Itô's integrated formula for strict local martingales with ... - HAL - INRIA

Itô’s integrated formula for strict local martingales with
jumps.
Oleksandr Chybiryakov ∗
Abstract
This note presents some properties of positive càdlàg local martingales which are not
martingales - strict local martingales - extending the results from [MY05] to local martingales
with jumps. Some new examples of strict local martingales are given. The construction
relies on absolute continuity relationships between Dunkl processes and absolute
continuity relationships between semi-stable Markov processes.
Key Words: Strict local martingales, local time, semi-stable Markov processes, Dunkl Markov
processes.
Mathematics Subject Classification (2000): 60G44, 60J75, 60J55, 60J25.
1 Main results.
Let (Ω, F, F t , P) be a filtered probability space. On Ω × R + we denote by O and P respectively
- the optional and predictable sigma fields and by B (R) the Borel sigma field. Consider (S t ) t≥0
- an R + valued local martingale with respect to the filtration (F t ) t≥0
. For the definitions of
local time for discontinuous local martingales we follow ([TL78], p. 17-22 (see also [Mey76] and
[Pro05]). For each a ∈ R there exists a continuous increasing process (L a t , t ≥ 0) , such that
Tanaka’s formula holds:
(S t − a) + = (S 0 − a) + +
which we write equivalently:
+ ∑
0a}dS u
0+
[
1{Su− >a} (S u − a) − + 1 {Su− ≤a} (S u − a) +] + 1 2 La t , (1)
(S t − a) + = (S 0 − a) + +
∫ t
0+
1 {Su− >a}dS u + 1 2 La t .
∗ Laboratoire de Probabilités et Modèles Aléatoires, Université Pierre et Marie Curie, 175 rue du Chevaleret,
F-75013 Paris. chyb@ccr.jussieu.fr
1

Furthermore, there exists a B (R)×O measurable version of L, a.s. càdlàg in t, and a B (R)×P
measurable version of L, which is a.s. continuous in t. We will only consider such versions. Also
note that for any f ≥ 0, Borel,
∫ t
0
f (X s ) d 〈X c , X c 〉 s
=
∫ +∞
−∞
f (a) L a t da.
We shall say that T is a (F t ) stopping time which reduces the local martingale S if (S t∧T ) is
a uniformly integrable martingale. We shall say that a process X is in class (D) if the family
{X τ , τ - a.s. finite (F t ) stopping time} is uniformly integrable.
The following Theorem is a straightforward generalization of Theorem 1 in [MY05].
Theorem 1 Let τ be an (F t ) stopping time such that τ < +∞ a.s. and K ≥ 0. Then there is
the following identity
where c S (τ) := E (S 0 − S τ ),
J K τ : = ∑
and ( L K t
)
t≥0
E (S τ − K) + = E (S 0 − K) + + EJ K τ + 1 2 ELK τ − c S (τ) , (2)
= 1 2
0K} (S u − K) − + ∑
is the (continuous) local time at K of S.
)
0K}dS u + ∑
0K} (S u − K) − + ∑
0K}dS u + S 0 − S t
0+
)
is a local martingale. Since (S t≥0 t − K) + − S t = −S t ∧ K, one has
−N K t = S t ∧ K − S 0 ∧ K + J K t + 1 2 LK t .
In order to get (2) it is enough to prove that Nt K is in class (D) , i.e. the family Nτ K 1 {τK}dS u .

Let (τ n ) n≥1
(τ n → +∞ a.s.) be a sequence of (F t ) stopping times which reduces both (S t )
( t≥0
∫ t
and 1 0+ {S u− >K}dS u . Then one gets
)t≥0
Finally, by Beppo-Levi:
EJ K t∧τ n
+ 1 2 ELK t∧τ n
= E [ (S t∧τ n
− K) + − (S 0 − K) +] ≤ ES t∧τ n
= ES 0 .
EJ K t + 1 2 ELK t ≤ ES 0
and
EJ K ∞ + 1 2 ELK ∞ ≤ ES 0 ,
then for any (F t ) stopping time τ
∣
∣N τ K 1 ∣
{τ λ = lim λ
λ→∞ 2 P [S, S] 1/2
τ
> λ .
Besides as a consequence of (2) one obtains
c S (τ) = lim
K→∞
< ∞
(
EJ K τ
Let τ be an a.s. finite (F t ) stopping time. Define
+ 1 2 ELK τ
)
.
C strict (K, τ) := lim
n→∞
E (S τ∧Tn − K) + , (4)
where T n → ∞ a.s., T n reduces (S t ) t≥0
. The following proposition shows that this limit exists
and does not depend on the reducing sequence T n , n ≥ 1.
Proposition 2 Let τ be an a.s. finite (F t ) stopping time. Then
C strict (K, τ) = E (S 0 − K) + + EJ K τ + 1 2 ELK τ . (5)
Furthermore, if the process (∆S t ) t≥0
is in class (D) , then
where S ∗ t := sup 0≤u≤t S u .
C strict (K, τ) = E [ (S τ − K) +] + lim
n→∞
nP (S ∗ τ > n) ,
3

Proof. By Tanaka’s formula
(S t − K) + − (S 0 − K) + =
∫ t
0+
1 {Su− >K}dS u + J K t + 1 2 LK t ,
where J K t
is defined by (3) . Since (S t ) t≥0
is a local martingale,
S K t :=
∫ t
1 {Su− >K}dS u
0+
is also a local martingale. We have seen in the proof of Theorem 1 that
N K t =
∫ t
0+
is a uniformly integrable martingale, then
1 {Su− >K}dS u + S 0 − S t
S K t := N K t + S t − S 0
is the sum of a uniformly integrable martingale and a local martingale (S t ) t≥0
. Therefore a
stopping time which reduces (S t ) t≥0
reduces ( )
St
K as well.
Let T be an (F t ) stopping time which reduces (S t ) t≥0
. Then S t∧T and St∧T K are uniformly
integrable martingales. For any τ - an a.s. finite (F t ) stopping time one gets
t≥0
E (S τ∧T − K) + = E (S 0 − K) + + EJ K τ∧T + 1 2 ELK τ∧T , (6)
now taking for T a stopping time T n , such that T n → ∞ a.s. and T n reduces (S t ) t≥0
, one
obtains that
C strict (K, τ) := lim
n→∞
E (S τ∧Tn − K) +
exists and does not depend on the sequence of stopping times reducing (S t ) t≥0
. Furthermore,
C strict (K, τ) = E (S 0 − K) + + EJ K τ + 1 2 ELK τ .
In order to move further suppose that the process (∆S t ) t≥0
is in class (D). Take
T n := inf {u > 0 |S u > n} .
Since (S t ) t≥0
is an adapted càdlàg process, T n is an (F t ) stopping time and T n → ∞ a.s. Since
|S t∧Tn | ≤ n + ∆S Tn ,
(S t∧Tn ) t≥0
is in class (D) and subsequently is a uniformly integrable martingale. In particular
T n is an (F t ) stopping time which reduces (S t ) t≥0
. Now one can get for any τ - an a.s. finite
(F t ) stopping time
E (S τ∧Tn − K) + = E [ (S τ − K) + 1 {τ≤Tn}]
+ E
[
(STn − K) + 1 {τ>Tn}]
.
4

The left hand side converges and equals C strict (K, τ) . The first expression on the right hand side
converges as well (by Beppo-Levi) to E [ (S τ − K) +] . Hence E [ (S Tn − K) + 1 {τ>Tn}]
converges
as well. Besides one has for n > K
(n − K) P (τ > T n ) ≤ E [ (S Tn − K) + 1 {τ>Tn}]
≤ (n − K) P (τ > Tn ) + E [ ]
∆S Tn 1 {τ>Tn}
and
E [ (S Tn − K) + [
1 {τ>Tn }]
− E ∆STn 1 {τ>Tn }]
≤ (n − K) P (τ > Tn ) ≤ E [ (S Tn − K) + 1 {τ>Tn }]
.
Since ( )
∆S Tn 1 {τ>Tn } is a uniformly integrable family E [ ∆S
n≥1 Tn 1 {τ>Tn }]
→ 0, as n → ∞, and
lim E [ (S Tn − K) + ]
1 {τ>Tn } = lim (n − K) P (τ > T n ) = lim nP (Sτ ∗ > n) .
n→∞ n→∞ n→∞
Finally
lim E (S τ∧T n
− K) + = E [ (S τ − K) +] + lim nP (Sτ ∗ > n) ,
n→∞ n→∞
where St ∗ := sup 0≤u≤t S u .
Remark 3 Note that from Theorem 1 and Proposition 2 for any positive local martingale S,
such that (∆S t ) t≥0
is in class (D) , and for any a.s. finite (F t ) stopping time τ
c S (τ) = lim
n→∞
nP (S ∗ τ > n) .
Remark 4 Under the conditions of Proposition 2
C strict (K, τ) =
sup E (S σ∧τ − K) + . (7)
σ-(F t ) stopping time
Proof. From (6) one obtains that for any pair of (F t ) stopping times τ, σ and a sequence
of stopping times R n , such that R n → ∞ a.s. and R n reduces (S t ) t≥0
Then by Fatou’s Lemma
E (S τ∧σ∧Rn − K) + = E (S 0 − K) + + EJ K τ∧σ∧R n
+ 1 2 ELK τ∧σ∧R n
.
E (S τ∧σ − K) + ≤ E (S 0 − K) + + lim inf
n→∞
Now (7) follows from (5) and (4) .
= E (S 0 − K) + + EJ K τ∧σ + 1 2 ELK τ∧σ
≤ E (S 0 − K) + + EJ K τ + 1 2 ELK τ .
[EJ K τ∧σ∧R n
+ 1 2 ELK τ∧σ∧R n
]
Remark 5 The original proof of Proposition 2 in [MY05] differs a little from ours. In order
to obtain (6) the fact, that the stopping time which reduces (S t ) t≥0
reduces as well ( )
St
K , is t≥0
not used. Let us go through this other proof and see that there is no contradiction.
5

Proof. Let T be an (F t ) stopping time which reduces (S t ) t≥0
and Tn K , n ≥ 1, Tn
K → ∞ be
a sequence of stopping times that reduce ( )
St
K . Then for any τ - an a.s. finite (F t≥0 t) stopping
time - one gets
E ( S τ∧T ∧T K n
− K )+ = E (S 0 − K) + + EJ K τ∧T ∧T K n + 1 2 ELK τ∧T ∧T K n .
On the right hand side one can pass to the limit as Tn
K → ∞ by Beppo-Levi and get a finite
limit as soon as we already know from the proof of Theorem 1 that
EJ K ∞ + 1 2 ELK ∞ ≤ ES 0 .
On the left hand side, ( S τ∧T ∧T K n
)n≥1
L 1 to S τ∧T . Finally one gets
is a uniformly integrable martingale, thus it converges in
which is the same as (6) .
E (S τ∧T − K) + = E (S 0 − K) + + EJ K τ∧T + 1 2 ELK τ∧T , (8)
For any µ - a finite measure on R + define
F µ (x) :=
∫ +∞
0
µ (dK) (x − K) +
and ¯µ := ∫ +∞
0
µ (dK) . As in [MY05] we have the following Proposition and Corollary (the
proofs are the same as in the continuous case).
Proposition 6 Under the notations and assumptions of Theorem 1
[∫ +∞
(
E [F µ (S τ )] = F µ (S 0 ) + E µ (dK) Jτ
K + 1 )]
2 LK τ − ¯µc S (τ) .
0
Corollary 7 The process
is a martingale.
F µ (S t ) − F µ (S 0 ) −
∫ +∞
0
(
µ (dK) Jt
K
+ 1 2 LK t
)
− ¯µ (S t − S 0 ) , t ≥ 0
2 Examples.
One can trivially construct strict ( local ) martingales from continuous strict ( local ) martingales:
indeed, M t := M (c)
t + M (d)
t and M (c)
t is a strict local martingale and M (d)
t is a uniformly
integrable martingale, then (M t ) is a strict local martingale.
We now obtain(
strict ) local martingales ( ) with jumps which are generalizations of the strict
local martingale , where is a Bessel process of dimension 3. As in the case of
1/R (3)
t
R (3)
t
6

(
1/R (3)
t
)
such strict local martingales can be obtained from absolute continuity relationships
between two Dunkl Markov processes instead of Bessel processes. For simplicity, we consider
here only one dimensional Dunkl Markov processes (see [GY05]).
The Dunkl Markov process (X t ) with parameter k is a Feller process with extended generator
given for f ∈ C 2 (R) by
L k f (x) = 1 ( )
1
2 f ′′ (x) + k
x f ′ f (x) − f (−x)
(x) − ,
2x 2
where k ≥ 0. Note that |X| is a Bessel process with index ν := k − 1 (k)
. Denote by P
2 x
of (X t ) started at x ∈ R, and by ( )
Ft
X the natural filtration of X.
the law
Proposition 8 Let 0 ≤ k < 1 ≤ 2 k′ and x > 0. Define T 0 := inf {s ≥ 0 |X s− = 0 or X s = 0} .
Then P x
(k) (T 0 < +∞) = 1 and there is the following absolute continuity relationship:
( ) k
P (k′ )
|Xt∧T0 |
′ −k ( ) k
′ Nt∧T0
x ∣ =
F
X
exp
(− (k′ ) 2 − k 2 ∫ )
t∧T0
ds
P (k) ∣
x
t |x| k
2
F
X , (9)
t
0
X 2 s
where N t denotes the number of jumps of X on [0, t] . Furthermore
( ) k |x|
′ −k ( ) Nt
k
((k ′ ) 2 − k 2 ∫ )
t
ds
M t :=
exp
|X t | k ′ 2 0
X 2 s
(10)
is a strict local martingale under P (k′ )
x , and
where E (k′ )
x is the expectation under P (k′ )
x .
P (k)
x (T 0 > t) = E (k′ )
x M t, (11)
(
Remark 9 Note that the law of T 0 under P x (k) is that of x 2 / 2Z (
1
gamma variable of a parameter 1 − k (see p.98 in [Yor01]).
2
2 −k) )
, where Z (
1
−k)
is a
2
Proof. Let X be a Dunkl Markov process. Note that ∆X s = X s − X s− = −2X s− , when
∆X s ≠ 0. Hence if X s = 0, then X s− = 0 and
T 0 = inf {s ≥ 0 |X s− = 0} = inf {s ≥ 0 ||X s | = 0} .
In order to prove (9) we proceed as in the proof of Proposition 4 in [GY05]. First we need to
extend Theorem 2 in [GY05] for k < 1 2 . Since |X| is a Bessel process with index ( k − 1 2)
, for
k < 1 2 , T 0 < +∞ a.s., and, for k ≥ 1 2 , T 0 = +∞ a.s. Denote
{
τ t := inf s ≥ 0
∣
∫ s
0
du
X 2 u
}
= t ,
7

then τ t is a continuous strictly increasing time change and τ ∞ = T 0 . Denote Y u := X τ u
. Since
for any f ∈ C 2 (R)
is a local martingale,
f (X t ) − f (X 0 ) −
f (Y u ) − f (Y 0 ) −
∫ t
∫ t
0
0
L k f (X s ) ds
Y 2
s L k f (Y s ) ds
is a local martingale. Then as in the proof of Proposition 4 in [GY05] one obtains that Y is in
the form
(
)
Y u = exp β (ν)
u + iπN u
(k/2) ,
( )
( )
where ν := k − 1, β (ν)
2 u is a Brownian motion with drift ν, N u
(k/2) is a Poisson process with
( )
parameter k/2 independent from . Denote
then τ At = t, for t < T 0 . Hence
β (ν)
u
A t :=
∫ t
0
du
,
X 2 u
X t = Y At , t < T 0 . (12)
Note also that differentiating the equality A τ t
= t with respect to time one gets
d
dt τ t = Yt
2
and A t = inf { s ≥ 0 ∣ ∫ s
0 Y u 2 du = t } , t < T 0 . Note that (9) is equivalent to
P x
(k) ∣
F
X
t
Indeed (9) is equivalent to
E (k′ )
x (F (X s , s ≤ t)) = E (k)
x
( ) k |x|
′ −k ( ) Nt
k
((k ′
= ) 2 − k 2 ∫ t
∩{t

where
(
D t : = exp (ν − ν ′ ) β (ν′ )
t − 1 (
ν 2 − (ν ′ ) 2) ) ( ) (k ′ /2) N (
k
t
t
exp − 1 )
2
k ′ 2 (k − k′ ) t
(
= exp (k − k ′ ) β (ν′ )
t − 1 (
k 2 − (k ′ ) 2) ) ( ) (k ′ /2) N k
t
t
2
k ′
{
and D At = M t , t < T 0 . Denote G t := σ β (ν′ )
G As∧u measurable and
(
F
E (k)
x
As u → +∞ one gets
(
F
E (k)
x
(
β (ν)
A s
, N (k/2)
A s
)
1 {As≤u}
)
s , N (k′ /2)
s
(
= E (k′ )
x
= E (k′ )
x
(
β (ν)
A s
, N (k/2)
A s
)
1 {As 0
( ) ( )
1
c X(x)
(d)
ct = X (xc−1 )
t
(
where
X (x)
t
t≥0
,
t≥0
)
denotes a semi-stable Markov process started at x > 0. Denote
T 0 := inf {s ≥ 0 |X s− = 0 or X s = 0} , (15)
then Lamperti in [Lam72] showed that: either T 0 = +∞ a.s., or T 0 < +∞ a.s. and X T0 − = 0
a.s., or T 0 < +∞ a.s. and X T0 − > 0 a.s. Furthermore this does not depend on the starting
point x > 0.
Note that for a semi-stable Markov process the following Lamperti relation is true.
suppose that there is no killing inside (0, ∞) .
Proposition 10 Let (ξ t ) be a one-dimensional Lévy process, starting at 0. Define
A (x)
t :=
∫ t
0
x exp (ξ s ) ds,
9
We

for any x > 0. Then the process (X u ) , defined implicitly by
is a semi-stable Markov process, starting at x, and
where T 0 is defined by (15) . The converse is also true.
Denote
τ (x)
t
x exp ξ t = X (x) A
, t < T 0 , (16)
t
A (x)
∞ = T 0 , (17)
:= inf { s ≥ 0 ∣ ∣A (x)
s = t } .
( )
Let F ξ t be the natural filtration of (ξ t ) and ( )
Ft
X be the natural filtration of (Xt ) . As
in [CPY94], using Proposition 10, one obtains the following absolute continuity relationship
between two semi-stable Markov processes.
Proposition 11 Suppose that (X t ) is a semi-stable Markov process associated with Lévy process
(ξ t ) via Lamperti relation (16) and E P e bξ t = e tρ(b) < ∞. Define Q by
( ) b ( ∫ t
Q| F
X
t ∩{t

( )
Note that (X t /x) b = exp bξ (x) τ
on {t < T 0 } . Differentiating (18) one gets that
t
Letting u tend to infinity, from (19) one gets
But from (17)
( { })
Q A ∩ τ (x)
t < +∞
d
dt τ (x)
t = 1 Xt
2 .
{ }
τ (x)
t < +∞ = {t < T 0 } . Hence
Q (A ∩ {t < T 0 }) = E P
(1 A∩{t

where ˜Π (dx) = e bx Π (dx) . Hence, in order to have Q (T 0 < +∞) = 1 and P (T 0 = +∞) = 1
one can choose b such that
∫
−a + xΠ (dx) ≥ 0 (20)
|x|>1
and
∫
−a + bσ 2 − x ( 1 − e bx) ∫
Π (dx) + xe bx Π (dx) < 0. (21)
|x|1
It is easy to see that (20) and (21) imply that b < 0. For example, for any given a and Π, such
that (20) is true, one can always choose b < 0 such that
∫
∫
bσ 2 − e −b |x| Π (dx) < a − xΠ (dx) , (22)
x1
which implies (21) . Note that condition (22) is more restrictive than (21) .
References
[BY05]
J. Bertoin and M. Yor, Exponential functionals of Lévy processes, To appear in Probability
Surveys (2005).
[CPY94] P. Carmona, F. Petit, and M. Yor, Sur les fonctionnelles exponentielles de certains
processus de Lévy, Stochastics Stochastics Rep. 47 (1994), no. 1-2, 71—101.
[GY05]
L. Gallardo and M. Yor, Some remarkable properties of the Dunkl martingales, To
appear in Séminaire de Probabilités XXXIX. Dedicated to Paul-André Meyer, Lecture
Notes in Math., Springer, Berlin, 2005.
[Lam72] J. Lamperti, Semi-stable Markov processes. I, Z. Wahrscheinlichkeitstheorie und Verw.
Gebiete 22 (1972), 205—225.
[LN05]
R. Liptser and A. Novikov, On tail distributions of supremum and quadratic variation
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(Seconde partie: Théorie des intégrales stochastiques, Univ. Strasbourg, Strasbourg,
année universitaire 1974/1975), Springer, Berlin, 1976, pp. 245—400. Lecture Notes in
Math., Vol. 511.
[MY05]
[Pro05]
D.B. Madan and M. Yor, Itô’s integrated formula for strict local martingales, To
appear in Séminaire de Probabilités XXXIX. Dedicated to Paul-André Meyer, Lecture
Notes in Math., Springer, Berlin, 2005.
P. E. Protter, Stochastic integration and differential equations, second ed., Version
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Stochastic Modelling and Applied Probability.
12

[TL78] Temps locaux, Astérisque, vol. 52, Société Mathématique de France, Paris, 1978,
Exposés du Séminaire J. Azéma-M. Yor, Held at the Université Pierre et Marie Curie,
Paris, 1976—1977, With an English summary.
[Yor01]
M. Yor, Exponential functionals of Brownian motion and related processes, Springer
Finance, Springer-Verlag, Berlin, 2001, With an introductory chapter by Hélyette
Geman, Chapters 1, 3, 4, 8 translated from the French by Stephen S. Wilson.
13