02141493v54n1p187

Publ. Mat. 54 (2010), 187–208
INTEGRATION WITH RESPECT TO LOCAL TIME
AND ITÔ’S FORMULA FOR SMOOTH
NONDEGENERATE MARTINGALES
Xavier Bardina and Carles Rovira
Abstract
We show an
R Itô’s formula for nondegenerate Brownian martingales Xt = 0t us dWs and functions F (x, t) with locally integrable
derivatives in t and x. We prove that one can express the additional term in Itô’s s formula as an integral over space and time
with respect to local time.
Introduction
We consider a continuous nondegenerate martingale X = {Xt , t ∈
Rt
[0, 1]} of the form Xt = 0 us dWs where W = {Wt , t ∈ [0, 1]} is a
standard Brownian motion and u is an adapted stochastic process. Let
F : R × [0, 1] → R be an absolutely continuous function with partial
derivatives satisfying some local integrability properties. The main aim
of this paper is to obtain an Itô’s formula for F (Xt , t) where the term
corresponding usually to the second order derivative is expressed as an
integral over space and time with respect to local time.
We will prove this results when u satisfies (locally) the assumptions
(H1) For all t ∈ [0, 1], ut belongs to the space D3,2 and for all p ≥ 2
E|ut |p + E|Ds ut |p + E
Z
1
r∨s
|Dr Ds uθ |2 dθ
+E
Z
1
r∨s∨v
p/2
|Dv Dr Ds uθ |2 dθ
p/2
≤ Kp ,
(H2) |ut | ≥ ρ > 0 for some constant ρ and for all t ∈ [0, 1].
Moret and Nualart [14] consider an Itô’s formula for this class of
nondegenerate martingales. Their main result reads as follows:
2000 Mathematics Subject Classification. 60H05, 60G44.
Key words. Martingales, integration wrt local time, Itô’s formula, local time.
188
X. Bardina, C. Rovira
Theorem 0.1 (Moret and Nualart, [14]). Let u be a process satisfying
Rt
(H1) and (H2). Set Xt = 0 us dWs . Then for any function f ∈ L2loc (R)
the quadratic covariation [f (X), X] exists and the following Itô’s formula
holds
Z t
1
F (Xt ) = F (0) +
f (Xs ) dXs + [f (X), X]t ,
2
0
Rx
for all t ∈ [0, 1], where F (x) = F (0) + 0 f (y) dy.
Moret [13] gave an extension of this last result for functions F depending also on t. She considers a new hypothesis on functions f :
(C) f (·, t) ∈ L2loc (R) and for all compact set K ⊂ R, f (x, t) is continuous in t as a function of [0, 1] to L2 (K).
Then, her result is the following:
Theorem 0.2 (Moret, [13]). Let u be a process satisfying (H1) and (H2).
Rt
Set X = 0 us dWs . Let F (x, t) be an absolutely continuous function in x
such that the partial derivative f (·, t) satisfies (C). Then, the quadratic
covariation [f (X, ·), X] exists and the following Itô’s formula holds
Z t
Z t
1
F (Xs , ds),
f (Xs , s) dXs + [f (X, ·), X] +
F (Xt , t) = F (0, 0) +
2
0
0
where
Z t
0
F (Xs , ds) ≡ lim
n→+∞
X
ti ∈Dn , ti ≤t
F (Xti+1 , ti+1 ) − F (Xti+1 , ti ) ,
exists uniformly in probability for (Dn )n a sequence of smooth partitions
of [0, 1].
In these two results, following the ideas of Föllmer, Protter and Shiryayev [11] for the Brownian motion, the additional term is written as a
quadratic covariation. Bardina and Jolis [2], [3] extended the results of
Föllmer et al. [11] to the case of the elliptic and hypoelliptic diffusions.
Nevertheless, it is important to point out the differences between [14]
and [11]. One of the keys of their proofs is to obtain some a priori estimates on the Riemann sums. In [11] these estimates are obtained using
the semimartingale expression of the time-reversed Brownian motion and
well-known bounds for the density of the Brownian motion. Moret and
Nualart [14] used another approach, using Malliavin calculus in order
to obtain sharp estimates for the density of the process Xt and avoiding
the time-reversed arguments.
Itô’s Formula for Nondegenerate Martingales
189
We want to express the quadratic variation term as an integral with
respect to the local time. We will refer the reader to the book of Rogers
and Williams [17] for the theory of local times for martingales.
There are several papers where the integrals with respect to local
time are used in Itô’s formula. In 1981, Bouleau and Yor [5] obtained
the following extension of the Itô’s formula:
Theorem 0.3 (Bouleau and Yor, [5]). Let X = (Xt )t≥0 be a continuous
semimartingale and let F : R → R be an absolutely continuous function
with derivative f . Assume that f is a mesurable locally bounded function.
Then:
Z
Z t
1
f (x) dx Lxt
F (Xt ) = F (X0 ) +
f (Xs ) dXs −
2 R
0
where dx Lxt is an integral with respect to x → Lxt .
Eisenbaum [8], [9] defined an integral in time and space with respect
to the local time of the Brownian motion. Using this integral, the quadratic covariation in the formula given in [11] can be expressed as an
integral with respect to the local time. She obtained the following result:
Theorem 0.4 (Eisenbaum, [8] and [9]). Let W = (Wt )0≤t≤1 be a standard Brownian motion and F a function defined on R × [0, 1] such that
∂F
there exist first order Radon-Nikodym derivatives ∂F
∂t and ∂x such that
for every A ∈ R+ ,
Z 1Z A 1
∂F
∂t (x, s) √s dx ds < +∞
0
−A
and
Then,
Z
1
0
Z
A
−A
F (Wt , t) = F (W0 , 0) +
2
1
∂F
(x, s) √ dx ds < +∞.
∂x
s
Z
0
t
t
∂F
(Ws , s) ds
0 ∂t
Z Z
1 t
∂F
−
(x, s) dLxs .
2 0 R ∂x
∂F
(Ws , s) dWs +
∂x
Z
This result has been extended by Bardina and Rovira [4] for elliptic
diffusion processes.
In our paper we will follow the ideas of Eisenbaum [8], [9], assuming on
the function F the hypothesis considered in Theorem 0.4. In the papers
of Eisenbaum [8], [9], as well as in [11] or in the extension of Bardina
190
X. Bardina, C. Rovira
and Rovira [4], one of the main ingredients is the study of the time
reversed process and the relationship between the quadratic covariation
and the forward and backward stochastic integrals. We show that we can
adapt the methods of Eisenbaum without using the time reversed process
and the backward integral. We will follow the methods of Moret and
Nualart [14] and we will use Malliavin calculus to obtain the necessary
estimates for the Riemann sums.
The existence of the quadratic covariation is not one of our main objectives. Nevertheless, it will be an important tool in our computations.
We recall its definition.
Definition 1. Given two stochastic processes Y = {Yt , t ∈ [0, 1]} and
Z = {Zt , t ∈ [0, 1]} we define their quadratic covariation as the stochastic process [Y, Z] given by the following limit in probability, if it
exists,
X
[Y, Z]t = lim
(Yti+1 − Yti )(Zti+1 − Zti ),
n
ti ∈Dn , ti <t
where Dn is a sequence of partitions of [0, 1] whose mesh tends to 0 as
n goes to ∞.
We will assume that the partitions Dn satisfy
(M) limn supti ∈Dn (ti+1 − ti ) = 0,
M := supn supti ∈Dn
ti+1
ti
< ∞.
We impose this condition in order to avoid certain possibly exploding
Riemann sums.
Other extensions for Itô’s formula has been obtained recently. Among
others, there is the paper of Dupoiron et al. [7] for uniformly elliptic diffusions and Dirichet processes, the work of Ghomrasni and Peskir [12] for
continuous semimartingales, the paper of Flandoli, Russo and Wolf [10]
for a Lyons-Zheng process or the work of Di Nunno, Meyer-Brandis,
Øksendal and Proske [6] for Lévy processes.
The paper is organized as follows. In Section 1 we give some basic
definitions and results on Malliavin calculus, recalling some results obtained in [14]. In Section 2 we define the space where we are able to
construct an integral in the plane with respect to the local time of a
nondegenerate Brownian martingale. Finally, Section 3 is devoted to
present our main result the extension of Itô’s formula.
Along the paper we will denote all the constants by C, Cp , K or Kp ,
unless they may change from line to line.
Itô’s Formula for Nondegenerate Martingales
191
1. Preliminaries
Let (Ω, F , P ) be the canonical probability space of a standard Brownian motion W = {Wt , 0 ≤ t ≤ 1}, that is, Ω is the space of all continuous
functions ω : [0, 1] → R vanishing at 0, P is the standard Wiener measure on Ω and F is the completion of the Borel σ-field of Ω with respect
to P .
Let S be the set of smooth random variables of the form
F = f (Wt1 , . . . , Wtn ),
(1)
f ∈ Cb∞ (Rn ) and t1 , . . . , tn ∈ [0, 1]. The Malliavin derivative of a smooth
random variable F of the form (1) is the stochastic process {Dt F, t ∈
[0, 1]} given by
Dt F =
n
X
∂f
(Wt1 , . . . , Wtn )I[0,ti ] (t),
∂xi
i=1
t ∈ [0, 1].
The Malliavin derivative of order N ≥ 2 is defined by iteration, as
follows. For F ∈ S, t1 , . . . , tN ∈ [0, 1],
DtN1 ,...,tN F = Dt1 Dt2 . . . DtN F.
For any real number p ≥ 1 and any integer N ≥ 1 we denote by DN,p
the completion of the set S with respect to the norm
"
kF kN,p = E(|F |p ) +
N
X
i=1
# p1
E(kDi F kpL2 ([0,1]i ) )
.
The domain of the derivative operator D is the space D1,2 .
The divergence operator δ is the adjoint of the derivative operator.
The domain of the operator δ, denoted by Dom δ, is the set of processes
u ∈ L2 ([0, 1] × Ω) such that there exists a square integrable random
variable δ(u) verifying
Z 1
E(F δ(u)) = E
Dt F ut dt ,
0
for any F ∈ S. The operator δ is an extension of Itô’s stochastic integral
R1
and we will make use of the notation δ(u) = 0 us dWs .
We will recall some useful results from [14]. We refer the reader to
this paper for their proof and a detailed account of these results. We
also refer to [16], [15] for any other property about operators D and δ.
192
X. Bardina, C. Rovira
Proposition 1.1. Let Y be a random variable in the space D1,2 such
Rb
that a (Ds Y )2 ds > 0 a.s. for some 0 ≤ a < b ≤ 1. Assume that
Rb
(DY / a (Ds Y )2 ds)I[a,b] belongs to Dom δ. Then Y has an absolutely
continuous distribution with density p that satisfies the inequality
Z
b
p(x) ≤ E a
Ds Y
Rb
a (Ds Y
)2 ds
!
dWs .
Proof: It follows from Proposition 1 and (2.6) in [14].
The following proposition is also a slight modification of Corollary 2
of [14].
Proposition
1.2. Let Y be a random variable in the space D1,2 such
R1
2
that 0 (Ds Y ) ds > 0 a.s. Let Z be a positive square integrable random
R1
variable such that (ZDY / 0 (Ds Y )2 ds)I[0,1] belongs to Dom δ. Then,
for any f ∈ L2 (R), we have
ZDY |E(f (Y )2 Z)| = kf k22 E δ
.
kDY k22 Proof: See Corollary 2 in [14]. The same proof works using a dominated
convergence argument.
Lemma 1.3. Fix p ≥ 1. Suppose that u satisfies hypotheses (H1) and
(H2). Let Z ∈ D1,2p . Then, we have, for 0 ≤ a < b ≤ 1:
Z
p
b
Dt Xb
E
ZRb
dWt a
2
(Dt Xb ) dt
a

1/2
≤ C0 (b − a)−p/2  E|Z|2p
+
Z
p !1/2 
b
,
E (Dt Z)2 dt
a
where C0 is a constant does not depend on Z.
Proof: See Lemma 10 in [14].
Itô’s Formula for Nondegenerate Martingales
193
2. Stochastic integration with respect to local time of
the martingale
Following the ideas of Eisenbaum [8], we consider first the space of
functions for whose elements we can define a stochastic integration with
respect to local time.
Let f be a measurable function from R × [0, 1] into R. We define the
norm k · k by
kf k =
Z
1
0
Z
2
f (x, s)
R
1
3
s4
dx ds
21
.
Consider the set of functions
H = {f : kf k < +∞}.
Since H is L2 ([0, 1] × R, s−3/4 dx ds), it is a Banach space, in fact, it is
a Hilbert space.
R t Let us consider X a nondegenerate martingale of the type Xt =
0 us dWs where u is an adapted stochastic process satisfying hypotheses (H1) and (H2). Let us show now how to define a stochastic integration
over the plane with respect to the local time L of the process X for the
elements of H.
Let f∆ be an elementary function,
f∆ (x, s) :=
X
fkl I(xk ,xk+1 ] (x)I(sl ,sl+1 ] (s),
(xk ,sl )∈∆
where (xk )1≤k≤m1 is a finite sequence of real numbers, (sl )1≤l≤m2 is a
subdivision of [0, 1], (fkl )1≤k≤m1 ; 1≤l≤m2 is a sequence of real numbers
and finally, ∆ = {(xk , sl ), 1 ≤ k ≤ m1 , 1 ≤ l ≤ m2 }. It is easy to check
that the elementary functions are dense in H.
We define the integration for the elementary function f∆ with respect
to the local time L of the martingale X as follows
Z
0
1
Z
f∆ (x, s) dLxs =
R
X
(xk ,sl )∈∆
k+1
k
fkl (Lxsl+1
− Lxslk+1 − Lxsl+1
+ Lxslk ).
194
X. Bardina, C. Rovira
Let f be a function of H. Let us consider (fn )n∈N a sequence of elementary
to f in H. We will check that the sequence
R Rfunctions converging
1
x
converges in L1 and that the limit does not de0 R fn (x, s) dLs
n∈N
pend on the choice of the sequence (fn )n∈N . So, we will use this limit as
R1R
the definition of the integral 0 R f (x, s) dLxs .
First of all, let us see two previous lemmas.
Lemma 2.1. For any locally bounded Borel measurable function f and
any t ∈ (0, 1] we have
Z
f (a) da Lat = − [f (X), X]t ,
R
where
da Lat
denotes the integral with respect to a → Lat .
Proof: It follows easily from Theorem 0.3 and Theorem 0.1.
Lemma 2.2. Consider f1 (x) := I(a,b] (x) and f2 (x) := I(c,d] (x), where
a < b and c < d are real numbers. Then for all ti < tj ≤ t,
where
E f1 (Xti+1 )f2 (Xtj+1 )(Xti+1 − Xti )(Xtj+1 − Xtj )
≤ E f1 (Xti+1 )f2 (Xtj+1 )Cij ,
(ti+1 − ti )(tj+1 − tj )
kCij k2 ≤ C p
,
ti+1 (tj+1 − ti+1 )
and C does not depend on f1 , f2 , i and j.
∞
Proof: When f1 = f2 = f ∈ CK
(R), this inequality is checked in the
proof of Proposition 14 in [14]. The same proof also works when f1 6= f2
∞
with f1 , f2 ∈ CK
(R). Now, fixed our functions f1 , f2 let us consider
n
∞
sequences f1 ↑ f1 and f2n ↑ f2 with fni ∈ CK
(R) for all n and i ∈
{1, 2}. Then, the result can be obtained by a dominated convergence
argument.
Theorem
R t R 2.3. Let f be a function of H. Then, there exists the integral 0 R f (x, s) dLxs for any t ∈ [0, 1].
Itô’s Formula for Nondegenerate Martingales
195
Proof: Let f∆ be an elementary function. From Theorem 0.2 and Lemma 2.1 it is easy to get that the quadratic covariation [f (X, .), X]t exists
and that
Z tZ
f∆ (x, s) dLxs = − [f∆ (X, ·), X]t .
0
R
The key of the proof is to check that for all elementary function f∆
Z t Z
x
E f∆ (x, s) dLs ≤ Ckf∆ k,
(2)
0
R
where the constant does not depend on f∆ .
Notice that,
Z t Z
E f∆ (x, s) dLxs 0
R
= E (|[f∆ (X, ·), X]t |)


X

= E lim
f∆ (Xti+1 , ti+1 )−f∆(Xti , ti ) Xti+1 −Xti 
n→∞ti ∈Dn ,ti ≤t




2 
X
(3)


f∆ (Xti+1 , ti+1 ) Xti+1 − Xti 
≤ 2 lim inf E 

ti ∈Dn , ti ≤t
 n→∞

2  12


 X
+ 2 lim inf E 
f∆ (Xti , ti ) Xti+1 − Xti 
n→∞


ti ∈Dn , ti ≤t
12
,
:= 2 lim inf I1 + 2 lim inf I2
n→∞
n→∞
where in the last inequality we have used Fatou’s lemma.
Along the study of I1 and I2 we will make use of the methods presented in the proofs of Propositions 13 and 14 in [14]. For the sake of
completeness, we will give the main steps of our proofs in the study of I2 .
196
X. Bardina, C. Rovira
By the isometry, and using Proposition 1.2 and Lemma 1.3 with Z =
R ti+1
u2s ds, b = ti and a = 0
ti

I2 = E 
=
X
2
f∆
(Xti , ti )
ti ∈Dn , ti ≤t
X
E
X
Z
ti ∈Dn , ti ≤t
≤
ti ∈Dn , ti ≤t
≤C
2
f∆
(Xti , ti )
ti+1
Z
ti+1
ti
ti
δ
2
f∆
(x, ti ) dxE
R
Z
X
− 12
R
ti ∈Dn , ti ≤t
2
f∆
(x, ti ) dx ti
s
Z
×  E X
Z X
m1 X
m2
R k=1 l=1
ti ∈Dn , ti ≤t
u2s
ds
!
Rt
( tii+1 u2s ds)DXti kDXti k22
0
ti+1
ti
u2s ds
2

dt
−1
2
fkl
I(xk ,xk+1 ] (x)I(sl ,sl+1 ] (ti ) ti 2 (ti+1 − ti ) dx
R k=1 l=1
=C
u2s ds
2 s Z ti Z
Dt
u2s ds + E

Z X
m1 X
m2
2
=C
fkl
I(xk ,xk+1 ] (x)
Z X
m1 X
m2

ti+1
ti
≤C
Z
X

− 12
I(sl ,sl+1 ] (ti ) ti (ti+1 −ti ) dx
ti ∈Dn , ti ≤t
2
fkl
I(xk ,xk+1 ] (x)
R k=1 l=1

×
Z
t
X
0 t ∈D , t ≤t
i
n i

− 12
I(sl ,sl+1 ] (ti ) ti I(ti ,ti+1 ] (s) ds dx.
Using the condition (M) over the partitions, we have that, by bounded
convergence,
lim
n→∞
Z
t
X
0 t ∈D , t ≤t
i
n i
−1
I(sl ,sl+1 ] (ti ) ti 2 I(ti ,ti+1 ] (s) ds =
Z
t
0
1
I(sl ,sl+1 ] (s) s− 2 ds,
Itô’s Formula for Nondegenerate Martingales
197
and then,
lim inf I2 ≤ C
n→∞
(4)
=C
Z X
m2
m1 X
2
fkl
I(xk ,xk+1 ] (x)
R k=1 l=1
Z
t
0
1
I(sl ,sl+1 ] (s) s− 2 ds dx
Z tZ
1
2
f∆
(x, s) √ dx ds
s
R
0
≤ Ckf∆ k2 .
On the other hand,

2 
X


I1 = E 
f∆ (Xti+1 , ti+1 ) Xti+1 − Xti 
ti ∈Dn , ti ≤t

(5)
=E
X
ti ∈Dn , ti ≤t

+ 2E 
2
f∆
(Xti+1 , ti+1 ) Xti+1 − Xti
X
2


f∆ (Xti+1 , ti+1 )f∆ (Xtj+1 , tj+1 )
ti ,tj ∈Dn , ti <tj ≤t
× Xti+1 − Xti
:= I1,1 + 2I1,2 .
Xtj+1

− Xtj 
Following now the methods of Proposition 14 of [14] and using again
Propositions 1.2 and 1.3 as we did in the study of I2 , we get that
I1,1 ≤ C
X
ti ∈Dn , ti ≤t
Z
R
−1
2
2
(ti+1 − ti ).
f∆
(x, ti ) dx ti+1
By similar computations to those of the term I2 we obtain that
(6)
lim inf I1,1 ≤ Ckf∆ k2 .
n→∞
198
X. Bardina, C. Rovira
Let us study now I1,2 . Using Lemma 2.2, notice that

X
I1,2 = E 
f∆ (Xti+1 , ti+1 )f∆ (Xtj+1 , tj+1 )
ti ,tj ∈Dn , ti <tj ≤t
× Xti+1 − Xti
≤
X
ti ,tj ∈Dn , ti <tj ≤t
Xtj+1

− Xtj 
E f∆ (Xti+1 , ti+1 )f∆ (Xtj+1 , tj+1 )Cij .
Following again the methods of the proof of Proposition 14 of [14] —more
precisely, the proof of inequalities (5.36) and (5.37)— the last expression
is bounded by
X
1
1
2 2
2
2
(Xtj+1 , tj+1 ) 2 E Cij
E f∆
(Xti+1 , ti+1 )f∆
ti ,tj ∈Dn , ti <tj ≤t
≤C
(ti+1 − ti )(tj+1 − tj )
X
3
(ti+1 (tj+1 − ti+1 )) 4
ti ,tj ∈Dn , ti <tj ≤t
×
(7)
≤C
Z
12 Z
2
f∆
(x, ti+1 ) dx
21
2
f∆
(x, tj+1 ) dx
R
R
(ti+1 − ti )(tj+1 − tj )
X
3
(ti+1 (tj+1 − ti+1 )) 4
ti ,tj ∈Dn , ti <tj ≤t
×
Z
2
f∆
(x, ti+1 ) dx
R
+
Z
2
f∆
(x, tj+1 ) dx
R
:= C(I1,2,1 + I1,2,2 ).
Since
X
tj ∈Dn , ti <tj ≤t
(tj+1 − tj )
3
(tj+1 − ti+1 ) 4
=
Z
t
X
1
3
ti+1 t ∈D , t <t ≤t (tj+1 − ti+1 ) 4
j
n i
j
I(tj ,tj+1 ] (s) ds,
Itô’s Formula for Nondegenerate Martingales
199
we get that
I1,2,1 ≤
Z
t
(ti+1 − ti )
Z
(ti+1 − ti )
X
3
4
ti+1
ti ∈Dn , ti <tj ≤t
≤C
X
ti+1
ds
3
ti+1
3
4
ti ∈Dn , ti <tj ≤t
1
(s − ti+1 ) 4
Z
R
2
f∆
(x, ti+1 ) dx
2
f∆
(x, ti+1 ) dx.
R
And this clearly yields that
(8)
lim inf I1,2,1 ≤ C
n→∞
Z tZ
0
2
f∆
(x, s)
R
1
3
s4
ds = Ckf∆ k2 .
Finally we have to consider I1,2,2 . First of all, notice that
I1,2,2 :=
X
(ti+1 − ti )(tj+1 − tj )
X
(ti+1 − ti )(tj+1 − tj )
3
ti ,tj ∈Dn , ti <tj ≤t
=
(ti+1 (tj+1 − ti+1 )) 4
Z X
m2
m1 X
tj ∈Dn , ti <tj ≤t
2
fkl
I(xk ,xk+1 ] (x)I(sl ,sl+1 ] (tj+1 ) dx
X
2
fkl
I(xk ,xk+1 ] (x)
X
!
R k=1 l=1
R k=1 l=1
×
R
(ti+1 (tj+1 − ti+1 )) 4
×
≤
2
f∆
(x, tj+1 ) dx
3
ti ,tj ∈Dn , ti <tj ≤t
Z X
m1 X
m2
Z
ti ∈Dn , ti <t
t
1
ti+1
(s − ti+1 ) 4
Z
3
(ti+1 − ti )
3
4
ti+1
I(sl ,sl+1 ] (tj+1 )I(tj ,tj+1 ] (s) ds dx.
From the obvious inequality
I(sl ,sl+1 ] (tj+1 )I(tj ,tj+1 ] (s) ≤ I(sl ,sl+1 ] (s)I(tj ,tj+1 ] (s)+ I(tj ,sl ] (s)I(tj ,tj+1 ] (sl ),
we obtain the bound
(9)
I1,2,2 ≤ I1,2,2,1 + I1,2,2,2 ,
200
X. Bardina, C. Rovira
where
I1,2,2,1 =
Z X
m1 X
m2
R k=1 l=1
×
I1,2,2,2 =
tj ∈Dn , ti <tj ≤t
t
1
ti+1
(s − ti+1 ) 4
Z
I(sl ,sl+1 ] (s)I(tj ,tj+1 ] (s) ds dx
3
(ti+1 − ti )
X
2
fkl
I(xk ,xk+1 ] (x)
R k=1 l=1
×
3
4
ti+1
ti ∈Dn ,ti <t
X
Z X
m1 X
m2
(ti+1 − ti )
X
2
fkl
I(xk ,xk+1 ] (x)
3
4
ti+1
ti ∈Dn , ti <t
X
tj ∈Dn , ti <tj ≤t
t
1
ti+1
(s − ti+1 ) 4
Z
I(tj ,sl ] (s)I(tj ,tj+1 ] (sl ) ds dx.
3
Now, since we can write
Z
Z
X
1
(ti+1 − ti ) t
2
f∆
(x, s)
I1,2,2,1 =
3
3 ds dx,
4
(s − ti+1 ) 4
R t ∈D , t <t
ti+1
ti+1
i
n i
using an argument of bounded convergence we have that
Z Z t
Z t
1
1
2
lim inf I1,2,2,1 ≤
f∆
(x, s)
3
3 ds du dx
n→∞
(s − u) 4
R 0 u4 u
=
(10)
t
Z Z
0
R
≤C
2
f∆
(x, s)
t
Z Z
0
R
Z
s
3
4
3
u (s − u) 4
0
2
f∆
(x, s)
1
1
1
1
s2
du ds dx
ds dx
≤ Ckf∆ k2 .
On the other hand, observe that fixed l, there exists only one j (that
we will denote by j(l)) such that tj(l) < sl ≤ tj(l)+1 . So,
X
(ti+1 − ti )
3
4
ti+1
ti ,tj ∈Dn , ti <tj ≤t
≤
t
1
ti+1
(s − ti+1 ) 4
Z
X
ti ∈Dn , ti <tj(l) ≤t
3
I(tj ,sl ] (s)I(tj ,tj+1 ] (sl ) ds
(ti+1 − ti )
3
4
ti+1
Z
tj(l)+1
tj(l)
1
3
(s − ti+1 ) 4
ds.
Itô’s Formula for Nondegenerate Martingales
201
Now, using that for i < j(l)
Z tj(l)+1
Z tj(l)+1
1
1
1
4
ds
≤
3
3 ds ≤ 4|Dn | ,
(s − ti+1 ) 4
(s − tj(l) ) 4
tj(l)
tj(l)
and that
(ti+1 − ti )
X
3
4
ti+1
ti ∈Dn , ti <tj(l) ≤t
≤
Z
0
1
1
3
s4
ds < ∞,
we obtain easily that
(11)
lim I1,2,2,2 = 0.
n→∞
So, putting together (3)–(11), we have proved (2).
Now, given f ∈ H, let us consider {fn }n∈N a sequence of elementary
functions converging to f in H, and we define
Z t Z
Z tZ
f (x, s) dLxs = L1 − lim
fn (x, s) dLxs .
0
n→∞
R
0
R
Clearly, this limit exists. Indeed, for any ε > 0 there exists n0 such that
for any n, m ≥ n0 , kfn − fm k < ε and using inequality (2) we obtain
that
Z t Z
Z tZ
E fm (x, s) dLxs fn (x, s) dLxs −
0
0
R
R
Z t Z
x
=E
(fn (x, s) − fm (x, s)) dLs ≤ kfn − fm k < ε.
0
R
Moreover, using again inequality (2), it is clear that the definition does
not depend on the choice of the sequence (fn ). Indeed, given (fn1 )n∈N
and (fn2 )n∈N two sequences converging to f in H, we have
Z t Z
Z tZ
E fn1 (x, s) dLxs −
fn2 (x, s) dLxs 0
0
R
R
≤ kfn1 − fn2 k ≤ kfn1 − f k + kf − fn2 k,
that goes to zero when n tends to infinity.
Remark 2.4. If f satisfies condition (C), from Theorem 0.2 we know
that the quadratic covariation [f (X, ·), X] exists. Moreover, if f ∈ H,
from the uniqueness of the extension in the construction of the integral
in Theorem 2.3 we get that
Z tZ
f (x, s) dLxs = − [f (X, ·), X]t .
0
R
202
X. Bardina, C. Rovira
The following result is an obvious consequence of Theorem 0.2 and
Remark 2.4.
Corollary 2.5. Let u be a process satisfying (H1) and (H2). Set X =
Rt
0 us dWs . Consider a sequence Dn of partitions of [0, 1] verifying conditions (M). Let F (x, t) be an absolutely continuous function in x such
that the partial derivative f (·, t) satisfies (C). Then, if f ∈ H, we have
the following extension for the Itô’s formula:
Z t
Z Z
Z t
1 t
F (Xs , ds).
f (x, s) dLxs +
f (Xs , s) dXs −
F (Xt , t) = F (0, 0)+
2 0 R
0
0
3. Itô’s formula extension
Now we can state the main result of this paper.
Theorem 3.1.
Hypothesis over the martingale:
1) Let u be an adapted process satisfying (H1) and (H2). Set Xt =
Rt
0 us dWs .
Hypothesis over the function:
1) Let F be a function defined on R × [0, 1] such that F admits first
order Radon-Nikodym derivatives with respect to each parameter.
Assume that these derivatives are measurable in both variables.
2) Assume also that these derivatives satisfy that for every A ∈ R,
Z 1Z A ∂F
dx √1 ds < +∞
(x,
s)
∂t
s
0
−A
Z
1
0
Z
A
−A
2
1
∂F
(x, s)
dx √ ds < +∞.
∂x
s
Then, for all t ∈ [0, 1],
F (Xt , t) = F (0, 0) +
Z
0
t
∂F
(Xs , s) dXs +
∂x
t
∂F
(Xs , s) ds
0 ∂t
Z Z
1 t
∂F
−
(x, s) dLxs .
2 0 R ∂x
Z
Remark 3.2. Notice that under the hypotheses of Theorem 3.1, it is possible that ∂F
∂x does not belong to the space H. In this case, using the localization arguments, we can always assume that ∂F
∂x (x, s)I(ε,t) (s), x ∈ R ,
Itô’s Formula for Nondegenerate Martingales
203
s ∈ [0, 1] belongs to H for any ε > 0 and we define, in the previous
theorem,
Z 1Z
Z tZ
∂F
∂F
(x, s) dLxs = lim
(x, s)I(ε,t) (s) dLxs .
ε→0
∂x
∂x
0
R
0
R
At the end of the proof, we will justify that this limit exists.
Proof: Using localization arguments we can assume that F has compact
support and
Z 1Z ∂F
1
∂t (x, s) dx √s ds < +∞
0
R
Z
1
0
Z R
2
∂F
1
dx √ ds < +∞.
(x, s)
∂x
s
LetR g ∈ C ∞ be a function with compact support from R to R+ such
that R g(s) ds = 1. We define, for any n ∈ N,
gn (s) = ng(ns)
and
Fn (x, t) =
Z
0
1
Z
R
F (y, s)gn (t − s)gn (x − y) dy ds.
Then Fn ∈ C ∞ (R × [0, 1]). Hence, by the usual Itô’s formula, for every ε > 0, we can write
Z t
Z t
∂Fn
∂Fn
(Xs , s) dXs +
(Xs , s) ds
(12) Fn (Xt , t) = Fn (Xε , ε) +
∂x
∂t
ε
ε
Z
1 t 2 ∂ 2 Fn
(Xs , s) ds.
u
+
2 ε s ∂x2
Using the arguments of Azéma et al. [1] we will study the convergence
of (12).
Since F is a continuous function with compact support, it is easy to
check that (Fn (Xt , t))n∈N converges in probability to F (Xt , t).
On the other hand
Z 1Z Z 1Z ∂F
∂F
1
∂t (x, s) dx ds ≤
∂t (x, s) dx √s ds < +∞.
0
∂F
∂t
0
R
1
R
∈ L (R×[0, 1]). Under our hypothesis over the martingale X,
Hence,
it follows from Proposition 1.1 and Lemma 1.3 that for any t ∈ [0, 1], the
204
X. Bardina, C. Rovira
random variable Xt is absolutely continuous with density pt satisfying
the estimate
C
pt (x) ≤ √ .
t
R
t
n
converges in proba(X
,
s)
ds
Then, it is easy to see that ε ∂F
s
∂t
n∈N
R
t
bility to ε ∂F
∂t (Xs , s) ds . Indeed,
Z t ∂F
∂Fn
(Xs , s) −
(Xs , s) ds
E ∂t
∂t
ε
Z tZ ∂Fn
∂F
ps (x) dx ds
≤
(x,
s)
−
(x,
s)
∂t
∂t
ε
R
Z tZ 1
∂Fn
∂F
√ dx ds
(x,
s)
−
(x,
s)
≤C
s
∂t
∂t
ε
R
Z tZ ∂Fn
∂F
C
(x, s) −
(x, s) dx ds,
≤ √
∂t
ε ε R ∂t
that goes to zero, when n tends to infinity, since
∂Fn
(x, t) =
∂t
Z
1
0
Z
R
∂F
∂t
∈ L1 (R × [0, 1]) and
∂F
(y, s)gn (t − s)gn (x − y) dy ds.
∂t
R
t ∂Fn
Similarly, we can prove that
ε ∂x (Xs , s) dXs n∈N converges in
R
t
probability to ε ∂F
∂x (Xs , s) dXs . Indeed, using the same arguments
we get that
E
∂F
∂x
∈ L2 (R × [0, 1]). Then,
2 !
Z t ∂F
∂F
n
(Xs , s) −
(Xs , s) dXs ∂x
∂x
ε
=E
=E
Z t 2 !
∂F
∂F
n
(Xs , s) −
(Xs , s) us dWs ∂x
∂x
ε
Z t
ε
!
2
∂F
∂Fn
2
(Xs , s) −
(Xs , s) us ds .
∂x
∂x
Itô’s Formula for Nondegenerate Martingales
205
Following the same ideas of Proposition 12 in [14], Proposition 1.2 and
Lemma 1.3 yield the following bound for the last expression
(13) C
Z tZ ε
R
2
∂F
1
∂Fn
(x, s) −
(x, s) √ dx ds
∂x
∂x
s
C
≤√
ε
Z tZ ε
R
2
∂Fn
∂F
(x, s) −
(x, s)
dx ds
∂x
∂x
2
that goes to zero when n tends to infinity, since ∂F
∂x ∈ L (R × [0, 1]) and
Z 1Z
∂F
∂Fn
(x, t) =
(y, s)gn (t − s)gn (x − y) dy ds.
∂x
0
R ∂x
So, letting n to infinity in (12), we get that the sequence
Z t
2
1
2 ∂ Fn
(Xs , s) ds
u
2 ε s ∂x2
n∈N
converges in probability to
F (Xt , t) − F (Xε , ε) −
Z
ε
t
∂F
(Xs , s) dXs −
∂x
Z
ε
t
∂F
(Xs , s) ds.
∂t
n
But, since ∂F
∂x (x, s)I(ε,t) (s) ∈ H, from Theorem 0.2 and Corollary 2.5,
we get that
Z t
2
∂Fn
∂Fn
2 ∂ Fn
us
(Xs , s) ds =
(X, ·), X −
(X, ·), X
∂x2
∂x
∂x
ε
t
ε
=−
Z
0
1
Z
R
∂Fn
(x, s)I(ε,t) (s) dLxs .
∂x
n
R,
The next step of the proof is to check that ∂F
∂x (x, s)I(ε,t) (s), x ∈
∂F
s ∈ [0, 1] n∈N converges in H to ∂x (x, s)I(ε,t) (s), x ∈ R, s ∈ [0, 1] . It
suffices to notice that,
Z tZ ε
R
2
1
∂Fn
∂F
(x, s) −
(x, s)
3 dx ds
∂x
∂x
s4
2
Z tZ ∂Fn
1
∂F
≤ 3
(x, s) −
(x, s)
dx ds
∂x
ε 4 ε R ∂x
206
X. Bardina, C. Rovira
that converges to zero when n tends to infinity. Then, we clearly have
proved that
Z 1 Z
∂Fn
(x, s)I(ε,t) (s) dLxs
0
R ∂x
n∈N
R 1 R ∂F
1
converges in L to 0 R ∂x (x, s)I(ε,t) (s) dLxs .
So, we have that for any ε > 0
Z t
Z t
∂F
∂F
(Xs , s) dXs +
(Xs , s) ds
(14) F (Xt , t) = F (Xε , ε) +
∂x
ε ∂t
ε
Z Z
1 1
∂F
−
(x, s)I(ε,t) (s) dLxs .
2 0 R ∂x
The last step is to let ε to zero. But we need to check that the limit of
the stochastic integral exists. Actually, it is enough to show that
Z t
∂F
E (Xs , s) ds < ∞
∂t
0
and that
E
Z
0
But,
Z
E t
0
t
∂F
(Xs , s) dXs
∂x
2
< ∞.
Z 1Z ∂F
∂F
dx √1 ds < +∞.
(Xs , s) ds ≤ C
(x,
s)
∂t
∂t
s
0
R
On the other hand, following the same type of arguments that in (13),
we are able to write
!
Z t
2
2
Z t
∂F
∂F
2
E
=E
(Xs , s) dXs
(Xs , s) us ds
∂x
0 ∂x
0
≤C
Z Z t
R
0
2
∂F
1
(x, s) √ ds dx < ∞.
∂x
s
Letting ε to zero, the proof is finished.
When ∂F
∂x does not belong to the space H, as we explained in Remark 3.2, we define
Z 1Z
Z tZ
∂F
∂F
(x, s) dLxs = lim
(x, s)I(ε,t) (s) dLxs .
ε→0
∂x
∂x
0
R
0
R
This limit clearly exists in probability since all the other limits in (14)
exist.
Itô’s Formula for Nondegenerate Martingales
207
Acknowledgements
This work was partially supported by DGES Grants MTM2006-01351
(Carles Rovira) and MTM2006-06427 (Xavier Bardina).
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Xavier Bardina:
Departament de Matemàtiques
Universitat Autònoma de Barcelona
08193-Bellaterra (Barcelona)
Spain
E-mail address: [email protected]
Carles Rovira:
Facultat de Matemàtiques
Universitat de Barcelona
Gran Via 585
08007-Barcelona
Spain
E-mail address: [email protected]
Primera versió rebuda el 6 de novembre de 2008,
darrera versió rebuda el 7 de maig de 2009.