In this section we assume that {W(t), t ∈ [0,1]} is a one-dimensional Brownian motion, defined in the canonical probability space (Ω,F, P).
We denote by π an arbitrary partition of the interval [0,1] of the form π={0 =t0< t1<ã ã ã< tn= 1}. We have to take limits (in probability, or inLp(Ω),p≥1) of families of random variablesSπ, depending onπ, as the norm ofπ(defined as|π|= sup0≤i≤n−1(ti+1−ti)) tends to zero. Notice first that this convegence is equivalent to the convergence along any sequence of partitions whose norms tend to zero. In most of the cases it suffices to consider increasing sequences, as the next technical lemma explains.
Lemma 3.1.1 Let Sπ be a family of elements of some complete metric space(V, d)indexed by the class of all partitions of[0,1]. Suppose that for any fixed partition π0 we have
|πlim|→0d(Sπ∨π0, Sπ) = 0, (3.1) whereπ∨π0 denotes the partition induced by the union of πandπ0. Then the familySπ converges to some elementS if and only if for any increasing sequence of partitions {π(k), k ≥ 1} of [0,1], such that |π(k)| → 0, the sequence Sπ(k)converges to S as ktends to infinity.
Proof: Clearly, the convergence of the familySπimplies the convergence of any sequenceSπ(k)with|π(k)| →0 to the same limit. Conversely, suppose that Sπ(k) → S for any increasing sequence π(k) with |π(k)| → 0, but there exists an ǫ > 0 and a sequence π(k) with |π(k)| → 0 such that d(Sπ(k), S)> ǫfor allk. Then we fixk0and by (3.1) we can find ak1 such that k1> k0and
d(Sπ(k0)∨π(k1), Sπ(k1))< ǫ 2. Next we choosek2> k1 large enough so that
d(Sπ(k0)∨π(k1)∨π(k2), Sπ(k2))< ǫ 2,
and we continue recursively. Setπ(n) = π(k0)∨π(k1)∨ ã ã ã ∨π(kn). Then after thenth step we have
d(Sπ(n), S)≥d(Sπ(kn), S)−d(Sπ(n), Sπ(kn))> ǫ 2.
Thenπ(n) is an increasing sequence of partitions such that the sequence of norms|π(n)|tends to zero butd(Sπ(n) , S)>2ǫ, which completes the proof
by contradiction.
Consider a measurable process u={ut, t∈[0,1]} such that 1
0 |ut|dt <
∞a.s. For any partitionπwe introduce the following step process:
uπ(t) =
n−1 i=0
1 ti+1−ti
ti+1
ti
usds
1(ti,ti+1](t). (3.2)
IfE1 0 |ut|dt
<∞we define the step process
uπ(t) =
n−1 i=0
1 ti+1−ti
ti+1
ti
E(us|F[ti,ti+1]c)ds
1(ti,ti+1](t). (3.3) We recall that F[tci,ti+1] denotes the σ-field generated by the increments Wt−Ws, where the interval (s, t] is disjoint with [ti, ti+1].
The next lemma presents in which topology the step processes uπ and
uπ are approximations of the processu.
Lemma 3.1.2 Suppose thatubelongs toL2([0,1]×Ω). Then, the processes uπ anduπ converge to the processuin the norm of the spaceL2([0,1]×Ω) as |π| tends to zero. Furthermore, these convergences also hold in L1,2 wheneveru∈L1,2.
Proof: The convergence uπ → u in L2([0,1]×Ω) as |π| tends to zero can be proved as in Lemma 1.1.3, but for the convergence ofuπwe need a different argument.
One can show that the families uπ and uπ satisfy condition (3.1) with V = L2([0,1]×Ω) (see Exercise 3.1.1). Consequently, by Lemma 3.1.1 it suffices to show the convergence along any fixed increasing sequence of partitionsπ(k) such that|π(k)|tends to zero. In the case of the familyuπ, we can regarduπ as the conditional expectation of the variable u, in the probability space [0,1]×Ω, given the productσ-field of the finite algebra of parts of [0,1] generated by π times F. Then the convergence of uπ to u in L2([0,1]×Ω) along a fixed increasing sequence of partitions follows from the martingale convergence theorem. For the familyuπ the argument of the proof is as follows.
Let π(k) be an increasing sequence of partitions such that |π(k)| → 0.
Set π(k) = {0 = tk0 < tk1 < ã ã ã < tknk = 1}. For any k we consider the σ-field Gk of parts of [0,1]×Ω generated by the sets (tki, tki+1]×F, where 0 ≤ i ≤ nk−1 and F ∈ F[tki,tki+1]c. Then notice that uπ(k) = E(u |Gk), where E denotes the mathematical expectation in the probability space [0,1]×Ω. By the martingale convergence theorem,uπ(k)converges to some element u in L2([0,1]×Ω). We want to show thatu=u. The difference v=uưuis orthogonal toL2([0,1]×Ω,Gk) for everyk. Consequently, for any fixedk≥1, such a processvsatisfies
I×Fv(t, ω)dtdP = 0 for anyF∈ F[tki,tki+1]c and for any intervalI⊂[tki, tki+1] inπ(m) withm≥k. Therefore,
E(v(t)|F[tki,tki+1]c) = 0 for all (t, ω) almost everywhere in [tki, tki+1]×Ω.
Therefore, for almost allt, with respect to the Lebesgue measure, the above conditional expectation is zero for any i, k such that t ∈ [tki, tki+1]. This implies thatv(t, ω) = 0 a.s., for almost allt, and the proof of the first part of the lemma is complete.
In order to show the convergence inL1,2we first compute the derivatives of the processes uπ anduπ using Proposition 1.2.8:
Druπ(t) =
n−1
i=0
1 ti+1−ti
ti+1
ti
Drusds
1(ti,ti+1](t), and
Druπ(t) =
n−1
i=0
1 ti+1−ti
ti+1
ti
E(Drus|F[ti,ti+1]c)ds
×1(ti,ti+1](t)1(ti,ti+1]c(r).
Then, the same arguments as in the first part of the proof will give the
desired convergence.
Now consider the Riemann sums associated to the preceding approxima- tions:
Sπ=
n−1
i=0
1 ti+1−ti
ti+1
ti
usds
(W(ti+1)−W(ti)) and
Sπ =
n−1
i=0
1 ti+1−ti
ti+1
ti
E(us|F[ti,ti+1]c)ds
(W(ti+1)−W(ti)).
Notice that from Lemma 1.3.2 the processes uπ are Skorohod integrable for any processuinL2([0,1]×Ω) and that
Sπ =δ(uπ).
On the other hand, for the process uπ to be Skorohod integrable we need some additional conditions. For instance, if u ∈ L1,2, then uπ ∈ L1,2 ⊂ Domδ, and we have
δ(uπ) =Sπ−
n−1 i=0
1 ti+1−ti
ti+1
ti
ti+1
ti
Dsutdsdt. (3.4) In conclusion, from Lemma 3.1.2 we deduce the following results:
(i) Let u∈L2([0,1]×Ω). If the familySπ converges inL2(Ω) to some limit, thenuis Skorohod integrable and this limit is equal toδ(u).
(ii) Let u∈L1,2. Then both families Sπ =δ(uπ) andδ(uπ) converge in L2(Ω) toδ(u).
Let us now discuss the convergence of the familySπ. Notice that Sπ =
1 0
utWtπdt, where
Wtπ =
n−1 i=0
W(ti+1)−W(ti)
ti+1−ti 1(ti,ti+1](t). (3.5) Definition 3.1.1 We say that a measurable process u={ut,0 ≤t ≤1} such that 1
0 |ut|dt < ∞ a.s. is Stratonovich integrable if the family Sπ converges in probability as|π| →0, and in this case the limit will be denoted by 1
0 ut◦dWt.
From (3.4) we see that for a given processuto be Stratonovich integrable it is not sufficient thatu∈L1,2. In fact, the second summand in (3.4) can be regarded as an approximation of the trace of the kernel Dsutin [0,1]2, and this trace is not well defined for an arbitrary square integrable kernel.
Let us introduce the following definitions:
Let X ∈ L1,2 and 1 ≤ p ≤ 2. We denote by D+X (resp. D−X) the element ofLp([0,1]×Ω) satisfying
nlim→∞
1 0
sup
s<t≤(s+1n)∧1
E(|DsXt−(D+X)s|p)ds= 0 (3.6) (resp.
nlim→∞
1 0
sup
(s−1n)∨0≤t<s
E(|DsXt−(D−X)s|p)ds= 0). (3.7) We denote byL1,2p+ (resp.L1,2p−) the class of processes inL1,2such that (3.6) (resp. (3.7)) holds. We setL1,2p =L1,2p+∩L1,2p− . ForX∈L1,2p we write
(∇X)t= (D+X)t+ (D−X)t. (3.8) Let X ∈ L1,2. Suppose that the mapping (s, t) ֒→ DsXt is continuous from a neighborhood of the diagonal Vε={|s−t|< ε} intoLp(Ω). Then X ∈L1,2p and
(D+X)t= (D−X)t=DtXt.
The following proposition provides an example of a process in the class L1,22 .
Proposition 3.1.1 Consider a process of the form Xt=X0+
t 0
usdWs+ t
0
vsds, (3.9)
where X0∈D1,2 andu∈D2,2(H), and v∈L1,2. Then, X belongs to the classL1,22 and
(D+X)t = ut +DtX0+ t
0
Dtvrdr+ t
0
DturdWr, (3.10) (D−X)t = DtX0+
t 0
Dtvrdr+ t
0
DturdWr. (3.11) Proof: First notice thatX belongs toL1,2, and for anyswe have
DsXt=us1[0,t](s) +DsX0+ t
0
Dsvrdr+ t
0
DsurdWr. Thus,
1 0
sup (s−n1)∨0<t≤s
E(|DsXt−(D−X)s|2)ds
≤ 2 n
1 0
s
(s−1n)∨0
E(|Dsvr|2)drds+ 2 1
0
s
(s−n1)∨0
E(|Dsur|2)drds +2
1 0
1 0
s
(s−n1)∨0
E(|DθDsur|2)drdsdθ,
and this converges to zero asntends to infinity. In a similar way we show that (D+X)t exists and is given by (3.10).
In a similar way,L1,2,fp− is the class of processesXinL1,2,f such that there exists an elementD−X∈Lp([0,1]×Ω) for which (3.7) holds. Suppose that Xt is given by (3.9), where X0 ∈ D1,2, u ∈ LF and v ∈ L1,2,f, then, X belongs to the classL1,2,f2− and (D−X)tis given by (3.11).
Then we have the following result, which gives sufficient conditions for the existence of the Stratonovich integral and provides the relation between the Skorohod and the Stratonovich integrals.
Theorem 3.1.1 Let u∈L1,21,loc. Thenuis Stratonovich integrable and 1
0
ut◦dWt= 1
0
utdWt+1 2
1 0
(∇u)tdt. (3.12)
Proof: By the usual localization argument we can assume thatu∈L1,21 . Then, from Eq. (3.4) and the above approximation results on the Skorohod integral, it suffices to show that
n−1
i=0
1 ti+1−ti
ti+1
ti
ti+1
ti
Dtusdsdt→ 1 2
1 0
(∇u)sds, in probability, as|π| →0. We will show that the expectation
E
n−1
i=0
1 ti+1−ti
ti+1
ti
dt ti+1
t
(Dtus)ds−1 2
1 0
D+u
tdt
converges to zero as|π| →0. A similar result can be proved for the operator D−, and the desired convergence would follow. We majorize the above expectation by the sum of the following two terms:
E
n−1 i=0
1 ti+1−ti
ti+1
ti
dt ti+1
t
(Dtus− D+u
t)ds
+E
ti+1
ti
n−1 i=0
ti+1−t ti+1−ti
D+u
tdt−1 2
1 0
D+u
tdt
≤ 1
0
sup
t≤s≤(t+|π|)∧1
E(|Dtus− D+u
t|)dt +E
1 0
D+u
t
n−1
i=0
ti+1−t ti+1−ti
1(ti,ti+1](t)−1 2
dt
. The first term in the above expression tends to zero by the definition of the class L1,21 . For the second term we will use the convergence of the functions n−1
i=0 ti+1−t
ti+1−ti1(ti,ti+1](t) to the constant 12 in the weak topology ofL2([0,1]). This weak convergence implies that
1 0
D+u
t
n−1
i=0
ti+1−t ti+1−ti
1(ti,ti+1](t)−1 2
dt
converges a.s. to zero as|π| →0. Finally, the convergence inL1(Ω) follows by dominated convergence, using the definition of the space L1,21 . Remarks:
1.If the mapping (s, t)֒→Dsut is continuous fromVε={|s−t|< ε}into L1(Ω), then the second summand in formula (3.12) reduces to1
0 Dtutdt.
2.Suppose thatX is a continuous semimartingale of the form (1.23). Then the Stratonovich integral of X exists on any interval [0, t] and coincides
with the limit in probability of the sums (1.24). That is, we have t
0
Xs◦dWs= t
0
XsdWs+1
2X, Wt,
where X, Wdenotes the joint quadratic variation of the semimartingale X and the Brownian motion. Suppose in addition thatX ∈L1,21 . In that case, we have (D−X)t= 0 (becauseX is adapted), and consequently, 1
0
D+X
tdt=X, W1= lim
|π|↓0 n−1
i=0
(X(ti+1)−X(ti))(W(ti+1)−W(ti)),
whereπdenotes a partition of [0,1]. In general, for processesu∈L1,21 that are continuous inL2(Ω), the joint quadratic variation of the processuand the Brownian motion coincides with the integral
1 0
( D+u
t− D−u
t)dt
(see Exercise 3.1.2). Thus, the joint quadratic variation does not coincide in general with the difference between the Skorohod and Stratonovich inte- grals.
The approach we have described admits diverse extensions. For instance, we can use approximations of the following type:
n−1
i=0
((1−α)u(ti) +αu(ti+1))(W(ti+1)−W(ti)), (3.13)
where αis a fixed number between 0 and 1. Assuming that u∈ L1,21 and E(ut) is continuous int, we can show (see Exercise 3.1.3) that expression (3.13) converges inL1(Ω) to the following quantity:
δ(u) +α 1
0
D+u
tdt+ (1−α) 1
0
D−u
tdt. (3.14) For α= 12 we obtain the Stratonovich integral, and forα= 0 expression (3.14) is the generalization of the Itˆo integral studied in [14, 30, 298].