3.2 Stochastic calculus for anticipating integrals
3.2.3 Itˆ o’s formula for the Skorohod
In this section we will show the change-of-variables formula for the indef- inite Skorohod integral. We start with the following version of this formula.
Denote byL2,2(4)the space of processesu∈L2,2such thatE(Du4L2([0,1]2))<
∞.
Theorem 3.2.2 Consider a process of the form Xt=X0+
t 0
usdWs+ t
0
vsds, (3.25)
where X0 ∈ D1,2loc, u ∈ L2,2(4),loc and v ∈ L1,2loc. Suppose that the process X has continuous paths. Let F :R→Rbe a twice continuously differentiable function. Then F′(Xt)ut belongs toL1,2loc and we have
F(Xt) = F(X0) + t
0
F′(Xs)dXs+1 2
t 0
F′′(Xs)u2s +
t 0
F′′(Xs)(D−X)susds. (3.26)
Proof: Suppose that (Ωn,1, X0n), (Ωn,2, un) and (Ωn,3, vn) are localizing sequences forX0,uandv, respectively. For each positive integerk letψk be a smooth function such that 0≤ψk ≤1,ψk(x) = 0 if|x| ≥k+ 1, and ψk(x) = 1 if|x| ≤k. Define
un,kt =untψk 1
0
(uns)2ds
. (3.27)
SetXtn,k =X0n+t
0un,ks dWs+t
0vnsdsand consider the familuy of sets Gn,k= Ωn,1∩Ωn,2∩Ωn,3∩
sup
0≤t≤1|Xt| ≤k
∩ 1
0
(uns)2ds≤k
. DefineFk =F ψk. Then, by a localization argument, it suffices to show the result for the processes X0n, un,k, andvn and for the function Fk. In this way we can assume that X0 ∈D1,2, u∈ L2,2(4),v ∈L1,2,1
0 u2sds≤k, and that the functionsF, F′ andF′′ are bounded.
Set tni = 2itn, 0 ≤ i ≤ 2n. As usual, the basic argument in proving a change-of-variables formula is Taylor development. Going up to the second order, we get
F(Xt) = F(X0) +
2n−1 i=0
F′(X(tni))(X(tni+1)−X(tni))
+
2n−1 i=0
1
2F′′(Xi)(X(tni+1)−X(tni))2,
whereXidenotes a random intermediate point betweenX(tni) andX(tni+1).
Now the proof will be decomposed in several steps.
Step 1. Let us show that
2n−1 i=0
F′′(Xi)(X(tni+1)−X(tni))2→ t
0
F′′(Xs)u2sds, (3.28) in L1(Ω), asntends to infinity.
The increment (X(tni+1)−X(tni))2 can be decomposed into tni+1
tni
usdWs
2 +
tni+1 tni
vsds 2
+ 2
tni+1 tni
usdWs
tni+1 tni
vsds
. The contribution of the last two terms to the limit (3.28) is zero. In fact, we have
E
2n−1 i=0
F′′(Xi)
tni+1 tni
vsds
2≤ F′′∞t2−n t
0
E(vs2)ds,
and
E
2n−1 i=0
F′′(Xi)
tni+1 tni
usdWs
tni+1 tni
vsds
≤ F′′∞
t2−n t
0
E(vs2)ds 12
uL1,2. Therefore, it suffices to show that
2n−1 i=0
F′′(Xi) tni+1
tni
usdWs
2
→ t
0
F′′(Xs)u2sds
in L1(Ω), as n tends to infinity. Suppose that n ≥ m, and for any i = 1, . . . , nlet us denote byt(m)i the point of themth partition that is closer
totni from the left. Then we have
2n−1 i=0
F′′(Xi)
tni+1 tni
usdWs
2
− t
0
F′′(Xs)u2sds
≤
2n−1 i=0
[F′′(Xi)−F′′(X(t(m)i ))]
tni+1 tni
usdWs
2
+
2m−1 j=0
F′′(X(tmj ))
i:tni∈[tmj ,tmj+1)
tni+1 tni
usdWs
2
− tni+1
tni
u2sds
+
2m−1 j=0
F′′(X(tmj )) tmj+1
tmj
u2sds− t
0
F′′(Xs)u2sds
= b1+b2+b3.
The expectation of the termb3 can be bounded by kE
sup
|s−r|≤t2−m|F′′(Xs)−F′′(Xr)|
,
which converges to zero as m tends to infinity by the continuity of the processXt. In the same way the expectation ofb1 is bounded by
E
sup
|s−r|≤t2−m|F′′(Xs)−F′′(Xr)|
2n−1 i=0
tni+1 tni
usdWs
2
. (3.29) Letting first n tends to infinity and applying Theorem 3.2.1, (3.29) con- verges to
E
sup
|s−r|≤t2−m|F′′(Xs)−F′′(Xr)| 1
0
u2sds
,
which tends to zero as m tends to infinity. Finally, the termb2 converges to zero in L1(Ω) as n tends to infinity, for any fixed m, due to Theorem 3.2.1.
Step 2. Clearly,
2n−1 i=0
F′(X(tni) tni+1
tni
vsds
→ t
0
F′(Xs)vsds (3.30) in L1(Ω) asntends to infinity.
Step 3. From Proposition 1.3.5 we deduce F′(X(tni))
tni+1 tni
usdWs= tni+1
tni
F′(X(tni))usdWs+ tni+1
tni
Ds[F′(X(tni))]usds.
Therefore, we obtain
2n−1 i=0
F′(X(tni)) tni+1
tni
usdWs=
2n−1 i=0
tni+1 tni
F′(X(tni))usdWs
+
2n−1 i=0
tni+1 tni
F′′(X(tni))DsX(tni)usds. (3.31)
Let us first show that
2n−1 i=0
tni+1 tni
F′′(X(tni))DsX(tni)usds→ t
0
F′′(Xs)(D−X)susds (3.32)
in L1(Ω) asntends to infinity. We have
E
2n−1 i=0
tni+1 tni
F′′(X(tni))DsX(tni)usds− t
0
F′′(Xs)(D−X)susds
≤ E
2n−1 i=0
F′′(X(tni)) tni+1
tni
!DsX(tni)−(D−X)s
"
usds +E
2n−1 i=0
tni+1 tni
[F′′(X(tni))−F′′(Xs)] (D−X)susds
≤ F′′∞
E t
0
u2sds 1/2
× 2 t
0
sup
s−t2−n≤r≤s
E
DsXr−(D−X)s2 ds
31/2
+E
sup
|s−r|≤t2−n|F′′(Xs)−F′′(Xr)| t
0
(D−X)sus
ds
: =dn1+dn2.
The term dn2 tends to zero asntends to infinity becauseEt
0|(D−X)sus| ds <∞. The termdn1 tends to zero asntends to infinity becauseX belongs toL1,22 by Proposition 3.1.1.
As a consequence of the convergences (3.28), (3.30) and (3.32) we have proved that the sequence
An :=
2n−1 i=0
tni+1 tni
F′(X(tni))usdWs
converges in L1(Ω) asntends to infinity to Φt : =F(Xt)−F(X0)−
t 0
F′(Xs)vsds−1 2
t 0
F′′(Xs)u2s
− t
0
F′′(Xs)(D−X)susds. (3.33) Step 4. The process utF′(Xt) belongs to L1,2 because u∈L2,2, v ∈L1,2, E(Du4L2([0,1]2))<∞, and the processesF′(Xt),F′′(Xt) and1
0 u2sdsare uniformly bounded. In fact, we have
Ds[utF′(Xt)] = utFt′′(Xt)
us1{s≤t}+DsX0
+ t
0
DsurdWr+ t
0
Dsvrdr
+F′(Xt)Dsut, and all terms in the right-hand side of the above expression are square inte- grable. For the third term we use the duality relationship of the Skorohod integral:
1 0
1 0
E
ut
t 0
DsurdWr
2 dsdt
= E
1 0
1 0
1 0
Dsur
&
2utDrut
t 0
DsurdWr
+u2tDsur
+u2t t
0
DrDsuθdWθ
' drdsdt
≤ cE
Du4L2([0,1]2)+D2uL2([0,1]3) .
Step 5. Using the duality relationship it is clear that for any smooth random variableG∈ S we have
n→∞lim E
G
2n−1 i=0
tni+1 tni
F′(X(tni))usdWs
=E
G t
0
F′(Xs)usdWs
. On the other hand, we have seen that thatAnconverges inL1(Ω) to (3.33).
Hence, (3.26) holds.
Remarks:
1. If the processX is adapted thenD−X = 0, and we obtain the classical Itˆo’s formula.
2. Also using the operator∇ introduced in (3.8) we can write F(Xt) =F(X0) +
t 0
F′(Xs)dXs+1 2
t 0
F′′(Xs)(∇X)susds.
3. A sufficient condition forX to have continuous paths is E
1
0 DutpHdt
<∞ for some p >2 (see Proposition 3.2.2).
4. One can show Theorem 3.2.2 under different types of hypotheses. More precisely, one can impose some conditions onXand modify the assumptions onX0,uandv. For instance, one can assume either
(a) u∈L1,2∩L∞([0,1]×Ω), v∈L2([0,1]×Ω), X ∈L1,22−, andX has a version which is continuous andXt∈D1,2for allt, or
(b) u ∈ L1,2 ∩L4(Ω;L2([0,1])), v ∈ L2([0,1]×Ω), X ∈ L1,22−, and X has a version which is continuous and Xt ∈ D1,2 for all t, and 1
0
1
0(DsXt)2dsdt+1
0(D−X)2tdt∈L4(Ω).
In fact, ifXhas continuous paths, by means of a localization argument it suffices to show the result for each functionFk, which has compact support.
On the other hand, the propertiesu∈L1,2,v∈L2([0,1]×Ω) and X ∈L1,22− imply the convergences (3.28), (3.30) and (3.32). The boundedness or integrability assumptions onu,DX andD−X are used in order to ensure that E(Φ2t)<∞, and utF′(Xt)∈L1,2.
5.If we assume the conditionsX0∈D1,2loc,u∈L2,2loc andv∈L1,2loc, andX has continuous paths, then we can conclude thatusF′(Xs)1[0,t](s)∈(Domδ)loc
and Itˆo’s formula (3.26) holds. In fact, steps 1, 2 and 3 of the proof are still valid. Finally, the sequence of processes
vn :=
2n−1 i=0
F′(X(tni))us1(tni,tni+1](s)
converges in L2([0,1]×Ω) to F′(Xs)us1[0,t](s) asntends to infinity, and by Step 3, δ(vn) converges in L1(Ω) to Φt. Then, by Proposition 1.3.6, usF′(Xs)1[0,t](s) belongs to the domain of the divergence and (3.26) holds.
In [10] the following version of Itˆo’s formula is proved:
Theorem 3.2.3 Suppose thatXt=X0+t
0usdWs+t
0vsds, whereX0∈ D1,2loc, u ∈
LF∩L∞(Ω;L2([0,1]))
loc, v ∈ L1,2,floc and the process X has continuous paths. Let F : R → R be a twice continuously differentiable function. Then F′(Xs)us1[0,t](s)belongs to(Domδ)loc and we have
F(Xt) = F(X0) + t
0
F′(Xs)dXs+1 2
t 0
F′′(Xs)u2s +
t 0
F′′(Xs)(D−X)susds. (3.34)
Proof: By a localization argument we can assume that the processes F(Xt),F′(Xt),F′′(Xt) are uniformly bounded and 1
0 u2sds≤k. Then we proceed as in the steps 1, 2 and 3 of the proof of Theorem 3.2.2, and we conclude using Proposition 1.3.6.
Notice that for the proof of the convergence (3.28) we have to apply Theorem 3.2.1 for u∈LF. Also, (3.31) holds thanks to Proposition 1.3.5
applied to the setA= [tni, tni+1].
Remarks:
1. A sufficient condition for the indefinite Skorohod integral t
0usdWs of a process u∈LF to have a continuous version isE1
0 |ut|pdt
<∞for some p >2 (see Exercise 3.2.12).
2. The fact that1
0 u2tdtis bounded is only used to insure that the right- hand side of (3.26) is square integrable, and it can be replaced by the conditions1
0 u2tdt,1
0 (D−X)2tdt∈L2(Ω).
3. If X0 is a constant and u and v are adapted processes such that 1
0 u2tdt < ∞ and 1
0 v2tdt < ∞ a.s., then these processes satisfy the hypotheses of Theorem 3.2.3 because u ∈
LF∩L∞(Ω;L2([0,1]))
loc by Proposition 1.3.18, v∈L1,2,floc (this property can be proved by a localiza- tion procedure similat to that used in the proof of Proposition 1.3.18), and the processXt=X0+t
0usdWs+t
0vsds has continuous paths because it is a continuous semimartingale. Furthermore, in this caseD−X vanishes and we obtain the classical Itˆo’s formula.
4. In Theorem 3.2.3 we can replace the conditionv ∈ L1,2,floc by the fact that we can localize v by processes such that 1
0 |vnt|dt ∈ L∞(Ω), and 4Vtn=t
0vsnds, t∈[0,1]5
∈L1,2,f2− . In this way the change-of-variable for- mula established in Theorem 3.2.3 is a generalization of Itˆo’s formula.
The following result is a multidimensional and local version of the change- of-variables formula for the Skorohod integral.
Theorem 3.2.4 Let W ={Wt, t∈[0,1]} be anN-dimensional Brownian motion. Suppose that X0i ∈D1,2,uij ∈L2,2(4) , and vi ∈L1,2, 1≤i≤M, 1≤j≤N are processes such that
Xti=X0i+ N j=1
t 0
uijsdWsj+ t
0
vsids, 0≤t≤1.
Assume that Xti has continuous paths. LetF :RM →R be a twice contin- uously differentiable function. Then we have
F(Xt) = F(X0) + M i=1
t 0
(∂iF)(Xs)dXsi
+1 2
M i,j=1
N k=1
t 0
(∂i∂jF)(Xs)uiksujks ds
+ M i,j=1
N k=1
t 0
(∂i∂jF)(Dk,−Xj)suiks ds,
where Dk denotes the derivative with respect to thekth compoment of the Wiener process.
Itˆo’s formula for the Skorohod integral allows us to deduce a change-of- variables formula for the Stratonovich integral. Let us first introduce the following classes of processes:
The setL2,4S is the class of processesu∈L1,21 ∩L2,2(4) continuous inL2(Ω) and such that∇u∈L1,2.
Theorem 3.2.5 Let F be a real-valued, twice continuously differentiable function. Consider a process of the formXt=X0+t
0us◦dWs+t 0vsds, where X0∈D1,2loc,u∈L2,4S,loc andv ∈L1,2loc. Suppose thatX has continuous paths. Then we have
F(Xt) =F(X0) + t
0
F′(Xs)vsds+ t
0
[F′(Xs)us]◦dWs.
Proof: As in the proof of the change-of-variable formula for the Skorohod integral we can assume that the processes F(Xt), F′(Xt), F′′(Xt) and 1
0 u2sdsare uniformly bounded,X0∈D1,2,u∈L2,4S andv∈L1,2. We know, by Theorem 3.1.1 that the processXt has the following decomposition:
Xt=X0+ t
0
usdWs+ t
0
vsds+ t
0
1
2(∇u)sds.
This process verifies the assumptions of Theorem 3.2.2. Consequently, we can apply Itˆo’s formula toX and obtain
F(Xt) = F(X0) + t
0
F′(Xs)vsds+1 2
t 0
F′(Xs)(∇u)sds +
t 0
F′(Xs)usdWs+1 2
t 0
F′′(Xs)(∇X)susds.
The process F′(Xt)ut belongs toL1,21 . In fact, notice first that as in the proof of Theorem 3.2.2 the boundedness of F′(Xt), F′′(Xt) and 1
0 u2sds and the fact that u∈ L2,2(4), v,∇u∈ L1,2 and X0 ∈ D1,2 imply that this process belongs toL1,2and
Ds[F′(Xt)ut] =F′(Xt)Dsut+F′′(Xt)DsXtut.
On the other hand, using that u ∈ L1,21 , u is continuous in L2(Ω), and X ∈L1,22 we deduce thatF′(Xt)utbelongs toL1,21 and that
(∇(F′(X)u))t=F′(Xt)(∇u)t+F′′(Xt)ut(∇X)t. Hence, applying Theorem 3.1.1, we can write
t 0
[F′(Xs)us]◦dWs = t
0
F′(Xs)usdWs+1 2
t 0
(∇(F′(X)u))sds
= t
0
F′(Xs)usdWs+1 2
t 0
F′(Xs)(∇u)sds +1
2 t
0
F′′(Xs)us(∇X)sds.
Finally, notice that
(∇X)t= 2(D−X)t+ut.
This completes the proof of the theorem.
In the next theorem we state a multidimensional version of the change- of-variable formula for the Stratonovich integral.
Theorem 3.2.6 Let W ={Wt, t∈[0,1]} be anN-dimensional Brownian motion. Suppose that X0i ∈D1,2,uij ∈L2,4S , and vi ∈L1,2, 1≤i≤M, 1≤j≤N are processes such that
Xti=X0i+ N j=1
t 0
uijs ◦dWsj+ t
0
vsids, 0≤t≤1.
Assume that Xti has continuous paths. LetF :RM →R be a twice contin- uously differentiable function. Then we have
F(Xt) =F(X0) + M i=1
N j=1
t 0
(∂iF)(Xs)uijs ◦dWsj+ M i=1
t 0
(∂iF)(Xs)visds.
Similar results can be obtained for theforward and backward stochastic integrals. In these cases we require some addtional continuity conditions.
Let us consider the case of the forward integral. This integral is defined as the limit in probability of the forward Riemann suns:
Definition 3.2.1 Let u={ut, t∈[0,1]} be a stochastic process. The for- ward stochastic integral 1
0 utd−Wt is defined as the limit in probability as
|π| →0 of the Riemann sums
n−1
i=1
uti(Wti+1−Wti).
The following proposition provides some sufficient conditions for the ex- istence of this limit.
Proposition 3.2.3 Letu={ut, t∈[0,1]} be a stochastic process which is continuous in the norm of the space D1,2. Suppose that u∈ L1,21−. Then u is forward integrable and
δ(u) = 1
0
utd−Wt+ 1
0
D−u
sds.
Proof: We can write, using Eq. (3.4) δ
n−1
i=1
uti1(ti,ti+1]
=
n−1
i=1
uti(Wti+1−Wti)−
n−1 i=1
ti+1
ti
Dsutids. (3.35) The processes
uπt−=
n−1
i=1
uti1(ti,ti+1](t)
converge in the norm of the spaceL1,2to the processu, due to the continuity of t→ut in D1,2. Hence, left-hand side of Equation (3.35), which equals toδ(uπ−), converges in L2(Ω) toδ(u). On the other hand,
E
n−1
i=1
ti+1
ti
Dsutids− 1
0
D−u
sds
≤
n−1 i=1
ti+1
ti
EDsuti− D−u
s
ds
≤ 1
0
sup
(s−|π|)∨0≤t<s
E(|DsXt−(D−X)s|)ds,
which tends to zero as |π tends to zero, by the definition of the space
L1,21−.
We can establish an Itˆo’s formula for the forward integral as in the case of the Stratonovich integral. Notice that the forward integral follows the rules of the Itˆo stochastic calculus. Define the setL2,4− as the class of processes u∈L1,21−∩L2,2(4) continuous inD1,2 and such thatD−u∈L1,2.
Theorem 3.2.7 Let F be a real-valued, twice continuously differentiable function. Consider a process of the formXt=X0+t
0usd−Ws+t 0vsds, whereX0∈D1,2loc,u∈L2,4−,loc andv∈L1,2loc. Suppose thatX has continuous paths. Then we have
F(Xt) =F(X0)+
t 0
F′(Xs)vsds+
t 0
[F′(Xs)us]d−Ws+1 2
t 0
F′′(Xs)u2sds.
(3.36) Proof: As in the proof of the change-of-variable formula for the Skorohod integral we can assume that the processes F(Xt), F′(Xt), F′′(Xt) and 1
0 u2sdsare uniformly bounded,X0∈D1,2,u∈L2,4− andv∈L1,2. We know, by Proposition 3.2.3 that the processXthas the following decomposition:
Xt=X0+ t
0
usdWs+ t
0
vsds+ t
0
(D−u)sds.
This process verifies the assumptions of Theorem 3.2.2. Consequently, we can apply Itˆo’s formula toX and obtain
F(Xt) = F(X0) + t
0
F′(Xs)vsds+ t
0
F′(Xs)(D−u)sds +
t 0
F′(Xs)usdWs+1 2
t 0
F′′(Xs)(∇X)susds.
The process F′(Xt)ut belongs toL1,21−. In fact, notice first that as in the proof of Theorem 3.2.2 the boundedness of F′(Xt), F′′(Xt) and 1
0 u2sds and the fact thatu∈ L2,2(4),v, D−u∈L1,2 andX0 ∈D1,2 imply that this process belongs toL1,2and
Ds[F′(Xt)ut] =F′(Xt)Dsut+F′′(Xt)DsXtut.
On the other hand, using that u∈L1,21−,uis continuous in L2(Ω), X has continuous paths, andX ∈L1,22 we deduce that F′(Xt)utbelongs to L1,21− and that
D−(F′(X)u)
t=F′(Xt)(D−u)t+F′′(Xt)ut(D−X)t. Hence, applying Proposition 3.2.3, we can write
t 0
[F′(Xs)us]d−Ws = t
0
F′(Xs)usdWs+ t
0
D−(F′(X)u)
sds
= t
0
F′(Xs)usdWs+ t
0
F′(Xs)(D−u)sds +
t 0
F′′(Xs)us(D−X)sds.
Finally, notice that
(∇X)t= 2(D−X)t+ut.
This completes the proof of the theorem.