fors,t ∈[0,T]and thus it is sufficient to prove the existence of the quadratic varia- tion ofSi(ω)andSi(ω) +Sj(ω)for 1≤i,j≤dwithi6= j. For typical price paths this can be done precisely as in the proof of Theorem 3.25with the only exception that the bounds (a)-(b) and (1)-(4) change by a multiplicative constant depending only on the dimensiond.
3.3 Föllmer’s Pathwise Integral and Model-Free
Definition 3.38. Let π and x be as above. We say that x possesses a quadratic variation along the sequence of partition π and this quadratic variation is defined as
[x]tπ :=ξ[0,t]. (3.3.2)
By the assumption that for anyt∈[0,T],ξ({t}) = (∆xt)2, the following decom- position is justified
[x]tπ := [x]tπ,cont.+
∑
0<s≤t
(∆xs)2 (3.3.3)
where[x]π,cont.will be called the continuous part of[x]π.
Theorem 7. [17] If x:[0,T]→Ris càdlàg, possesses the quadratic variation along the sequence of partitionsπand f :R→Ris of the class C2, then
f(xT) = f(x0) + Z T
0+
f0(xs−)dxs+1 2
Z T
0+
f00(xs−)d[x]π,cont.s
+
∑
0<s≤T
{∆f(xs)−f0(xs−)∆xs},
where the integral Z T
0+
f0(xs−)dxsis defined as the limit Z T
0+
f0(xs−)dxs= lim
n→+∞
Nn
i=1
∑
f0
xtn
i−1
x(tin)−x(ti−1n )
. (3.3.4)
The following result appear in our paper [20].
Theorem 8. Letω,ω˜ :[0,T]→R be two càdlàg paths. Assume thatω˜ has finite total variation and let us consider two integrals
1. the integral Z
(0,T]
ω(t−)d ˜ω(t)understood as the Lebesgue-Stieltjes integral (with respect to the measured ˜ω given byd ˜ω(a,b]:=ω˜(b)−ω˜(a));
2. the integral (F) Z
(0,T]
ω(t−)d ˜ω(t) understood as Föllmer’s integral along the sequence of partitions(τn)n∈Nsuch that for n∈N,τncontains nth Lebesgue partition of the pathω, i.e.
(F) Z
(0,T]
ω(t−)d ˜ω(t):= lim
n→+∞
kn
i=1
∑
ω τi−1n n ω τ˜ in
−ω τ˜ i−1n o ,
whereτn:=n
0=τ0n<τ1n< . . . <τknn−1<T =τknn <+∞=τknn+1=τknn+2=. . .o andπn(ω)⊂τn.
These two integrals coincide, provided that the latter exists.
Proof. Step 1. First, let us notice that for anyε>0 we may uniformly approximate ω by a step function
ωε(t):=
N i=1
∑
ω(ti−1)1[ti−1,ti)(t) +ω(T)1{T}(t), where 0=t0<t1< . . . <tN=T,such that
ωε−ω ∞≤ε.
To obtain suchωεwe simply definet0:=0,ti:=infn
t>ti−1:
ω(t)−ω(ti−1) >ε
o fori=1,2, . . .such thatti−1<+∞(we apply the convention that inf /0= +∞) and N:=max
i∈ {1,2, . . .}:ti−1<T . Step 2. We have the estimate
Z
(0,T]
ω(t−)d ˜ω(t)− Z
(0,T]
ωε(t−)d ˜ω(t)
≤ Z
(0,T]
ω(t−)−ωε(t−)
d ˜ω(t)
≤ε Z
(0,T]
d ˜ω(t)
=εTV ˜ω,[0,T]
, (3.3.5)
where TV ˜ω,[0,T]
denotes the total variation of ˜ω.Moreover Z
(0,T]
ωε(t−)d ˜ω(t) =
N i=1
∑
ωε(ti−1) ω˜(ti)−ω˜(ti−1) . We also have
kn
∑
i=1
ω τi−1n n
˜ ω τin
−ω τ˜ i−1n o
−
kn
∑
i=1
ωε τi−1n n
˜ ω τin
−ω τ˜ i−1n o
≤
kn
∑
i=1
ω τi−1n
−ωε τi−1n ω τ˜ in
−ω τ˜ i−1n
≤εTV ˜ω,[0,T]
. (3.3.6)
Step 3. Now let us consider the difference Z
(0,T]ωε(t−)d ˜ω(t)−
kn
∑
i=1
ωε τi−1n n ω τ˜ in
−ω τ˜ i−1n o
=
N i=1
∑
ωε(ti−1) ω˜(ti)−ω˜(ti−1)
−
kn i=1
∑
ωε τi−1n n ω τ˜ in
−ω τ˜ i−1n o
.(3.3.7)
Letτn(t)denotes the first pointτinin the partitionτnsuch thatτin≥t.From the definition of thenth Lebesgue partition ofω it follows that fort<T
lim sup
n→+∞ τn(t)≤inf
u>t :ω(u)6=ω(t−) .
By the definition of timest1,t2, . . . ,tN−1 we have that for anyt∈ {t1,t2, . . . ,tN−1}, ω(t)6=ω(t−)orω(t) =ω(t−)butω is not constant on any interval of the form [t,u],u∈(t,T]and lim
n→+∞πn(t) =t.Thus, for sufficiently largens ti≤τn(ti)<ti+1fori=1,2, . . . ,N−1.
Now, denoting by kn(t)such index that τn(t) =τkn
n(t), for sufficiently largens andi=2, . . . ,N we haveti−2≤τkn
n(ti−1)−1<ti−1. Thusωε
τkn
n(ti−1)−1
=ωε(ti−2) and sinceωε is constant on[ti−1,ti),i=1,2, . . . ,N,we obtain
N i=1
∑
ωε(ti−1) ω˜(ti)−ω˜(ti−1)
−
kn i=1
∑
ωε τi−1n n ω τ˜ in
−ω τ˜ i−1n o
=
N
∑
i=2
ωε(ti−1)−ωε(ti−2)
ω τ˜ n(ti−1)
−ω˜(ti−1)
. (3.3.8)
Since lim
n→+∞τn(t) =t fort ∈ {t1,t2, . . . ,tN−1} and ˜ω is càdlàg, we finally get that the difference (3.3.7) tends to 0 asn→+∞.
From this and (3.3.5), (3.3.6) (taking ε as close to 0 as we wish) we get the assertion.
The multi-dimensional Föllmer’s integration by parts formula is given by:
SiT(ω)STj(ω)−Si0(ω)S0j(ω) = Z
(0,T]
St−i (ω)dStj(ω) +
Z
(0,T]
St−j (ω)dSti(ω) + [Si,Sj](ω), (3.3.9) where f(x1,· · ·,xd) =xixj, i,j∈ {1,2,· · · }and the integral
Z
(0,T]
Sit−(ω)dStj(ω)
denotes Föllmer’s pathwise integral, see [17]. The corresponding model-free ver- sion is given by:
ωi(t)ωj(t)−ωi(0)ωj(0) = (F) Z
(0,t]
ωi(s−)dωj(s) + (F) Z
(0,t]
ωj(s−)dωi(s)
+[Si,Sj]t(ω), (3.3.10)
where (F) Z
(0,t]
. . .denotes Föllmer’s pathwise integral (along the sequence of the
Lebesgue partitions), t ∈[0,T] and ω ∈D
[0,T],Rd
possesses quadratic varia- tion along the same partitions. This means that the sequence of discrete quadratic (co)variation, see (3.2.9), converges for i,j=1,2,· · ·,d in the uniform topology [Si,Sj](ω).
Notice also that for typical price pathω, Föllmer’s integral in (3.3.10) coincides with the model-free Itô integral
Z
(0,t]
Sis−(ω)dSsj(ω). The latter may be expressed as the limit of the sums of the form
∞
∑
k=1
Siπn
k−1(ω)Sπjn
k−1∧t,πkn∧t(ω) and
∞ k=1
∑
Siπn
k−1(ω)1(πk−1n ,πkn](t) converges uniformly to St−i (ω) fort ∈(0,T]. Using Theorem3.23, in particular, continuity results (3.2.12), the distanced∞,ψε between Föllmer’s integral and the model-free Itô integral is 0. This implies that for typical price paths the two integrals coincide. Thus (3.3.10) may be also viewed as the integration by parts formula for the model-free Itô integral.
Next we prove the multi-dimentional model-free Itò fourmula for càdlàg price paths.
Lemma 3.39. LetS˜:Ω×[0,T]→Rd be such that the processS is adapted (to the˜ filtration(Ft)t∈[0,T]), and for anyω∈Ω,ω˜ :=S˜(ω)is finite variation, càdlàg path on[0,T].Then for typical pathω ∈Ωand i,j=1,2, . . . ,d, t ∈[0,T], the following integration by parts formula holds
ωi(t)ω˜j(t)−ωi(0)ω˜j(0) = Z
(0,t]
S˜s−j (ω)dSis(ω) + Z
(0,t]ωi(s−)d ˜ωj(s)
+
∑
0<s≤t
∆ωi(s)∆ω˜j(s), (3.3.11) where
Z
(0,t]
S˜s−j (ω)dSis(ω)denotes the model-free Itô integral and Z
(0,t]
ωi(s−)d ˜ωj(s) denotes the Lebesgue-Stieltjes integral which coincides with the Föllmer integral.
Proof. Consider Föllmer’s pathwise integration by parts formula (3.3.10). Using the fact that for typical price paths, model-free Itô integral
Z
(0,t]
S˜s−j (ω)dSis(ω)
coincides with Föllmer’s integral (along the sequence of the Lebesgue partitions (πn)n∈Nof(ω,ω˜)∈R2d) and that since (by Lemma8), Föllmer’s integral
(F) Z
(0,t]
ωi(s−)d ˜ωj(s)
(along the same sequence of partitions) coincides with the classical Lebesgue-Stieltjes integral
Z
(0,t]
ωi(s−)d ˜ωj(s),
to prove this Lemma, we need only to show that fori,j=1,2, . . . ,d,t ∈[0,T], the sequence of discrete quadratic (co)variation
∞
∑
k=1
Siπn
k−1∧t,πkn∧t(ω)Sj
πk−1n ∧t,πkn∧t(ω˜) =
∞
∑
k=1
Siπn
k−1∧t,πkn∧t(ω)S˜j
πk−1n ∧t,πkn∧t(ω) converges to
0<s≤t
∑
∆ωi(s)∆ω˜j(s)
uniformly int. The prove of this Lemma relies on the properties of Lebesgue par- titions and the Schwartz inequality. Let ε >0 be such that there is no jump of ˜ω of size equalε and let Iε,n,n∈N, be the sequence of all indicesk∈Nfor which
Sj
πk−1n ∧t,πkn∧t(ω˜)
>ε.We have
∞ k=1
∑
Siπn
k−1∧t,πkn∧t(ω)Sπjn
k−1∧t,πkn∧t(ω˜)−
∑
k∈Iε,n
Siπn
k−1∧t,πkn∧t(ω)Sπjn
k−1∧t,πkn∧t(ω˜)
=
∑
k∈N\Iε,n
Siπn
k−1∧t,πkn∧t(ω)Sπjn
k−1∧t,πkn∧T(ω˜)
≤ s
∑
k∈N\Iε,n
Si
πk−1n ∧t,πkn∧t(ω)2 s
∑
k∈N\Iε,n
Sj
πk−1n ∧t,πkn∧t(ω˜)2
≤√ ε
s ∞
∑
k=1
Sj
πk−1n ∧t,πkn∧t(ω)˜
s ∞
∑
k=1
Si
πk−1n ∧t,πkn∧t(ω)2. Notice that
∞ k=1
∑
Sj
πk−1n ∧t,πkn∧t(ω˜)
is bounded by the total variation of ˜ω while
∞
∑
k=1
Siπn
k−1∧t,πkn∧t(ω)2
!
t∈[0,T]
converges to the quadratic variation ofωi asn→+∞(the convergence still holds though partitions (πn)n∈N may be finer than the Lebesgue partitions ofω, see the proofs of [31, Corollary 3.11], [47, Theorem 2]). Finally notice that
n→∞lim
∑
k∈Iε,n
Sπin
k−1∧t,πkn∧t(ω)Sπjn
k−1∧t,πkn∧t(ω) =˜
∑
0<s≤t,|∆ω˜(s)|>ε
∆ωi(s)∆ω˜j(s). (recall that there is no jump of ˜ω of size equalε). Since there is only finite number of jumps of ˜ω of size greater thanε and ε >0 may be as close to 0 as we wish (since ˜ω has only countable number of jumps), the result follows.