= 0 a.s.
t
∞
Hint: write (supn |Bn | < ∞) = ∪ (supn |Bn | < k) so that the probability of the event on the left is bounded
k=1
by the series of the probabilities on the right. But
P(sup |Bn | < k) ≤ lim P(|Bn | < k) = lim P(|B1 | < k/n1/2 ) . . .
n
n
n
Let us consider the following consequence of (3.1). Writing Wt = tB1/t as t → 0+ , for almost
every path there exist two sequences {tn } and {sn } converging to 0 such that tn+1 < sn+1 < tn < sn
and W (tn ) > 0, W (sn ) < 0. Since the Brownian paths are almost surely continuous, there exists a
sequence {τn } with tn < τn < sn and W (τn ) = 0. We have thus proved that
Corollary 3.4. With probability one, Bronwian motion returns to the origin infinitely often.
36
3 Sample paths of the Brownian motion
Let us introduce the random set
Z(ω) = {t ≥ 0 : Bt (ω) = 0}.
Notice that 0 ∈ Z(ω) for every ω, a.s., and previous corollary shows that, with probability one, 0
is an accumulation point for Z(ω).
Theorem 3.5. With probability one, the set Z(ω) is closed, of Lebesgue measure 0, nowhere dense in
R+ .
Proof. For simplicity, we suppress the phrase “with probability one” in the following. Since Z(ω) =
B(ω, ·)−1 (0) is the inverse of a closed set through a continuous mapping, it follows that it is closed.
Let λ denote the Lebesgue measure on R+ ; using Fubini’s theorem
Z ∞
P(ω : Bt (ω) = 0) dt = 0
E[λ(Z(ω))] = (λ × P){(t, ω) : Bt (ω) = 0} =
0
since Bt takes every given value with probability zero.
Let I be an interval of the positive real line; since t 7→ Bt (ω) is continuous on I, if the set
Z(ω) ∩ I were dense in I then necessarily Bt ≡ 0 on I, which is absurd, since (otherwise Bt would be
differentiable). Hence Z(ω) is not dense in any interval I.
2
3.1 Regolarity of sample paths
Of particular interest is the study of variations of sample paths of Brownian motion; this is connected
with the possibility of defining a (Lebesgue-Stieltjes) integral with respect to each path.
3.1.1 Quadratic variation
Let f : R+ → R be a given function; for any interval [s, t] ⊂ R+ and partition π = t0 = s < t1 < · · · <
tn = t we set
n 2
X
qV (f, π, [s, t]) =
f (ti ) − f (ti−1 ) .
i=1
We denote kπk = max{ti − ti−1 }, ti−1 , ti ∈ π, the size of the partition; then we call f a bounded
quadratic variation function on [s, t] if the following limit exists
qV (f, [s, t]) = lim qV (f, π, [s, t]) < ∞.
kπk→0
We begin presenting a link between quadratic variation and H¨
older continuity.
Proposition 3.6. Fix an interval [s, t] ⊂ R+ ; then every function f : [s, t] → R H¨
older continuous of
parameter α > 1/2 verifies qV (f, [s, t]) = 0.
3.1 Regolarity of sample paths
37
Proof. Take a partition π of [s, t]; then
2
n
n 2 X
X
2α f (ti ) − f (ti−1 ) (ti − ti−1 ) f (ti ) − f (ti−1 ) =
α
(ti − ti−1 ) i=1
i=1
≤
n
X
i=1
≤C
2
C 2 (ti − ti−1 )2α
max |ti − ti−1 |
2α−1
ti−1 ,ti ∈π
n
X
i=1
|
hence
|ti − ti−1 |
{z
}
=t−s
qV (f, π, [s, t]) ≤ C 2 (t − s)kπk2α−1
and, passing to the limit as kπk → 0, the thesis follows.
2
The next step is the study of quadratic variations for the trajectories t 7→ B t (ω) of a Brownian
motion on an interval [s, t]. If we set A[s,t] (ω) = qV (B· (ω), [s, t]), it results to be a random variable
Ft -measurable. Surprisingly enough, it happens that this random variable is constant with probability
1, and it holds A[s,t] = t − s.
In order to simplify the proof, we give a weaker result, considering convergence in L 2 (from which
convergence in probability and convergence almost surely for a subsequences follows).
Proposition 3.7. Let B = {Bτ , τ ≥ 0} be a Brownian motion, [s, t] ⊂ R+ be a given interval and π
a partition of [s, t]. Then
lim qV (B· (ω), π, [s, t]) = t − s
kπk→0
Proof. Since t − s =
P
k tk
in L2 .
− tk−1 we have
qV (B· , π, [s, t]) − (t − s) =
Xh
k
(Btk − Btk−1 )2 − (tk − tk−1 )
i
and
2 X nh
io
ih
E (Btk − Btk−1 )2 − (tk − tk−1 ) (Btj − Btj−1 )2 − (tj − tj−1 )
EqV (B· , π, [s, t]) − (t − s) =
j,k
i2
X h
=
E (Btk − Btk−1 )2 − (tk − tk−1 )
k
since random variables (Btk − Btk−1 )2 − (tk − tk−1 ) and (Btj − Btj−1 )2 − (tj − tj−1 ), for j 6= k, are
independent with mean 0, the expected value of their product is 0; hence
38
3 Sample paths of the Brownian motion
"
#2
2 X
Btk − Btk−1 2
2
√
EqV (B· , π, [s, t]) − (t − s) =
(tk − tk−1 ) E
−1
tk − tk−1
k
X
=c
(tk − tk−1 )2
k
where, for each k,
Btk −Btk−1
√
tk −tk−1
c=E
"
Then we have
as kπk → 0.
2
is a standard Gaussian distribution, so that
Btk − Btk−1
√
tk − tk−1
2
−1
#2
=
Z
2
1
(x2 − 1)2 √ e−x /2 dx = 2.
2π
R
2
EqV (B· , π, [s, t]) − (t − s) ≤ 2kπk(t − s) → 0
3.1.2 Bounded variation
Let f : R+ → R be a given function; for any interval [s, t] and partition π = t0 = s < t1 < · · · < tn = t
we set
n X
V (f, π, [s, t]) =
f (ti ) − f (ti−1 ).
i=1
We call f a function with bounded variation on [s, t] if the following limit exists
V (f, [s, t]) = lim V (f, π, [s, t]) < ∞.
kπk→0
We call f a function with total bounded variation if there exists V (f, [0, ∞)) < ∞.
We also set
V + (f, π, [s, t]) =
n h
X
i=1
f (ti ) − f (ti−1 )
i+
,
V − (f, π, [s, t]) =
n h
X
i=1
f (ti ) − f (ti−1 )
i−
,
and similarly
V + (f, [s, t]) = lim V + (f, π, [s, t]),
kπk→0
V − (f, [s, t]) = lim V − (f, π, [s, t]).
kπk→0
Take a function f with bounded variation, with f (0) = 0; we set V + (x) = V + (f, [0, x]) and V − (x) =
V − (f, [0, x]), V (x) = V (f, [0, x]). Then we can write f as a difference between two increasing functions
f (x) = V + (x) − V − (x)
while the variation of f is given by V (x) = V + (x) + V − (x).
3.2 The law of the iterated logarithm
39
Proposition 3.8. Trajectories of Brownian motion are, with probability 1, of unbounded variation on
every interval.
Proof. Fix [a, b] ⊂ R+ and a partition π of [a, b]; the quadratic variation of B· (ω) is
X
qV (B· , π, [a, b]) =
(Btk − Btk−1 )2
k
≤ max |Btk − Btk−1 |
k
X
k
|Btk − Btk−1 |
(3.4)
Since trajectories are continuous we have
lim max |Btk − Btk−1 | = 0,
kπk→0
k
a.s.
0
Assume that there exists an event Ω[a,b]
∈ F, with positive probability, such that trajectories B· (ω)
0
have bounded variation for ω ∈ Ω[a,b] . Then for such ω, quadratic variation of the trajectory should
be 0 according to (3.4), having a contradiction with Proposition 3.7.
Hence there exists Ω[a,b] ∈ F with P(Ω[a,b] ) = 1 such that for ω ∈ Ω[a,b] the path t 7→ Bt (ω) has
unbounded variation on [a, b]. We consider the sequence of intervals [a, b] ⊂ R+ with rational extreme
points. Define the set
[
¯=
Ω
Ω[a,b]
0≤a≤b∈Q
¯ ∈ F and P(Ω)
¯ = 1. Therefore, for each ω 6∈ Ω,
¯ Bt (ω) has unbounded variation on
it holds that Ω
every nontrivial interval [s, t] with real valued extreme points (each such interval contains an interval
[a, b] with rational end points, and the variation increases with the interval). This concludes the proof.
2
The above results, together with Proposition 3.6, imply the following property.
Corollary 3.9. There exists a set of probability 1 such that the corresponding trajectories t 7→ B t (ω)
are nowhere α-H¨
older continuous for arbitrary exponent α > 21 .
3.2 The law of the iterated logarithm
We present in this section the fine results about regularity of Brownian sample paths, following the
beautiful analysis of P. L´
evy. We will present there the law of iterated logarithm, and we show that
the H¨
older exponent γ < 12 cannot be improved.
The first result (the law of the iterated logarithm) describes the oscillations of the Brownian sample
paths near zero and infinity.
Theorem 3.10. For ω in a set of probability 1
max lim p
t→0+
Bt (ω)
2t log log(1/t)
= 1.
(3.5)
40
3 Sample paths of the Brownian motion
Before we proceed with the proof, we shall notice the following consequences of the theorem. By
simmetry, since −Bt is again a Brownian motion,
min lim p
t→0+
Bt (ω)
2t log log(1/t)
= −1,
and time inversion provides the following
max lim √
t→+∞
Bt (ω)
= 1,
2t log log t
min lim √
t→+∞
Bt (ω)
= −1.
2t log log t
For almost every trajectory, we can construct two sequences {tn } and {sn }, increasing and diverging
to +∞, with sn < tn < sn+1 , such that
p
p
B(sn ) ≤ − sn log log sn .
B(tn ) ≥ tn log log tn ,
Then we see that the oscillations of every path increase more and more; further, due to continuity, we
see that the trajectories touch each real value infinitely often.
In preparation for the proof, we propose the following exercise.
Problem 3.3. Compute the tail of the normal distribution: for every x > 0 we have
Z ∞
2
2
1
x
−x2 /2
e
≤
e−y /2 dy ≤ e−x /2 .
1 + x2
x
x
(3.6)
The next lemma will also be useful.
Lemma 3.11. Consider on (Ω, F, P) a sequence {Xn , n ∈ N} of real Gaussian random variables,
with zero mean, and define the partial sums Sn = X1 + · · · + Xn . Then, for any x > 0, it holds
P(max Si ≥ x) ≤ 2P(Sn ≥ x).
i≤n
Proof. Note that
max Si ≥ x
i≤n
where
=
n
[
Ai ,
i=1
Ai = {ω ∈ Ω : S1 < x, . . . , Si−1 < x, Si ≥ x}.
Define also the set B = {ω : Sn ≥ x}.
Since Ai ∩ Aj = ∅, i 6= j, using independence, we find
P(Ai ∩ B) ≥ P(Ai ∩ {ω : Sn ≥ Si }) = P(Ai )P({ω : Sn ≥ Si }) = P(Ai )P(Xi+1 + . . . + Xn ≥ 0).
It is easy to check that
P(Xi+1 + . . . + Xn ≥ 0) =
therefore
n
X
n
1
;
2
1
1X
P(Ai ) ≥ P
P(B) ≥
P(Ai ∩ B) ≥
2
2
i=1
i=1
and this gives the assertion.
2
n
[
i=1
Ai
!
3.2 The law of the iterated logarithm
41
Corollary 3.12. (Maximal inequality for Brownian motion) For a Brownian motion {B t , t ≥ 0} we
have:
P(sup Br > x) ≤ 2P(Br > x),
t ≥ 0.
(3.7)
r≤t
Proof. Brownian motion has continuous paths, hence the probability on the left in (3.7) can be studied
for τ varying in the set of dyadic rational numbers between s and t. Using that the increments of
Brownian motion are independent and Gaussian distributed we obtain the assertion from the previous
lemma.
2
Proof (Theorem 3.10). We shall prove first that
max lim p
t→0+
Bt (ω)
2t log log(1/t)
≤ 1.
(3.8)
p
With the notation h(t) = log log(1/t) and fixed δ ∈ (0, 1), we choose θ ∈ (0, 1) such that λ =
θ(1 + δ)2 > 1. Define a sequence of times tn = θn decreasing to 0 and consider the events
An =
max Bt > (1 + δ)h(t) .
t∈[tn+1 ,tn ]
If we prove that
P
P(An ) is converging, we have from Borel-Cantelli lemma that for ω outside a set
n≥0
of measure 0, there exists n0 = n0 (ω) such that for any n ≥ n0
max
t∈[tn+1 ,tn ]
Bt (ω) < (1 + δ)h(t).
This gives that
max lim
t→0+
Bt (ω)
≤ (1 + δ)
h(t)
and from the arbitrariety of δ it follows (3.8).
To estimate the probability of An we notice the following inclusion
An ⊂ max Bt > (1 + δ)h(tn+1 ) .
t∈[0,tn ]
Using (3.7) we have the following estimate on the probability of this event:
Bt
(1 + δ)h(tn+1 )
√
P({ max Bt > (1 + δ)h(tn+1 )}) ≤ 2P(Btn > (1 + δ)h(tn+1 )) ≤ 2P( √ n >
)
t∈[0,tn ]
tn
tn
the random variable on the left hand side has a Gaussian standard distribution, so we can use (3.6):
)
√ n+1 and get
set for simplicity xn = (1+δ)h(t
tn
P({ max Bt > (1 + δ)h(tn+1 )}) ≤
t∈[0,tn ]
p
2/π
1 −x2n /2
e
.
xn
42
3 Sample paths of the Brownian motion
Compute x2n : using the above notation we have xn = 2λ log[(n + 1) log(1/θ)] hence
P(An ) ≤ C
1
.
(n + 1)λ
We prove next the converse inequality. The proof is based on an application of the second part of
the Borel-Cantelli lemma. With the same notation of the first part, choose first ε and θ ∈ (0, 1); we
define the events
A0n = {Btn − Btn+1 ≥ (1 − ε)h(tn )}.
P
Let us prove that n P(A0n ) diverges. The idea is to use the left inequality of (3.6), which yields, for
any x, the estimate
Bt − Btn+1
1 1 −x2 /2
P( n
> x) ≥ √
e
tn − tn+1
2 2π x
n)
so, taking x = xn = (1 − ε) √th(t
, we have
n −tn+1
P(
Btn − Btn+1
1
p
.
> x) ≥ C
2 /1−θ
(1−ε)
tn − tn+1
n
log(n)
If we take θ < 1 − (1 − ε)2 this term is the general term of a diverging series. Then by Borel-Cantelli
lemma we have
P(Btn − Btn+1 ≥ (1 − ε)h(tn ) infinitely often) = 1.
By (3.8) applied to the Brownian motion −Bt , t ≥ 0, we know that Btn+1 ≥ −(1 + ε)htn+1 definitely
as n → ∞. Hence we get for infinite indices n
ht
Btn ≥ (1 − ε)h(tn ) − (1 + ε)htn+1 = h(tn ) 1 − ε − (1 + ε) n+1
htn
√
√
h
= θ and choosing ε + (1 + ε) θ < δ we have
Using the limit lim hθn+1
θn
n→∞
Btn ≥ (1 − δ)h(tn ) infinitely often.
2
3.2.1 Modulus of continuity
We have seen that Brownian sample paths are α-H¨
older continuous for every α < 12 and, by Corollary
3.9, they are nowhere α-H¨
older continuous for every α > 12 . This section will treat the limit case
1
α = 2.
The modulus of continuity of a real function f : I → R, where I ⊂ R is a given interval, is defined
as the function
w(δ) =
sup
|f (x) − f (y)|.
x,y∈I, |x−y|≤δ
Of course, the modulus of continuity of the Brownian sample paths is bounded above by Cδ α for any
older continuous for such α. On the other hand, it shall
α < 21 , since the trajectories are nowehere α-H¨
p
be larger that log log(1/δ) by the law of the iterated logarithm. The exact modulus of continuity
for the Brownian sample paths is determined from the following theorem. We omit its proof but refer
the interested reader, for instance, to [KS88, Theorem 2.9.5].
3.3 The canonical space
Theorem 3.13 (L´
evy). For every T > 0, on the interval I = [0, T ] we have
(
)
|Bt − Bs |
P lim+
sup
= 1 = 1.
δ→0 s<t∈I,|t−s|≤δ (2δ log(1/δ))1/2
43
(3.9)
This result proves that, letting w(·, ω) be the modulus of continuity of Bt (ω) for t ∈ [0, T ], then it
follows (almost surely) that
w(δ, ω)
= 1.
lim
δ→0+ (2δ log(1/δ))1/2
Therefore, every rajectory cannot be H¨
older continuous of exponent 1/2 on the time interval [0, T ],
for every T . It also holds the following
Corollary 3.14. There exists an event N of zero probability such that each Brownian sample path
outsde this set is never H¨
older continuous of exponent α ≥ 1/2 in any time interval I ⊂ R + having
non empty interior.
Proof. Taking J = [q, r], for q < r ∈ Q+ ,
lim
sup
δ→0+ s<t∈J,|t−s|≤δ
|Bt − Bs |
= 1 a.s.
(2δ log(1/δ))1/2
Therefore, if Nq,r is the negligible set where (3.10) is not verified, we obtain that N =
(3.10)
∪
q<r∈Q+
Nq,r is
again an event having probability zero.
Now, the thesis follows: if ω outside N is such that the Brownian path t 7→ Bt (ω) is H¨
older
continuous with exponent α ≥ 21 in an interval I with non empty internal part, there exists an interval
[q, r] contained in I such that the Brownian path is H¨
older continuous with exponent α ≥ 12 in [q, r],
and we have a contraddiction.
2
3.3 The canonical space
Let us consider a real, standard Brownian motion {Bt , t ∈ [0, 1]}, defined on the space (Ω, F, {Ft }, P).
Having chosen a finite interval helps to simplify some technical details and may be removed if necessary.
Without loss of generality, we can assume that {Bt } has continuous paths (by taking a modification of
the process, if necessary). We consider the mapping ξ : ω → {t 7→ Bt (ω)} which associates to each ω
the trajectory, as an element in the space C = C0 ([0, 1]) of real valued, continuous functions vanishing
at 0. C, endowed with the distance
d(ω, η) = sup |ω(t) − η(t)|
t∈[0,1]
is a complete, separable metric space. Let G 0 be the Borel σ-field generated by the open sets of C.
Define on the space C, the coordinate functions (or projections) X(t), X(t)(ω) = ω(t), for every
t ∈ [0, 1] and ω ∈ C. We consider on C also the σ-algebra G generated by the finite-dimensional sets
44
3 Sample paths of the Brownian motion
{ω ∈ C : ω(t1 ) ∈ A1 , . . . , ω(tn ) ∈ An },
with n ∈ N, t1 , . . . , tn ∈ [0, 1], Ai ∈ B(R), i = 1, . . . , n. It has the property that all the coordinate
functions X(t) are G-measurable, and G is the smallest σ-field having this property.
Lemma 3.15. G = G 0 .
Proof. The inclusion G ⊂ G 0 is obvious; let us check the other one. The space C is separable, hence
d(ω, η) =
sup
t∈Q∩[0,1]
|ω(t) − η(t)|;
this implies that the balls U = {ω ∈ C : |ωt − γt | ≤ ε ∀ t ∈ [0, 1]} are in G. But every open set A ∈ G 0
can be written as a countable union of balls, hence it is in G.
2
As a corollary we have the following result, whose prove we let to the reader.
Problem 3.4. The mapping ξ : (Ω, F) → (C, G) is measurable, i.e., ξ is a random variable taking
values in C.
˜ = P ◦ B −1 of
Thanks to the above result, on the space (C, G) we can define the image measure P
P under {Bt , t ≥ 0}, i.e.,
˜
P(G)
= P(ω ∈ Ω : t 7→ Bt (ω) ∈ G),
G ∈ G.
˜ is called the Wiener measure. Define, on the probability space (C, G, P), the coordinate mapping
P
process X(t)(ω) = ω(t) for every t ∈ [0, 1] and ω ∈ C. X is equivalent to B and is often called the
canonical Brownian motion.
Addendum. A second construction of Brownian motion
Here we propose an explicit way to construct a real Brownian motion. This construction goes back to
to L´evy and Ciesielski (see, for instance, Karatzas and Shreve [KS88] or Zabczyk [Za04]). It does
not require to appeal to the Kolmogorov existence theorem.
We proceed in some steps and ask to complete the details as exercise.
I step. We first define the Haar functions hk : [0, 1] → R, k ∈ N,
h0 (t) = 1, t ∈ [0, 1],
(
1
for t ∈ [0, 1/2],
h1 (t) =
−1 for t ∈ (1/2, 1]
and, if n ∈ N and 2n ≤ k < 2n+1 ,
hk (t) =

n/2


2
−2



0
n/2
for t ∈
for t ∈
h
i
n
k−2n k−2 +1/2
,
n ,
n
2
2
i
k−2n +1/2 k−2n +1
, 2n
,
2n
otherwise.
3.3 The canonical space
45
The sequence {hk } forms an orthonormal and complete system in L2 (0, 1).
II step. Define the Schauder functions sk : [0, 1] → R, k ∈ N,
Z t
sk (t) =
hk (s)ds,
k ∈ N.
0
Note that all sk are non-negative; moreover we have
ksk k∞ = sup |sk (t)| = 2n/2
t∈[0,1]
1
= 2−n/2−1 , 2n ≤ k < 2n+1 .
2n+1
(3.11)
III step. Let us consider a probability space (Ω, F, P) on which it is well defined a sequence {X n , n ∈
N} of independent real Gaussian random variables such that the law of each Xn is N (0, 1).
Q
For instance, we can take Ω =
Ri the product of infinite copies of the real line Ri = R, i ∈ N,
i∈N
endowed with the product σ-algebra F = B(R) ⊗ . . . ⊗ B(R) × . . .. Moreover, we take as measure P the
infinite product of Gaussian laws N (0, 1). We define Xn (ω) = ωn , for any ω ∈ Ω, n ∈ N. It is clear
that (Ω, F, P, {Xn }) satisfies our assertion.
IV step. Consider the previous sequence of independent Gaussian random variables {X n }. One has
that
p
|Xk (ω)| is O( log k) for k → ∞, a.s.
√
Indeed, we have P(|Xk | > 4 log k) ≤ ce−4 log k = kc4 , k ∈ N, and, applying the
√ Borel-Cantelli lemma,
for any ω a.s., there exists k0 = k0 (ω) such that, if k ≥ k0 then |Xk (ω)| ≤ 4 log k.
V step. Define
Bt (ω) =
X
Xk (ω)sk (t),
k≥0
ω ∈ Ω, t ∈ [0, 1].
One verifies that {Bt , t ∈ [0, 1]} is a Brownian motion (note that B0 (ω) = 0 for any ω).
VI step. We extend the previous definition of Bt when t ∈ [0, ∞).
Consider a sequence of infinite copies of the same probability space:
(Ωk , Fk , Pk ) = (Ω, F, P),
k≥0
where (Ω, F, P) is the probability space on which is defined Bt , t ∈ [0, 1]. Let us introduce the product
probability space
Y
O
O
Ω˙ =
Ωk ,
F˙ =
Fk ,
P˙ =
Pk .
k≥0
Let us denote by
Btk
k≥0
k≥0
the Brownian motion on (Ωk , Fk , Pk ), t ∈ [0, 1]. Define in a recursive way:
(
B˙ t = Bt1 ,
t ∈ [0, 1],
n+1
˙
˙
Bt = Bn + Bt−n , t ∈ [n, n + 1], n ≥ 1.
46
3 Sample paths of the Brownian motion
˙
Hence, for ω = (ωk ) ∈ Ω,
B˙ t (ω) = Bt1 (ω1 ),
t ∈ [0, 1],
1
2
B˙ t (ω) = B (ω1 ) + B (ω2 ),
1
t−1
...
B˙ t (ω) =
n
X
t ∈ [1, 2],
n+1
B1k (ωk ) + Bt−n
(ωn+1 ),
k=1
t ∈ [n, n + 1].
˙
˙ F,
˙ P).
One checks that {B˙ t , t ≥ 0} is a real Brownian motion on (Ω,
Remark 3.16. Let {Bt , t ≥ 0} be a real standard Brownian motion defined on the probability space
(Ω, F, P). The above construction suggests that, in order to have a n-dimensional Brownian motion,
it is possible to proceed similarly.
Consider a sequence of copies of the same probability space:
(Ωk , Fk , Pk ) = (Ω, F, P),
k = 1, . . . , n
where (Ω, F, P) is the probability space on which is defined {Bt , t ≥ 0}.
On the product probability space
Ω˙ =
n
Y
k=1
Ωk ,
F˙ =
we define
Wt = (Bt (ω1 ), . . . , Bt (ωn )),
n
O
k=1
Fk ,
P˙ =
n
O
Pk
k=1
˙
ω = (ω1 , . . . , ωn ) ∈ Ω,
t ≥ 0.
Then it is an easy computation to verify that {Wt , t ≥ 0} is an n-dimensional Brownian motion
according to Definition 2.5.
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