Model-Free Approach with Càdlàg Price Paths

In particular, for k=−1,0,1,2,· · ·, the sequence of Lebesgue partition is de- fined by:

T₋₁ⁿ (ω):=0,

T₀ⁿ(ω):=inf{t≥0|ω(t)∈Dn},

T_kⁿ(ω):=inf{t≥T_k−1ⁿ |ω(t)∈Dnandω(t)6=ω(T_k−1ⁿ )}, k=1,2,· · · and the set of grids is defined as follows,

Dn:={k2⁻ⁿ|k∈Z}.

Proposition 3.9. For typicalω, the function

[ω]:t∈[0,∞)→[ω]_t:= [ω]_t= [ω]

t (3.1.8)

exists, and is an increasing element ofΩ, satisfying[ω]₀=0.

Proposition3.9 follows from Theorem3.25 which is proved in Section3.2. It states that quadratic variation,[ω]_t, exists along[0,∞)where

[ω]_t:=lim sup

n→∞

[ω]_tⁿ, [ω]

t:=lim inf

n→∞ [ω]ⁿ_t.

It represents the integrated volatility of ω over the time period [0,t]. However, Definition3.8relies on the arbitrary choiceDnof sequence of grids, [46]. Moreover, price paths of typicalω are continuous functions. We now suspend the continuous model free approach and turn our attention to the càdlàg case.

The properties of continuous price paths for model free approach are well un- derstood in the literature, see [34,35,44, 45,46]. For example, typical continuous model free price paths reveals several properties of local martingales. More pre- cisely, Vovk and Shafer in the recent book [50] defined the set Gλ of nonnegative simple capital processes, closed under the limit of the infimum, to be nonnegative supermartingales with respect to the filtration, see also [48]. When this set is closed under limit, the notion of supermartingales coincide with the notion of continuous martingales.

downwards, that isω = (ω¹, . . . ,ω^d)∈Ω_ψ if fori=1, . . . ,dit satisfies

∆ωⁱ(t):=ωⁱ(t)−ωⁱ(t−)≥ −ψ sup

s∈[0,t)

|ω(s)|

!

, (3.2.1)

wheret∈(0,T]andωⁱ(t−):= lim

s→t,s<tω(s).

The following sample spaces are examples ofΩ:

i. Ω_c:=C([0,T],R^d), the space of all continuous functionsω: [0,T]→R^d, ii. Ω₊:=D([0,T],R^d+), the space of all non-negative càdlàg functionsω: [0,T]→

R^d+ (hereψ(x) =x),

iii. ˜Ω_ψ which is defined as the subset of all càdlàg functionsω: [0,T]→R^d such that

|ω(t)−ω(t−)| ≤ψ sup

s∈[0,t)

|ω(s)|

!

, t∈(0,T],

andψ: R+→(0,∞)is a fixed non-decreasing function.

The choice ofΩrests on the investor’s beliefs.

We say that an event happena.s. if the trader has a strategy that multiplies his capital by an infinite factor if the event fails. Stopping timesτ :Ω→[0,T]∪ {∞}

with respect to the filtration (Ft)_t∈[0,T] and the corresponding σ-algebra Fτ are defined as usual, see Definition 2.1. In particular, we consider the sequence of partitions 0≤τ₁≤τ₂≤ · · ·. Then the correspondingn^thpartition is:

τⁿ:=0≤τ₁ⁿ<τ₂ⁿ<· · ·τ_kⁿ_n−1<T =τ_kⁿ_n <+∞=τ_kⁿ_n+1=τ_kⁿ_n+2=· · · (3.2.2) for somek=1,2,· · · We also denote by

h_k:Ω→R^d (3.2.3)

theFτ_k-measurable bounded functions. Recall that the first step to construct a clas- sical stochastic integral is to define step-functions (also known as simple process).

Definition 3.10. Let us consider the process H :Ω×[0,T]→ R^d. Then H is called a simple strategy if there exist stopping times 0=τ₀≤τ₁ ≤. . . and Fτk- measurable bounded functions h_k :Ω→R^d, k∈N0, such that for each ω ∈Ω, τ_K(ω) =τ_K+1(ω) =. . .∈[0,T]∪ {∞}from some K∈N0on, and

H_t(ω) =h₀(ω)1{0}(t) +

+∞

k=0

∑

h_k(ω)1(^τk(ω),τk+1(ω)] (t).

Definition 3.11. Let H be a simple trading strategy and λ >0. Then the corre- sponding simple capital (integral) process with the initial capitalλ is

(H·S)^λ_t (ω):=λ+

∞

∑

n=0

h_n(ω)(S_τn+1∧t(ω)−S_τn∧t(ω))

=

∞

∑

n=0

h_k(ω)·S_{τk∧t,τk+1∧t}(ω). (3.2.4) This integral is well defined for all ω ∈ Ω and for all t ∈[0,T]. The scalar product inR^dis denoted by “·” andS_u,v:=S_v−S_u. Forλ=0, we will simply write

(H·S)^λ = (H·S). (3.2.5)

We denote byH the family of simple trading strategies.

Definition 3.12. Let H be a simply trading strategy andλ ∈R. Then H is called (strongly)λ-admissible if

(H·S)^λ_t (ω)≥0 for allω ∈Ωand all t ∈[0,T]. (3.2.6) The family ofλ-admissible strategies is denoted byHλ.

Definition 3.13. Vovk’s outer measureP¯ of a setA ⊆Ωis defined as the minimal superhedging price for1A, that is

P¯(A):=inf{λ ∈R:∃(Hⁿ)_n∈N⊂Hλ s.t∀ω ∈Ω: lim inf

n→∞ (Hⁿ·S)^λ_T(ω)≥1_A(ω)}.

Definition3.13is a slightly modified Vovk’s outer measure, see [46, Definition 2.3] . It denotes the minimal super-hedging price of a set of possible scenarios A ∈Ω. Also, it dominates all local martingales measures on the sample spaceΩ [31]. Moreover, it is related to the notion of the no-arbitrage opportunities of the first kind (NA1). This precludes very large profit with small risk.

Definition 3.14. A setA ∈Ωis null ifP¯(A) =0.

Naturally,A in Definition3.14maybe deemed an arbitrage opportunity scenar- ios in the classical mathematical finance, see [31, Proposition 2.6].

Definition 3.15. A property ofω ∈ΩholdsP¯-almost surely or for typicalω if the setA ofω where it fails is null.

Definition 3.16. A probability measureP onΩadmits no arbitrage opportunities of the first kind if

c→+∞lim sup

H∈H¹P

(H·S)¹_T ≥c

=0. (3.2.7)

If the coordinate process on(Ω,(Ft)_0≤t≤T,P)satisfies (NA1), andA ∈FT is null, i.e. (P¯(A) =0), thenP(A) =0.

Since we will need some continuity results of the model-free integrals, we will use measure of Definition3.13, rather than original Vovk’s outer measure. However, this measure is defined with the use of strongly λ-admissible strategies. A more useful version of weaklyλ-admissible strategies in the proof of quadratic variation is defined below.

Definition 3.17. Letλ ≥0 and H be a simple trading strategy. Then H is called weaklyλ-admissible strategy if for all(t,ω)∈[0,T]×Ω_ψ, the following holds:

(H·S)_t(ω)≥ −λ(1+|S_{ρ λ(H)}(ω)|1[ρ λ(H)(ω),T](t)), (3.2.8) where

ρ λ(H)(ω):=inf{t∈[0,T]:(H·S)_t(ω)≤ −λ} and H_t(ω) =H_t(ω)1[0,ρ λ(H)(ω)∧T](ω).

Intuitively, usingHas a strategy, one expects a payoff larger than−λ. However, there is a risk of losing all previous gains plusλ(|S_t|)through one large jump. Now the class of weaklyλ-admissible strategies is denoted byGλ. Let us now introduce the notion ofnth Lebesque partitionπⁿ,n∈N, adapted to càdlàg setting which we will use in the sequel.

Forn∈Nwe define

Dⁿ:={k2⁻ⁿ : k∈Z}.

For f ∈D [0,T],R

, πⁿ(f)consists of points in time at which the path f crosses (in space) a dyadic number from Dⁿ which is not the same as the dyadic number crossed (as the last number from Dⁿ) at the preceding time. This idea is made precise in the following definition.

Definition 3.18. Let n∈Nand letDⁿbe the n^th generation of dyadic numbers. For f ∈D [0,T],R

its n^th Lebesgue partitionπⁿ(f):={π_kⁿ(f) : k∈N0}is given by the sequence of times(π_kⁿ(f))_k∈N0 inductively defined by

π₀ⁿ(f):=0 and Dⁿ₀(f):=max{Dⁿ∩(−∞,S₀(f)]}, and for every k∈Nwe further set

π_kⁿ(f):=inf{t∈[π_k−1ⁿ (f),T] : Jf π_k−1ⁿ (f)

,f(t)K∩(Dⁿ\

Dⁿ_k−1(f) )6= /0}

Dⁿ_k(f):= arg min

D∈Jf(π_k−1ⁿ (f))^,f(π_kⁿ(f))K∩(Dⁿ\n

Dⁿ_k−1(f)o )

|D−f π_kⁿ(f)

|

with the conventioninf /0=∞andJu,vK:= [min(u,v),max(u,v)].

Next we define the sequence of Lebesgue partitions for d-dimensional càdlàg functionω∈Ω.

Definition 3.19. For n∈Nandω ∈D

[0,T],R^d

its Lebesgue partitionπⁿ(ω):=

{π_kⁿ(ω) : k∈N0}is iteratively defined byπ₀ⁿ(ω):=0and π_kⁿ(ω):=min

(

t >π_k−1ⁿ (ω) : t∈

d

[

i=1

πⁿ(ωⁱ)∪

d

[

i,j=1,i6=j

πⁿ(ωⁱ+ω^j) )

, k∈N,

where ω = (ω¹, . . . ,ω^d)andπⁿ(ωⁱ)andπⁿ(ωⁱ+ω^j)are the Lebesgue partitions ofωⁱandωⁱ+ω^jas introduced in Definition3.18, respectively.

Note that πⁿ defined in Definition 3.19 is indeed a stopping times, see [47, Lemma 3].

Definition 3.20. Letω∈Ω. Then the sequence of discrete quadratic (co)variations along the sequence of Lebesgue partitions is:

h Sⁱ,S^j

i

t(ω):=

∞

∑

k=1

Sⁱ_πn

k−1∧t,π_kⁿ∧t(ω)S_π^jn

k−1∧t,π_kⁿ∧t(ω), t∈[0,T]. (3.2.9) For typical price pathsω ∈Ω, the sequence of quadratic variation (3.2.9) con- verges fori,j=1,2, . . . ,din the uniform topology to some (càdlàg) function,[0,T]3 t7→[Sⁱ,S^j]_t(ω), which is proven in Subsection3.2.2, which we will call the quadratic (co)variation ofSⁱandS^j.

We will use the following notation:

[S]_T(ω) =





d i,

∑

j=1

[Sⁱ,S^j]²_T(ω)





1 2

.

ForZ:Ω×[0,T]→R^r (r=1,2, . . .) let us define Z(ω)

∞:= sup

0≤t≤T

Z_t(ω) ,

where| · | denotes the Euclidean norm onR^r. Following [31] we will identify two processesX,Y :Ω×[0,T]→R^r if

P¯

ω ∈Ω:

X(ω)−Y(ω)

∞>0

=0.

This defines an equivalence relation, and we will write L₀(R^r) (orL₀in short) for the space of its equivalence classes. We equip the spaceL₀(R^r)with the distance

d_∞(X,Y):=E[kX−Yk_∞∧1], (3.2.10) whereEdenotes an expectation operator defined forZ: Ω→[0,∞]by

E[Z]:=inf

λ >0 :∃(Hⁿ)_n∈N⊆Hλ s.t.∀ω∈Ω_ψ : lim inf

n→∞ (λ+ (Hⁿ·S)_T(ω))≥Z(ω)

.

It can be shown that(L₀(R^r),d_∞)is a complete metric space and(D(R^r),d_∞)is a closed subspace, where D(R^r)are those processes inL₀(R^r)which have a càdlàg representative.

Definition 3.21. Letε >0. A sequence X_n∈L₀ converges in the outer measureP¯ on a set A⊂Ωto X ∈L₀if

n→+∞lim P¯

ω ∈A:

X_n(ω)−X(ω) ∞>ε

=0.

Definition 3.22. Let q,M>0. Then Ωq,M:=n

ω ∈Ω :|[S]_T(ω)| ≤q, S(ω)

∞≤M o

.

Note that if for anyq,M>0,X_nconverges in outer measure ¯Pon the setΩ_q,Mto X∈L₀thenXis unique. For fixedq,M>0 let us now introduce the pseudo-distance d_∞,q,M onL₀which is given by

d_∞,q,M(X,Y):=E[kX−Yk_∞∧1Ωq,M].

Now let us define for some fixedε ∈(0,1)the following metric onL₀ d_∞,ψ^ε (X,Y):=

∞ n,m=1

∑

2−(n/2+m)(1+ε)(ψ(2^m)∨2^m)⁻¹d_∞,2ⁿ_,2^m(X,Y) (3.2.11) One of the main results of [31] is the following Theorem.

Theorem 3.23. There exists two metric spaces (H₁,d_H

1) and (H₂,d_H

2) such that the (equivalence classes of) step functions (simple strategies) are dense in H₁, H₂ embedds intoD(R^d)and the integral map I: H 7→(H·S) =

Z

(0,t]

H_sdS_s

!

t∈[0,T]

, defined for simple strategies in Equation (3.2.4), has a continuous extension that maps(H₁,d_H

1)to(H₂,d_H

2). Moreover, one has the following continuity estimate d_∞,ψ^ε ((F·S),(G·S)).d_∞(F,G)^1/3, (3.2.12) and one can take d_H

1 =d_∞and d_H

2=d_∞,ψ^ε . The metric space (H₁,d_H

1) can be chosen to contain the left-continuous ver- sions of adapted càdlàg processes, see [31, Remark 4.3]. If we replace the fil- tration (Ft)_t∈[0,T] by its right-continuous version, we can define (H₁,d_H

1) such that it contains at least the càdlàg adapted processes and, furthermore, such that if (F_n)_n∈N⊂H₁ is a sequence with sup

ω∈Ω

kF_n(ω)−F(ω)k_∞→0 asn→+∞, then F ∈H₁and there exists a subsequence(F_nk)with

k→∞limk(F_nk·S)(ω)−(F·S)(ω)k_∞=0

for typical price paths ω ∈Ω. By [31, Corollary 4.9] and Inequality (3.2.12) we also get

Corollary 3.24. For a,q,m,M>0and any H∈H₁one has P¯

nk(H·S)k_∞≥ao

∩n

kHk_∞≤mo

∩n

|[S]_T| ≤qo

∩n

kSk_∞≤Mo

≤(1+3dM+2dψ(M))6√

q+2+2M

a m.

3.2.1 Pathwise Quadratic Variation of Càdlàg Price Paths

In this subsection we show detailed proof of the existence of quadratic variation for typical càdlàg price paths. We follow closely [31]. Recall the definition of the quadratic (co)variation, see Equation (3.2.9). The corresponding one-dimensional quadratic variation is given by:

[ω]_tⁿ:=

∞

∑

k=1

S_τⁿ

k∧t(ω)−S_τⁿ

k−1∧t(ω) 2

.

Thus we state the following theorem.

Theorem 3.25. For typical price pathsω∈Ω_ψ⊆D([0,T],R)the discrete quadratic variation

[S]ⁿ_t(ω):=

∞

∑

k=1

S_τⁿ

k∧t(ω)−S_τⁿ

k−1∧t(ω)2

, t∈[0,T],

along the Lebesgue partitions (π_n(ω))_n∈N converges in the uniform metric to a function[S](ω)∈D([0,T],R+).

To prove the existence of this quantity, several auxiliary results are involved.

First we define an auxiliary set of the class of trading strategies in the spirit of the minimal superhedging price.

Definition 3.26. Letλ >0and the setA ∈Ω_ψ. The set functionQ is given by:¯ Q(¯ A):=inf

n

λ >0 :∃(Hⁿ)_n∈N⊂G_λs.t.∀ω∈Ω_ψ :

n→∞liminf(λ+ (Hⁿ·S)_T(ω) +λ111·S_{ρ λ(H}n)(ω)1{ρ λ(Hⁿ)<∞}(ω))

≥1A(ω)o

, (3.2.13)

where111 := (1,· · ·,1)∈R^d.

Lemma 3.27. IfA ∈Ωψ,K:={ω ∈Ω_ψ :||ω||_∞≤K}for K∈R+, then Q(¯ A)≤P¯(A)≤(1+3dK+2dψ(K))Q(¯ A).

We refer to [31, Lemma 2.9] for the proof of this Lemma. Next we analyse the crossing behaviour of typical price paths with respect to the dyadic level Dⁿ. This process rest on the Doob’s inequality.

Definition 3.28. Let f :[0,T]→Rbe a càdlàg function,(a,b)⊂Rbe an open non- empty interval and t ∈[0,T]. The number U_t^(a,b)(f)of upcrossings of the interval (a,b)by the function f during the time interval[0,t]is given by

U_t^(a,b)(f):=sup

n∈N

sup

0≤s1<t1<···<sn<tn≤t n

∑

i=1

I(f(s_i),f(t_i)), where

I(f(s_i),f(t_i)):=

(1 if f(s_i)≤a and f(t_i)≥b, 0 if otherwise.

The number D^(a,b)_t (f)of downcrossings is defined analogously. For h>0we also introduce the accumulated number of upcrossing respectively downcrossings by

U_t(f,h):=

∑

k∈Z

U(kh,(k+1)h)

t (f) and D_t(f,h):=

∑

k∈Z

D(kh,(k+1)h)

t (f).

The following stopping times γK(ω):=inf

t∈[0,T] : |S_t(ω)| ≥K (3.2.14) forω ∈Ω_ψ andK∈Nplay a vital role in the derivation of Doob’s inequality.

Lemma 3.29. Let K >0. For each n∈N, there exists a strongly 1-admissible simple strategy Hⁿ∈H1such that

(Hⁿ·S)_t(ω)≥ U_t(ω,₂¹n) 2²ⁿ

2K(2K+ψ(K))−1 (3.2.15) for all t ∈[0,T]and everyω ∈ {ω∈Ω_ψ : kωk_∞<K} ⊆D([0,T],R).

The proof of this lemma follows a recursive algorithm on the open interval (a,b)⊂ R. The trader in this case buys one unit immediately when he realizes that the security price,S_t(ω), drops below theaand sell the security when the price is aboveb.

Proof. LetU_t^(a,b)(ω)be Doob’s upcrossings over an interval(a,b)⊆[−K,K]. Foll- wing the algorithm:

i. Buy one unit immediately whenS_t(ω)<a.

ii. Sell whenS_t(ω)>b.

Carry out this process until the terminal timeTor until we leave the interval(−K,K), whatever occurs first, we obtain a simple strategyH^(a,b)∈Ha+K+ψ(K)with

a+K+ψ(K) + (H^(a,b)·S)_t∧γK(ω)≥(b−a)U_t∧γ^(a,b)_K(ω), (t,ω)∈[0,T]×Ω_ψ. For a formal construction of H^(a,b) we refer to [47, Lemma 4.5]. Note that we need the predictable bound of the jump size given by ψ to guarantee the strong admissibility ofH^(a,b). Set now

Hⁿ:= ∑k∈Z,(k+1)2⁻ⁿ<K,k2⁻ⁿ>−KH^{(k2−n,(k+1)2−n)} K2ⁿ⁺¹(2K+ψ(K)) .

SinceH^{(k2−n,(k+1)2−n)}∈Hk2⁻ⁿ+K+ψ(K)⊆H2K+ψ(K) for allkwith(k+1)2⁻ⁿ<K andk2⁻ⁿ>−K, we haveHⁿ∈H1, and

1+ (Hⁿ·S)_t(ω)(ω)≥

∑_k∈Z,(k+1)2⁻ⁿ_<K,

k2⁻ⁿ>−K

2⁻ⁿU^{(k2−n,(k+1)2−n)}

t(ω) (ω)

K2ⁿ⁺¹(2K+ψ(K))

= U_t(ω,2⁻ⁿ) 2²ⁿ2K(2K+ψ(K)) for eacht∈[0,T]and allω ∈ {ω ∈Ω_ψ : kωk_∞<K}.

The following corollary proves that both the upward crossing and downward crossing are bounded. The prove rest on the following lemma called Borel-Cantelli lemma.

Lemma 3.30. Let(A_j)_j∈N⊆Ω_ψ be a sequence of events. If

∞ j=1

∑

P¯(A_j)<∞, then

P¯





∞

\

i=1

∞

[

i=j

A_j



≤lim

i→∞inf ¯P





∞

[

j=i

A_j



≤lim

i→∞inf

∞

∑

j=i

P¯(A_j) =0. (3.2.16)

Thus, the pathwise version of Doob’s upcrossing lemma enables us to control the number of level crossings of typical càdlàg price paths belonging toΩ_ψ. Corollary 3.31. For typical price paths ω ∈ Ω_ψ ⊆ D([0,T],R) there exists an N(ω)∈Nsuch that

U_T(ω,2⁻ⁿ)≤n²2²ⁿ and DT(ω,2⁻ⁿ)≤n²2²ⁿ for all n≥N(ω).

Proof. Since for eachk∈Z,U_t^{(k2−n,(k+1)2−n)}(ω)and D^(k2

−n,(k+1)2⁻ⁿ)

t (ω)differ by

no more than 1, we have|U_T(ω,2⁻ⁿ)−DT(ω,2⁻ⁿ)| ∈[0,2ⁿ⁺¹K]for alln∈Nand for every ω ∈Ωψ with sup

t∈[0,T]

|S_t(ω)|<K. So if we show that P(B_K) =0 for all K∈N, where

B_K:= ^\

m∈N

[

n≥m

A_K,n with

A_K,n= (

ω ∈Ω_ψ : sup

t∈[0,T]

|S_t(ω)|<KandU_T(ω,2⁻ⁿ)≥n²2²ⁿ 2

) ,

then our claim follows from the countable subadditivity ofP. But using Lemma3.29 we immediately obtain that

P(A_K,n)≤n⁻²[2K(2K+ψ(K))]

and applying the Borel-Cantelli Lemma3.30, we obtain thatP(B_K) =0, since it is summable.

To prove the convergence of the discrete quadratic variation processes([S]ⁿ)_n∈N, we shall show that the sequence ([S]ⁿ)_n∈N is a Cauchy sequence in the uniform metric onD([0,T],R+). For this purpose, we define the auxiliary sequence(Zⁿ)_n∈N by

Z_tⁿ:= [S]ⁿ_t −[S]ⁿ⁻¹_t , t∈[0,T].

Similarly as in Vovk [47], the proof of Theorem3.25 is based on the sequence of integral processes(K ⁿ)_n∈Ngiven by

Ktⁿ:=n⁴2⁻²ⁿ+2⁻ⁿ⁺⁵(K+ψ(K))²+ (Z_tⁿ)²−

∞ k=1

∑

(Z_τⁿn

k∧t−Z_τⁿn

k−1∧t)², t∈[0,T], (3.2.17) forK∈N, and the stopping times

σ_Kⁿ:=min (

τ_kⁿ :

k

∑

i=1

Z_τⁿn

i −Z_τⁿn i−1

2

>n⁴2⁻²ⁿ )

∧minn

τ_kⁿ : Z_τⁿn k >Ko

, n∈N.

(3.2.18) The next lemma states that eachK ⁿ is indeed an integral process with respect to a weakly admissible simple strategy, cf. [47, Lemma 5].

Lemma 3.32. For each n∈Nand K∈N, there exists a weakly admissible simple strategy L^K,n∈Gn⁴2⁻²ⁿ+2⁻ⁿ⁺⁵(K+ψ(K))² such that

K_γⁿ_K∧σKⁿ_∧t=n⁴2⁻²ⁿ+2⁻ⁿ⁺⁵(K+ψ(K))²+ (L^K,n·S)_t, t∈[0,T].

Proof. For eachK∈Nand each n∈N[47, Lemma 5] shows the equality for the strategy

L_t^K,n:=1(0,γ_K∧σ_Kⁿ](t)

∑

k

(−4)Z_τⁿn k(S_τⁿ

k−S_χn−1(τ_kⁿ))1(τ_kⁿ,τ_k+1ⁿ ](t), t∈[0,T], where

χⁿ⁻¹(t):=max n

τ_kⁿ⁻¹0 : τ_kⁿ⁻¹0 ≤t o

. SinceL^K,nis obviously a simple strategy, it remains to prove that

L^K,n∈Gn⁴2⁻²ⁿ+2⁻ⁿ⁺⁵(K+ψ(K))². (3.2.19) First we observe up to time ˜τⁿ:=max{τ_kⁿ : τ_kⁿ<γ_K∧σ_Kⁿ}that

t∈[0,minτ˜ⁿ]K_γⁿ_K∧σKⁿ_∧t≥2⁻ⁿ⁺⁵(K+ψ(K))², (3.2.20) which follows directly from the definition of K ⁿ and Equation (3.2.18). Fort ∈ (τ˜ⁿ,γ_K∧σ_Kⁿ]notice that

|S_τ˜ⁿ−S_χn−1(τ˜ⁿ)| ≤2⁻ⁿ⁺², (3.2.21) since we either have ˜τⁿ∈π_n−1, which impliesχⁿ⁻¹(τ˜ⁿ) =τ˜ⁿand|S_τ˜ⁿ−S_χn−1(τ˜ⁿ)|= 0, or we have ˜τⁿ∈/π_n−1, which implies (3.2.21) as ˜τⁿ<τ_kⁿ⁻¹0+1and

|S_τ˜ⁿ−S_χn−1(τ˜ⁿ)| ≤ |S_τ˜ⁿ−Dⁿ⁻¹_k0 |+|S_χn−1(τ˜ⁿ)−Dⁿ⁻¹_k0 | ≤2⁻ⁿ⁺¹+2⁻ⁿ⁺¹, where k⁰ is such that χⁿ⁻¹(τ˜ⁿ) =τ_kⁿ⁻¹0 . Using Equation (3.2.21), |Z_τⁿ_˜n| ≤K and

|S_τ˜ⁿ| ≤K, we estimate

|4Z_τⁿ_˜n(Sτ˜ⁿ−S_χn−1(τ˜ⁿ))(S_t−S_τ˜ⁿ)| ≤4K2⁻ⁿ⁺²(|S_t|+K) =2⁻ⁿ⁺⁴(K|S_t|+K²), which together with (3.2.20) gives weak admissibility as claimed in (3.2.19).

Corollary 3.33. For typical price pathsω∈Ω_ψ⊆D([0,T],R)there exist an N(ω)∈ Nsuch that

K_γⁿ_K∧σKⁿ_∧t(ω)<n⁶2⁻ⁿ, t∈[0,T], for all n≥N(ω).

Proof. Consider the events A_n,m:=

(

ω ∈Ω_ψ :∃t∈[0,T]s.t. K_γⁿ_K∧σⁿ

K∧t(ω)≥n⁶2⁻ⁿand sup

t∈[0,T]

|S_t(ω)| ≤m )

forn,m∈N. By the countable subadditivity ofPand the Borel-Cantelli lemma the claim follows once we have shown that

∑

n

P(A_n,m)<∞ for everym∈N. To that end, we define the stopping times

ρⁿ:=inf n

t ∈[0,T] : K_γⁿ_K∧σKⁿ_∧t≥n⁶2⁻ⁿ o

, n∈N,

so that A_n,m=

ω∈Ω_ψ : n⁻⁶2ⁿK_γⁿ_K∧σKⁿ_∧ρⁿ_∧T(ω)≥1 and sup

t∈[0,T]

|S_t(ω)| ≤m

.

Now it follows directly from Lemma3.32that

Q(A_n,m)≤n⁻⁶2ⁿ(n⁴2⁻²ⁿ+2⁻ⁿ⁺⁵(K+ψ(K))²) =n⁻²2⁻ⁿ+n⁻⁶2⁵(K+ψ(K))², which is summable. Since P(A_n,m)≤(1+3m+2ψ(m))Q(A_n,m) by Lemma3.27, the proof is complete.

Finally, we have collected all necessary ingredients to prove the main result of this section, namely Theorem 3.25. More precisely, we shall show that ([S]ⁿ− [S]ⁿ⁻¹)_n∈N is a Cauchy sequence. This implies Theorem 3.25 since the uniform metric onD([0,T],R+)is complete.

Proof of Theorem3.25. ForK∈Nlet us define A_K:=

(

ω∈Ω_ψ : sup

t∈[0,T]

|S_t(ω)| ≤Kand sup

t∈[0,T]

|Z_tⁿ(ω)| ≥n³2⁻ⁿ² for infinitely manyn∈N )

and B:=

(

ω ∈Ωψ :∃N(ω)∈Ns.t. K_γⁿ_K∧σ_Kⁿ∧t(ω)<n⁶2⁻ⁿ, t ∈[0,T],

U_T(ω,2⁻ⁿ)≤n²2²ⁿandDT(ω,2⁻ⁿ)≤n²2²ⁿ, n≥N(ω) )

.

Thanks to the countable subadditivity of P it is sufficient to show that P(A_K) =0 for everyK∈N. Moreover, again by the subadditivity ofPwe see

P(A_K)≤P(A_K∩B) +P(A_K∩B^c).

By Corollary3.31 and Corollary3.33 it is already known thatP(A_K∩B^c) =0. In the following we show thatA_K∩B= /0.

For this purpose, let us fix anω ∈Bsuch that sup

t∈[0,T]

|S_t(ω)| ≤K. Sinceω ∈B there exits anN(ω)∈Nsuch that for allm≥N(ω):

(a) The number of stopping times inπmdoes not exceed 2m²2^2m+2≤3m²2^2m. (b) The number of stopping times inπ_msuch that

∆S_τ^m

k(ω) :=

S_τ^m

k(ω)− lim

s→τ_k^m,s<τ_k^mS_s(ω)

≥2^−m+1, τ_k^m∈πm, is less or equal to 2m²2^2m.

As sup

t∈[0,T]

|S_t(ω)| ≤K, notice thatγK(ω) =T and that fort∈[0,T]we have Z_τⁿn

k+1∧t(ω)−Z_τⁿⁿ

k∧t(ω) =

[S]ⁿ_τⁿ

k+1∧t(ω)−[S]ⁿ_τⁿ

k∧t(ω)

− [S]ⁿ⁻¹_τn

k+1∧t(ω)−[S]ⁿ⁻¹_τn k∧t(ω)

=

S_τⁿ

k+1∧t(ω)−S_τⁿ

k∧t(ω) 2

− S_τⁿ

k+1∧t(ω)−S_χn−1(τ_kⁿ∧t)(ω) 2

− S_τⁿ

k∧t(ω)−S_χn−1(τ_kⁿ∧t)(ω) 2

=−2 S_τⁿ

k∧t(ω)−S_χn−1(τ_kⁿ∧t)(ω) S_τⁿ

k+1∧t(ω)−S_τⁿ

k∧t(ω) ,

where we recall thatχⁿ⁻¹(t):=max{τ_kⁿ⁻¹0 : τ_kⁿ⁻¹0 ≤t}. Therefore, keeping (3.2.21) in mind, the infinite sum in (3.2.17) can be estimated by

∞

∑

k=0

Z_τⁿn

k+1∧t(ω)−Z_τⁿn k∧t(ω)

!2

(3.2.22)

=4

∞

∑

k=0

S_τⁿ

k∧t(ω)−S_χn−1(τ_kⁿ∧t)(ω)2 S_τⁿ

k+1∧t(ω)−S_τⁿ

k∧t(ω)2

≤2⁶⁻²ⁿ

∞

∑

k=0

S_τⁿ

k+1∧t(ω)−S_τⁿ

k∧t(ω)2

. (3.2.23)

Forn≥N=N(ω)andt ∈[0,T]we observe for the summands in (3.2.22) the following bounds, which are similar to the bounds (A)-(E) in the proof of [47, The- orem 1]:

1. Ifτ_k+1ⁿ ∈/πn−1, then one has χⁿ⁻¹(τ_k+1ⁿ ) =χⁿ⁻¹(τ_kⁿ) =τ_kⁿ⁻¹0 for somek⁰and thus

S_τⁿ

k+1∧t(ω)−S_τⁿ

k∧t(ω) ≤ |S_τⁿ

k+1∧t(ω)−Dⁿ⁻¹

k⁰ |+|S_τⁿ

k∧t(ω)−Dⁿ⁻¹

k⁰ | ≤2²⁻ⁿ. The number of such summands is at most 3n²2²ⁿ.

2. Ifτ_k+1ⁿ ∈πn−1and|∆S_τⁿ

k+1| ≤2⁻ⁿ⁺¹, then one has S_τⁿ

k+1∧t(ω)−S_τⁿ

k∧t(ω)

≤2¹⁻ⁿ+2⁻ⁿ⁺¹=2²⁻ⁿ and the number of such summands is at most 3n²2²ⁿ.

3. Ifτ_k+1ⁿ ∈πn−1and|∆S_τⁿ

k+1| ∈[2^−m+1,2^−m+2), for somem∈ {N,N+1, . . . ,n}

than one has that S_τⁿ

k+1∧t(ω)−S_τⁿ

k∧t(ω)

≤2¹⁻ⁿ+2^−m+2. and the number of such summands is at most 2m²2^2m. 4. Ifτ_k+1ⁿ ∈π_n−1and∆S_τⁿ

k+1≥2^−N+2, then one has S_τⁿ

k+1∧t(ω)−S_τⁿ

k(ω)∧t

≤2K

and the number of such summand is bounded by a constant C =C(ω,K) independent ofn.

Using the bounds derived in (1)-(4), the estimate (3.2.22) can be continued by

∞ k=0

∑

Z_τⁿn

k+1∧t(ω)−Z_τⁿn k∧t(ω)

!2

≤2⁶⁻²ⁿ 6n²2²ⁿ2⁴⁻²ⁿ+

n

∑

m=N

2m²2^2m(2¹⁻ⁿ+2^−m+2)²+4CK²

! ,

and thus there exists an ˜N=N(ω˜ )∈Nsuch that

∞

∑

k=0

Z_τⁿn

k+1∧t(ω)−Z_τⁿn k∧t(ω)

!2

≤2⁻²ⁿn⁴, t∈[0,T],

for alln≥N˜. Combining the last estimate with the definition ofK ⁿ(cf. (3.2.17)), we obtain

K_σⁿ_Kⁿ_∧t(ω)≥ Z_σⁿn

K∧t(ω)2

, t∈[0,T], for alln≥N. Moreover, by assumption on˜ ω one has

K_σⁿ_Kⁿ_∧t(ω)<n⁶2⁻ⁿ, t∈[0,T], for alln≥N∨N. In particular, we conclude that sup˜

t∈[0,T]

Zⁿ_σn

K∧γK∧t(ω)

<Kwhenever nis large enough and thus

n⁶2⁻ⁿ>

Z_tⁿ(ω)2

, t ∈[0,T],

for all sufficiently large n. Finally, we have sup

t∈[0,T]

|Z_tⁿ(ω)|<n³2⁻ⁿ² for all large n and thereforeω ∈/A_K∩B.

Remark 3.34. As proven by [47, Proposition 3], the existence of quadratic vari- ation in Theorem 3.25 is equivalent to the existence of quadratic variation in the sense of Föllmer. Thus, Theorem3.25 allow us to use Föllmer’s pathwise Itô for- mula [17] to typical price paths belonging to the sample spaceΩ_ψand in particular to define the pathwise integral

Z

f⁰(S_s)dS_s

for f ∈C²or for more general path-dependent functionals as in [2,9,23].

3.2.2 Extension to Multi-dimensional Price Paths

In order to extend the existence of quadratic variation from one-dimensional to multi-dimensional typical price paths, we consider now the sample space Ω_ψ ⊆ D([0,T],R^d) and introduce a d-dimensional version of the Lebesgue partitions ford∈N.

Definition 3.35. For n∈Nand a d-dimensional càdlàg functionω: [0,T]→R^dits Lebesgue partition πn(ω):={τ_kⁿ(ω) : k≥0}is iteratively defined by τ₀ⁿ(ω):=0 and

τ_kⁿ(ω):=min (

τ>τ_k−1ⁿ (ω) : τ∈

d

[

i=1

π_n(ωⁱ)∪

d

[

i,j=1,i6=j

π_n(ωⁱ+ω^j) )

, k∈N,

where ω = (ω¹, . . . ,ω^d) andπ_n(ωⁱ) andπ_n(ωⁱ+ω^j) are the Lebesgue partitions ofωⁱandωⁱ+ω^j, respectively.

To state the existence of quadratic variation for typical price paths in Ω_ψ, we define the canonical projection on Ω_ψ by S_tⁱ(ω):=ωⁱ(t) forω = (ω¹, . . . ,ω^d)∈ Ω_ψ,t∈[0,T]andi=1, . . . ,d.

Corollary 3.36. Let d∈N and 1≤i,j≤d. For typical price paths ω ∈Ω_ψ the discrete quadratic variation

h Sⁱ,S^j

in

t (ω):=

∞ k=1

∑

Sⁱ_τn

k∧t(ω)−S_τⁱn

k−1∧t(ω) S^j

τ_kⁿ∧t(ω)−S^j

τ_k−1ⁿ ∧t(ω)

, t∈[0,T], converges along the Lebesgue partitions (π_n(ω))_n∈N in the uniform metric to a function[Sⁱ,S^j](ω)∈D([0,T],R).

Proof. To show the convergence of h

Sⁱ,S^j in

·(ω) for a path ω ∈Ω_ψ, we observe that

Sⁱ_s,t(ω)S_s,t^j (ω) =1 2

(Sⁱ_t(ω) +S_t^j(ω))−(Sⁱ_s(ω) +S_s^j(ω))2

−(Sⁱ_s,t(ω))²−(S_s,t^j (ω))²

fors,t ∈[0,T]and thus it is sufficient to prove the existence of the quadratic varia- tion ofSⁱ(ω)andSⁱ(ω) +S^j(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