Theorem 4.15. Let W be a Brownian motion. Then the augmented filtration F
W is continuous and W is an FW−Brownian motion.
Proof. The left-continuity of FW is a direct consequence of the path continuity of W . The inclusion FW0 ⊂ FW0+ is trivial and, by Theorem 4.14, we have F0+W ⊂ σ(N (F )) ⊂ FW0 . Similarly, by the independent increments property, Ft+W =FWt for all t ≥ 0. Finally, W is an FW−Brownian motion as it satisfies
all the required properties of Definition 4.1. ♦
In the rest of these notes, we will always work with the augmented filtration, and we still denote it as FW := FW.
4.6 Small/large time behavior of the Brownian sample paths
The discrete-time approximation of the Brownian motion suggests thatWtt tends to zero at least along natural numbers, by the law of large numbers. With a little effort, we obtain the following strong law of large numbers for the Brownian motion.
Theorem 4.16. For a Brownian motion W , we have Wt
t −→ 0 P − a.s. as t → ∞ . Proof. We first decompose
Wt
t = Wt− Wbtc
t +btc
t Wbtc
btc By the law of large numbers, we have
Wbtc btc = 1
btc
btc
X
i=1
(Wi− Wi−1) −→ 0 P − a.s.
4.6. Asymptotics of Brownian paths 49 We next estimate that
Wt− Wbtc
t ≤ btc
t ∆btc
btc , where ∆n:= sup
n−1<t≤n
(Wt− Wn−1) , n ≥ 1 .
Clearly, {∆n, n ≥ 1} is a sequence of independent identically distributed random variables. The distribution of ∆n is explicitly given by Proposition 4.12. In particular, by a direct application of the Chebychev inequality, it is easily seen that P
n≥1P[∆n ≥ nε] = P
n≥1P[∆1 ≥ nε] < ∞. By the Borel Cantelli Theorem, this implies that ∆n/n −→ 0 P−a.s.
Wt− Wbtc
btc −→ 0 P − a.s. as t → ∞ ,
and the required result follows from the fact that t/btc −→ 1 as t → ∞. ♦ As an immediate consequence of the law of large numbers for the Brownian motion, we obtain the invariance property of the Brownian motion by time inversion:
Proposition 4.17. Let W be a standard Brownian motion. Then the process B0 = 0 and Bt := tW1
t for t > 0 is a Brownian motion.
Proof. All of the properties (ii)-(iii’)-(iv) of the definition are obvious, except for the sample path continuity at zero. But this is equivalent to the law of large
numbers stated in Proposition 4.16. ♦
The following result shows the path irregularity of the Brownian motion.
Proposition 4.18. Let W be a Brownian motion in R. Then, P−a.s. W changes sign infinitely many times in any time interval [0, t], t > 0.
Proof. Observe that the random times
τ+:= inf {t > 0 : Wt> 0} and τ−:= inf {t > 0 : Wt< 0}
are stopping times with respect to the augmented filtration FW. Since this filtration is continuous, it follows that the event sets {τ+ = 0} and {τ− = 0}
are in F0W. By the symmetry of the Brownian motion, its non degeneracy on any interval [0, t], t > 0, and the fact that F0W is trivial, it follows that P [τ+= 0] = P [τ−= 0] = 1. Hence for a.e. ω ∈ Ω, there are sequences of random times τn+& 0 and τn−% 0 with Wτ+
n > 0 and Wτ−
n < 0 for n ≥ 1. ♦ We next state that the sample path of the Brownian motion is not bounded P−a.s.
Proposition 4.19. For a standard Brownian motion W , we have lim sup
t→∞
Wt = ∞ and lim inf
t→∞ Wt = −∞, P − a.s.
Proof. By symmetry of the Brownian motion, we only have to prove the limsup result.
Step 1 The invariance of the Brownian motion by time inversion of Proposition 4.17 implies that
lim sup
t→∞
Wt= lim sup
u→0
1
uBu where Bu:= uW1/u1{u6=0}
defines a Brownian motion. Then, it follows from the Zero-One law of Theo-rem 4.14 that C0 := lim supt→∞Wt is deterministic. By the symmetry of the Brownian motion, we see that C0∈ R+∪ {∞}.
By the translation invariance of the Brownian motion, we see that C0= lim sup
t→∞
(Wt− Ws) in distribution for every s ≥ 0.
Then, if C0< ∞, it follows that e−λC0 = Eh
e−λC0+λWsi
= e−λC0+λ2s/2
which can not happen. Hence C0= ∞. ♦
Another consequence is the following result which shows the complexity of the sample paths of the Brownian motion.
Proposition 4.20. For any t0≥ 0, we have lim inf
t&t0
Wt− Wt0
t − t0 = −∞ and lim sup
t&t0
Wt− Wt0
t − t0 = ∞ .
Proof. From the invariance of the Brownian motion by time translation, it is sufficient to consider t0 = 0. From Proposition 4.17, Bt := tW1/t defines a Brownian motion. Since Wt/t = B1/t, it follows that the behavior of Wt/t for t & 0 corresponds to the behavior of Bu for u % ∞. The required limit result
is then a restatement of Proposition 4.19. ♦
We conclude this section by the law of the iterated logarithm for the Brown-ian motion. This result will not be used in our applications to finance, we report here for completeness, and we organize its proof in the subsequent problem set.
Theorem 4.21. For a Brownian motion W , we have lim sup
t→0
Wt q
2t ln(ln1t)
= 1 and lim inf
t→0
Wt q
2t ln(ln1t)
= −1
4.6. Asymptotics of Brownian paths 51 In particular, this result shows that the Brownian motion is nowhere12−H¨older continuous, see Exercise 4.4.
Exercise 4.22. (Law of Iterated Logarithm)
Let W be a Brownian motion, and h(t) := 2t ln(ln(1/t)). We want to proove the Law of Iterated Logarithm :
lim sup
t&0
Wt
ph(t) = 1 a.s.
1. (a) For λ, T > 0 with 2λT < 1, prove that P
0≤t≤Tmax{Wt2− t} ≥ α ≤ e−λαEeλ(WT2−T ).
(b) For θ, η ∈ (0, 1), and λn:=2θn(1 + η)−1
, deduce that:
P
0≤t≤θmaxn{Wt2− t} ≥ (1 + η)2h(θn)
≤ e−1/2(1+η)(1+η−1)1/2|n ln θ|−(1+η).
(c) By the Borel Cantelli Lemma, justify that lim supt&0h(t)Wt2 ≤ (1+η)θ 2, P−a.s.
(d) Conclude that lim supt&0 √Wt
h(t) ≤ 1, P−a.s.
2. For θ ∈ (0, 1), consider the event sets An :=Wθn− Wθn+1 ≥√
1 − θp
h(θn) , n ≥ 1.
(a) Using the inequalityR∞
x e−u2/2du ≥ xe1+x−x2 /22 , show that for some con-stant C
P[An] ≥ C
npln(n) for n sufficiently large.
(b) By the Borel-Cantelli Lemma, deduce that Wθn−Wθn+1 ≥√
1 − θph(θn), P−a.s.
(c) Combining with question (1d), show that lim supt&0√Wt
h(t) ≥√ 1 − θ−
4θ, P−a.s.
(d) Deduce that lim supt&0 √Wt
h(t) = 1, P−a.s.
Solution of Exercise 4.22 1. We first show that
lim sup
t&0
Wt
ph(t) ≤ 1 a.s. (4.6)
Let T > 0 and λ > 0 be such that 2λT < 1. Notice that {Wt2− t, t ≤ T } is a martingale. Then {eλ(Wt2−t), t ≤ T } is a nonnegative submartingale, by the Jensen inequality. It follows from the Doob maximal inequality for submartin-gales that for all α ≥ 0,
k≥0k−(1+η) < ∞, it follows from the Borel-Cantelli lemma that, for almost all ω ∈ Ω, there exists a natural number Kθ,η(ω) such that for all
and the required result follows by letting θ tend to 1 and η to 0 along the rationals.
2. We now show the converse inequality. Let θ ∈ (0, 1) be fixed, and define:
An :=Wθn− Wθn+1 ≥√ 1 − θp
h(θn) , n ≥ 1.
Using the inequality R∞
x e−u2/2du ≥ xe1+x−x2 /22 with x = xn := the events An’s are independent, it follows from the Borel-Cantelli lemma that
4.7. Quadratic variation 53
We finally send θ & 0 along the rationals to conclude that lim supt&0√Wt
h(t)≥ 1. where we set tn0 := 0, and we define the discrete quadratic variation:
QVπtn(W ) :=X
As we shall see shortly, the Brownian motion has infinite total variation, see (4.9) below. In particular, this implies that classical integration theories are not suitable for the case of the Brownian motion. The key-idea in order to define an integration theory with respect to the Brownian motion is the following result which states that the quadratic variation defined as the L2−limit of (4.8) is finite.
Before stating the main result of this section, we observe that the quadratic variation (along any subdivision) of a continuously differentiable function f converges to zero. Indeed, P
ti≤t|f (ti+1) − f (ti)|2 ≤ kf0k2
L∞([0,t])
P
ti≤t|ti+1− ti|2 −→ 0. Because of the non-differentiability property stated in Proposition 4.20, this result does not hold for the Brownian motion.
Proposition 4.23. Let W be a standard Brownian motion in R, and (πn)n≥1
a partition as in (4.7). Then the quadratic variation of the Brownian motion is finite and given by:
hW it := L2− lim
n→∞QVπtn(W ) = t for all t ≥ 0.
Proof. We directly compute that:
E