Jeanblanc M., Yor M., Chesney M. Mathematical Methods for Financial Markets
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4.1 Local Time 217
h
g
t
Y
t
= h
0
Y
0
+
t
0
h
g
s
dY
s
for every predictable, locally bounded process h.
Proof: By the MCT, it is enough to show this formula for processes of the
form h
u
= 1
[0,τ]
(u), where τ is a stopping time. In this case,
h
g
t
= 1
{g
t
≤τ}
= 1
{t≤d
τ
}
where d
τ
=inf{s ≥ τ : Y
s
=0}.
Hence,
h
g
t
Y
t
= 1
{t≤d
τ
}
Y
t
= Y
t∧d
τ
= Y
0
+
t
0
1
{s≤d
τ
}
dY
s
= h
0
Y
0
+
t
0
h
g
s
dY
s
.
Let Y
t
= B
t
, then from the balayage formula we obtain that
h
g
t
B
t
=
t
0
h
g
s
dB
s
is a local martingale with increasing process
t
0
h
2
g
s
ds.
Exercise 4.1.6.2 Let ϕ : R
+
→ R be a locally bounded real-valued
function, and L the local time of the Brownian motion at level 0. Prove that
(ϕ(L
t
)B
t
,t≥ 0) is a Brownian motion time changed by
t
0
ϕ
2
(L
s
)ds.
Hint: Note that for h
s
= ϕ(L
s
), one has h
s
= h
g
s
, then use the balayage
formula. Note also that one could prove the result first for ϕ ∈ C
1
and then
pass to the limit.
4.1.7 Skorokhod’s Reflection Lemma
The following real variable lemma will allow us in particular to view local
times as supremum processes.
Lemma 4.1.7.1 Let y be a continuous function. There is a unique pair of
functions (z,k) such that
(i) k(0) = 0, k is an increasing continuous function
(ii) z(t)=−y(t)+k(t) ≥ 0
(iii)
t
0
1
{z(s)>0}
dk(s)=0,
This pair is given by
k
∗
(t)= sup
0<s≤t
(y(s)) ∨ 0,z
∗
(t)=−y(t)+k
∗
(t).
218 4 Complements on Brownian Motion
Proof: The pair k
∗
(t)=sup
0<s≤t
(y(s))∨0,z
∗
(t)=−y(t)+k
∗
(t) satisfies the
required properties. Let us prove that the solution is unique. Let (z
1
,k
1
)and
(z
2
,k
2
) be two pairs of solutions. Then, since z
1
− z
2
has bounded variation,
from the integration by parts formula,
0 ≤ (z
1
− z
2
)
2
(t)=2
t
0
(z
1
(s) − z
2
(s)) d(k
1
(s) − k
2
(s)) .
From (iii), the right-hand side of the above equality is equal to
−2
t
0
z
2
(s) dk
1
(s) − 2
t
0
z
1
(s) dk
2
(s)
which is negative. Hence, z
1
= z
2
.
Note that, if y is increasing, then z = 0. We now give an important
consequence of the Skorokhod lemma:
Theorem 4.1.7.2 (L´evy’s Equivalence Theorem.) Let B be a Brownian
motion starting at 0, L its local time at level 0 and M
t
=sup
s≤t
B
s
. The two-
dimensional processes (|B|,L) and (M −B, M) have the same law, i.e.,
(|B
t
|,L
t
; t ≥ 0)
law
=(M
t
− B
t
,M
t
; t ≥ 0) .
Proof: Tanaka’s formula implies that
|B
t
| =
t
0
sgn(B
s
)dB
s
+ L
0
t
where L
0
,thelocaltimeofB, is an increasing process. Therefore, (|B|,L
0
)
is a solution of Skorokhod’s lemma associated with the Brownian motion
β
t
= −
t
0
sgn(B
s
)dB
s
. Hence, L
0
t
=sup
s≤t
β
s
. By denoting M
t
=sup
s≤t
B
s
,
we obtain the decompositions
|B
t
| = −β
t
+ L
0
t
M
t
− B
t
=(−B
t
)+M
t
.
The pair (M −B,M) is a solution to Skorokhod’s lemma associated with the
Brownian motion B, because M increases only on the set M −B = 0. Hence,
the processes (|B|,L)and(M −B,M) have the same law.
Comments 4.1.7.3 (a) We have proved, in Proposition 3.1.3.1 that, for any
fixed t, M
t
law
= |B
t
|. Here, we obtain that the processes M −B and |B| have the
same law. In particular, for fixed t, M
t
−B
t
law
= |B
t
|.WealsohaveM
t
law
= L
t
.
4.1 Local Time 219
(b) As a consequence of Skorokhod’s lemma, if β
t
=
t
0
sgn(B
s
)dB
s
,then
it is easily shown that σ(β
s
,s ≤ t)=σ(|B
s
|,s ≤ t). See Subsection 5.8.2
for comments. This may be contrasted with the equality obtained in 3.1.4.3:
σ(M
s
− B
s
,s≤ t)=σ(B
s
,s≤ t) .
(c) There are various identities in law involving the BM and its maximum
process. From L´evy’s theorem, one obtains that
(|B
t
| + L
t
; t ≥ 0)
law
=(2M
t
− B
t
; t ≥ 0) .
From Exercise 4.1.7.12, we obtain that, for every t,2M
t
−B
t
law
= R
t
where
R is a BES
3
process (see Chapter 6 if needed). Pitman [712] has extended
this result at the level of processes, proving that
(2M
t
− B
t
,M
t
; t ≥ 0)
law
=(R
t
,J
t
; t ≥ 0)
where R is a BES
3
process and J
t
=inf
s≥t
R
s
(see Section 5.7). Hence, it
also holds that
(|B
t
| + L
t
,L
t
; t ≥ 0)
law
=(R
t
,J
t
; t ≥ 0) .
We now present further consequences of L´evy’s theorem:
Example 4.1.7.4 Let (τ
, ≥ 0) be the inverse of the local time (L
0
t
,t ≥ 0)
defined as τ
=inf{t : L
0
t
>},andletT
x
be the first hitting time of x.Then
(T
x
,x ≥ 0)
law
=(τ
x
,x ≥ 0). Indeed, from L´evy’s equivalence Theorem 4.1.7.2
(M
t
,t ≥ 0)
law
=(L
t
,t ≥ 0). Hence the same equality holds for the inverse
processes. As a consequence, we note that
(L
x
t
,t≥ 0)
law
=
(L
0
t
−|x|)
+
,t≥ 0
.
Indeed, on the one hand
(L
x
t
,t≥ 0) = (L
x
T
x
+(t−T
x
)
+
,t≥ 0)
law
=(L
0
(t−T
x
)
+
,t≥ 0)
where L
0
and T
x
are independent. On the other hand
(L
0
τ
+(t−τ
)
+
− )
+
,t≥ 0
law
=(L
0
(t−bτ
)
+
,t≥ 0)
where τ
is independent of (L
0
t
,t≥ 0).Toconclude,weuseτ
law
= T
, and
take = |x|.
220 4 Complements on Brownian Motion
Example 4.1.7.5 For fixed t,letθ
t
be the first time at which the Brownian
motion reaches its maximum over the time interval [0,t]:
θ
t
:=inf{s ≤ t |B
s
= M
t
} =inf{s ≤ t |B
s
=sup
u≤t
B
u
}.
If t = 1, we obtain
(θ
1
≤ u)=
sup
u≤s≤1
B
s
≤ sup
s≤u
B
s
=
sup
u≤s≤1
(B
s
− B
u
)+B
u
≤ sup
s≤u
B
s
=
sup
0≤v≤1−u
B
v
+ B
u
≤ M
u
,
where
B is a BM independent of (B
s
,s≤ u). Setting
M
u
=sup
s≤u
B
s
,weget
from L´evy’s Theorem 4.1.7.2 and Proposition 3.1.3.1
P(θ
1
≤ u)=P(
M
1−u
≤ M
u
− B
u
)=P(|
B
1−u
|≤|B
u
|)
= P(
√
1 − u|
B
1
|≤
√
u|B
1
|)=P
|B
1
|
|
B
1
|
≥
√
1 − u
√
u
= P
C
2
≥
1 − u
u
= P
u ≥
1
1+C
2
where C follows the standard Cauchy law (see Appendix A.4.2). Hence,
for u ≤ 1,
P(θ
1
≤ u)=
2
π
arc sin
√
u.
Finally, by scaling, for s ≤ t,
P(θ
t
≤ s)=
2
π
arc sin
s
t
,
therefore, θ
t
is Arcsine distributed on [0,t]. Note the non-trivial identity in
law θ
t
law
= A
+
t
where A
+
t
=
t
0
1
{B
s
>0}
ds (see Subsection 2.5.2). As a direct
application of L´evy’s equivalence theorem, we obtain
θ
t
law
= g
t
=sup{s ≤ t : B
s
=0}.
Proceeding along the same lines, we obtain the equality
P(M
t
∈ dx, θ
t
∈ du)=
x
πu
u(t − u)
exp
−
x
2
2u
1
{0≤x,0≤u≤t}
du dx
(4.1.10)
and from the previous equalities and using the Markov property
P(θ
1
≤ u|F
u
)=P(sup
u≤s≤1
(B
s
− B
u
)+B
u
≤ sup
s≤u
B
s
|F
u
)
= P(
M
1−u
≤ M
u
− B
u
|F
u
)=Ψ(1 − u, M
u
− B
u
) .
4.1 Local Time 221
Here,
Ψ(u, x)=P(
M
u
≤ x)=P(|B
u
|≤x)=
2
√
2π
x/
√
u
0
exp
−
y
2
2
dy .
Note that, for x>0, the density of M
t
at x can also be obtained from the
equality (4.1.10). Hence, we have the equality
t
0
du
x
π
u
3
(t − u)
exp
−
x
2
2u
=
2
πt
e
−x
2
/(2t)
. (4.1.11)
We also deduce from L´evy’s theorem that the right-hand side of (4.1.10)is
equal to P(L
t
∈ dx, g
t
∈ du).
Example 4.1.7.6 From L´evy’s identity, it is straightforward to obtain that
P(L
a
∞
= ∞)=1.
Example 4.1.7.7 As discussed in Pitman [713], the law of the pair (L
x
1
,B
1
)
may be obtained from L´evy’s identity: for y>0,
P(L
x
1
∈ dy, B
1
∈ db)=
|x| + y + |b − x|
√
2π
exp
−
1
2
(|x| + y + |b − x|)
2
dydb .
Proposition 4.1.7.8 Let ϕ be a C
1
function. Then, the process
ϕ(M
t
) − (M
t
− B
t
)ϕ
(M
t
)
is a local martingale.
Proof: As a first step we assume that ϕ is C
2
. Then, from integration by
parts and using the fact that M is increasing
(M
t
− B
t
)ϕ
(M
t
)=
t
0
ϕ
(M
s
) d(M
s
− B
s
)+
t
0
(M
s
− B
s
)ϕ
(M
s
)dM
s
.
Now, we note that
t
0
(M
s
− B
s
)ϕ
(M
s
)dM
s
=0,sincedM
s
is carried by
{s : M
s
= B
s
},andthat
t
0
ϕ
(M
s
)dM
s
= ϕ(M
t
) − ϕ(0). The result follows.
The general case is obtained using the MCT.
Comment 4.1.7.9 As we mentioned in Example 1.5.4.5, any solution of
Tanaka’s SDE X
t
= X
0
+
t
0
sgn(X
s
)dB
s
is a Brownian motion. We can check
that there are indeed weak solutions to this equation: start with a Brownian
motion X and construct the BM B
t
=
t
0
sgn(X
s
)dX
s
. This Brownian motion
is equal to |X|−L,soB is adapted to the filtration generated by |X| which
is strictly smaller than the filtration generated by X. Hence, the equation
X
t
= X
0
+
t
0
sgn(X
s
)dB
s
has no strong solution. Moreover, one can find
infinitely many solutions, e.g.,
g
t
X
t
,where is a ±1-valued predictable
process, and g
t
=sup{s ≤ t : X
s
=0}.
222 4 Complements on Brownian Motion
Exercise 4.1.7.10 Prove Proposition 4.1.7.8 as a consequence of the bal-
ayage formula applied to Y
t
= M
t
− B
t
.
Exercise 4.1.7.11 Using the balayage formula, extend the result of Propo-
sition 4.1.7.8 when ϕ
is replaced by a bounded Borel function.
Exercise 4.1.7.12 Prove, using Theorem 3.1.1.2, that the joint law of the
pair (|B
t
|,L
0
t
)is
P(|B
t
|∈dx, L
0
t
∈ d)=1
{x≥0}
1
{≥0}
2(x + )
√
2πt
3
exp
−
(x + )
2
2t
dx d .
Exercise 4.1.7.13 Let ϕ be in C
1
b
.Provethat(ϕ(L
0
t
) −|B
t
|ϕ
(L
0
t
),t ≥ 0)
is a martingale. Let T
∗
a
=inf{t ≥ 0:|B
t
| = a}.ProvethatL
0
T
∗
a
follows the
exponential law with parameter 1/a.
Hint: Use Proposition 4.1.7.8 together with L´evy’s Theorem. Then, compute
the Laplace transform of L
0
T
∗
a
by means of the optional stopping theorem.
The second part may also be obtained as a particular case of Az´ema’s
lemma 5.2.2.5.
Exercise 4.1.7.14 Let y be a continuous positive function vanishing at 0:
y(0) = 0. Prove that there exists a unique pair of functions (z, k) such that
(i) k(0) = 0, where k is an increasing continuous function
(ii) z(t)+k(t)=y(t),z(t) ≥ 0
(iii)
t
0
1
{z(s)>0}
dk(s)=0
(iv) ∀t, ∃d(t) ≥ t, z(d(t)) = 0
Hint: k
∗
(t)=inf
s≥t
(y(s)).
Exercise 4.1.7.15 Let S be a price process, assumed to be a continuous
local martingale, and ϕ a C
1
concave, increasing function. Denote by S
∗
the
running maximum of S. Prove that the process X
t
= ϕ(S
∗
t
)+ϕ
(S
∗
t
)(S
t
−S
∗
t
)
is the value of the self-financing strategy with a risky investment given by
S
t
ϕ
(S
∗
t
), which satisfies the floor constraint X
t
≥ ϕ(S
t
).
Hint: Using an extension of Proposition 4.1.7.8, X is a local martingale. It
is easy to check that X
t
= X
0
+
t
0
ϕ
(S
∗
s
)dS
s
. For an intensive study of this
process in finance, see El Karoui and Meziou [305]. The equality X
t
≥ ϕ(S
t
)
follows from concavity of ϕ.
4.1.8 Local Time of a Semi-martingale
As mentioned above, local times can also be defined in greater generality for
semi-martingales. The same approach as the one used in Subsection 4.1.2 leads
to the following:
4.1 Local Time 223
Theorem 4.1.8.1 (Occupation Time Formula.) Let X be a continuous
semi-martingale. There exists a family of increasing processes (Tanaka-
Meyer local times) (L
x
t
(X),t≥ 0;x ∈ R) such that for every bounded
measurable function ϕ
t
0
ϕ(X
s
) dX
s
=
+∞
−∞
L
x
t
(X)ϕ(x) dx . (4.1.12)
There is a version of L
x
t
which is jointly continuous in t and right-continuous
with left limits in x.(IfX is a continuous martingale, its local time may be
chosen jointly continuous.) In the sequel, we always choose this version. This
local time satisfies
L
x
t
(X) = lim
→0
1
t
0
1
[x,x+[
(X
s
)dX
s
.
If Z is a continuous local martingale,
L
x
t
(Z) = lim
→0
1
2
t
0
1
]x−,x+[
(Z
s
)dZ
s
.
The same result holds with any random time in place of t.
For a continuous semi-martingale X = Z + A,
L
x
t
(X) − L
x−
t
(X)=2
t
0
1
{X
s
=x}
dX
s
=2
t
0
1
{X
s
=x}
dA
s
. (4.1.13)
In particular,
L
0
t
(|X|) = lim
→0
1
t
0
1
]−,[
(X
s
)dX
s
= L
0
t
(X)+L
0−
t
(X),
hence
L
0
t
(|X|)=2L
0
t
(X) − 2
t
0
1
{X
s
=0}
dA
s
.
Example 4.1.8.2 A Non-Continuous Local Time. Let Z be a continuous
martingale and X be the semi-martingale
X
t
= aZ
+
t
− bZ
−
t
=
t
0
dZ
s
(a1
{Z
s
>0}
+ b1
{Z
s
<0}
)+
a − b
2
L
0
t
(Z) .
Then, it follows from (4.1.13)thatL
0
t
(X) − L
0−
t
(X)=(a − b)L
0
t
(Z). In
particular, for the reflected BM, i.e., for X when Z
t
= B
t
,a =1,b = −1,
we get L
0
(|B|) − L
0−
(|B|)=2L
0
(B). Note that L
0−
(|B|) = 0, hence
L
0
(|B|)=2L
0
(B).
224 4 Complements on Brownian Motion
Example 4.1.8.3 Let Y
t
= |B
t
|. Tanaka’s formula gives:
|B
t
| =
t
0
sgn(B
s
)dB
s
+ L
t
where (L
t
,t ≥ 0) denotes the local time of (B
t
; t ≥ 0) at y =0. By an
application of the balayage formula, we obtain
h
g
t
|B
t
| =
t
0
h
g
s
sgn(B
s
)dB
s
+
t
0
h
s
dL
s
having used the fact that L
g
s
= L
s
. Consequently, replacing, if necessary, h by
|h|, we see that the process
t
0
|h
s
|dL
s
is the local time at 0 of (h
g
t
B
t
,t≥ 0).
Tanaka-Meyer Formulae
As before we set sgn(x)=1forx>0 and sgn(x)=−1forx ≤ 0. Let X be
a continuous semi-martingale. For every (t, x),
|X
t
− x| = |X
0
− x| +
t
0
sgn (X
s
− x) dX
s
+ L
x
t
(X) ,
(4.1.14)
(X
t
− x)
+
=(X
0
− x)
+
+
t
0
1
{X
s
>x}
dX
s
+
1
2
L
x
t
(X) ,
(4.1.15)
(X
t
− x)
−
=(X
0
− x)
−
−
t
0
1
{X
s
≤x}
dX
s
+
1
2
L
x
t
(X) .
(4.1.16)
In particular, |X − x|, (X − x)
+
and (X − x)
−
are semi-martingales.
Proposition 4.1.8.4 (L´evy’s Equivalence Theorem for Drifted Brow-
nian Motion.) Let B
(ν)
be a BM with drift ν, i.e., B
(ν)
t
= B
t
+ νt,and
M
(ν)
t
=sup
s≤t
B
(ν)
s
.Then
(M
(ν)
t
− B
(ν)
t
,M
(ν)
t
; t ≥ 0)
law
=(|X
(ν)
t
|,L
t
(X
(ν)
); t ≥ 0) (4.1.17)
where X
(ν)
is the (unique) strong solution of
dX
t
= dB
t
− ν sgn(X
t
) dt, X
0
=0.
4.1 Local Time 225
Proof: Let X
(ν)
be the strong solution of
dX
t
= dB
t
− ν sgn(X
t
) dt, X
0
=0
(see Theorem 1.5.5.1 for the existence of X) and apply Tanaka’s formula.
Then,
|X
(ν)
t
| =
t
0
sgn (X
(ν)
s
)
dB
s
− ν sgn (X
(ν)
s
) ds
+ L
0
t
(X
(ν)
)
where L
0
(X
(ν)
) is the Tanaka-Meyer local time of X
(ν)
at level 0. Hence,
setting β
t
=
t
0
sgn X
(ν)
s
dB
s
,
|X
(ν)
t
| =(β
t
− νt)+L
0
t
(X
(ν)
)
and the result follows from Skorokhod’s lemma.
Comments 4.1.8.5 (a) Note that the processes |B
(ν)
| and M
(ν)
− B
(ν)
do
not have the same law (hence the right-hand side of (4.1.17) cannot be replaced
by (|B
(ν)
t
|,L
t
(B
(ν)
),t≥ 0)). Indeed, for ν>0, B
(ν)
t
goes to infinity as t goes
to ∞,whereasM
(ν)
t
− B
(ν)
t
vanishes for some arbitrarily large values of t.
Pitman and Rogers [714] extended the result of Pitman [712] and proved that
(|B
(ν)
t
| + L
(ν)
t
,t≥ 0)
law
=(2M
(ν)
t
− B
(ν)
t
,t≥ 0) .
(b) The equality in law of Proposition 4.1.8.4 admits an extension to the
case dB
(a)
t
= a
t
(B
(a)
t
)dt + dB
t
and X
(a)
the unique weak solution of
dX
(a)
t
= dB
t
− a
t
(X
(a)
t
)sgn(X
(a)
t
)dt, X
(a)
0
=0
where a
t
(x) is a bounded predictable family. The equality
(M
(a)
− B
(a)
,M
(a)
)
law
=(|X
(a)
|,L(X
(a)
))
is proved in Shiryaev and Cherny [792].
We discuss here the Itˆo-Tanaka formula for strict local continuous
martingales, as it is given in Madan and Yor [614].
Theorem 4.1.8.6 Let S be a positive continuous strict local martingale, τ
an F
S
-stopping time, a.s. finite, and K a positive real number. Then
E((S
τ
− K)
+
)=(S
0
− K)
+
+
1
2
E(L
K
τ
) − E(S
0
− S
τ
)
where L
K
is the local time of S at level K.
226 4 Complements on Brownian Motion
Proof: We prove that
M
t
=
1
2
L
K
t
− (S
t
− K)
+
+ S
t
=
1
2
L
K
t
+(S
t
∧ K)
is a uniformly integrable martingale. In a first step, from Tanaka’s formula, M
is a (positive) local martingale, hence a super-martingale and E(L
K
t
) ≤ 2S
0
.
Since L is an increasing process, it follows that E(L
K
∞
) ≤ 2S
0
and the process
M is a uniformly integrable martingale. We then apply the optimal stopping
theoremattimeτ.
Comment 4.1.8.7 It is important to see that, if the discounted price process
is a martingale under the e.m.m., then the put-call parity holds: indeed, taking
expectation of discounted values of (S
T
−K)
+
= S
T
−K +(K −S
T
)
+
leads
to C(x, T )=x − Ke
−rT
+ P (x, T). This is no more the case if discounted
prices are strict local martingales. See Madan and Yor [614], Cox and Hobson
[203], Pal and Protter [692].
4.1.9 Generalized Itˆo-Tanaka Formula
Theorem 4.1.9.1 Let X be a continuous semi-martingale, f a convex
function, D
−
f its left derivative and f
(dx) its second derivative in the
distribution sense. Then,
f(X
t
)=f(X
0
)+
t
0
D
−
f(X
s
)dX
s
+
1
2
R
L
x
t
(X)f
(dx)
holds.
Corollary 4.1.9.2 Let X be a continuous semi-martingale, f a C
1
function
and assume that there exists a measurable function h, integrable on any finite
interval [−a, a] such that f
(y) − f
(x)=
y
x
h(z)dz.Then,Itˆo’s formula
f(X
t
)=f(X
0
)+
t
0
f
(X
s
)dX
s
+
1
2
t
0
h(X
s
)dX
s
holds.
Proof: In this case, f is locally the difference of two convex functions and
f
(dx)=h(x)dx. Indeed, for every ϕ ∈ C
∞
b
,
f
,ϕ = −f
,ϕ
= −
dxf
(x)ϕ
(x)=
dzh(z)ϕ(z) .
In particular, if f is a C
1
function, which is C
2
on R\{a
1
,...,a
n
},forafinite
number of points (a
i
), then