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arXiv:1311.6187v1 [math.PR] 24 Nov 2013

Pathwise stochastic integrals for model free ο¬nance β

Nicolas Perkowskiβ

CEREMADE & CNRS UMR 7534

UniversitΒ΄e Paris-Dauphine

perkowski@ceremade.dauphine.fr

David J. PrΒ¨omelβ‘

Humboldt-UniversitΒ¨at zu Berlin

Institut fΒ¨ur Mathematik

proemel@math.hu-berlin.de

November 26, 2013

Abstract

We show that Lyonsβ rough path integral is a natural tool to use in model free ο¬nancial

mathematics by proving that it is possible to make an arbitrarily large proο¬t by investing

in those paths which do not have a rough path associated to them. We also show that in

certain situations, the rough path integral can be constructed as a limit of Riemann sums,

and not just as a limit of compensated Riemann sums which are usually used to deο¬ne it.

This proves that the rough path integral is really an extension of FΒ¨ollmerβs pathwise ItΛo

integral. Moreover, we construct a βmodel free ItΛo integralβ in the spirit of Karandikar.

1 Introduction

In this paper, we use Vovkβs [Vov12] game-theoretic approach to mathematical ο¬nance to set

up an integration theory for βtypical price pathsβ. Vovkβs approach is based on an outer

measure, which is given by the cheapest pathwise superhedging price. A property is said

to hold for typical price paths if the set of paths where the property is violated has outer

measure is zero. Roughly speaking, this means that it is possible to make an arbitrarily large

proο¬t by investing in those paths which violate the property.

We slightly adapt the deο¬nition of Vovkβs outer measure, to obtain an object which in

our opinion has a slightly more natural ο¬nancial interpretation, and with which it is easier to

work. For this outer measure, we prove a sort of βmodel free ItΛo isometryβ, based on which we

construct pathwise ItΛo integrals. We show that for typical price paths Sand c`adl`ag integrands

F, the integral RFdSexists. We also obtain an explicit, pathwise rate for the convergence

of (RFndS) to RFdS, where (Fn) are step functions which approximate Funiformly. Note

that for our model free ItΛo integral, the set of typical price paths for which RFdSexists

depends on F, and therefore it is not possible to exclude one set of paths from the beginning

and to construct all ItΛo integrals on the remaining set of paths.

However, the established speed of convergence allows us to show that to every typical price

path S, there is a naturally associated rough path (S, RSdS) in the sense of Lyons [Lyo98]. So

βWe would like to thank Mathias BeiglbΒ¨ock for suggesting us to look at the work of Vovk and at possible

connections to rough paths. We are grateful to Peter Imkeller for helpful discussions on the sub ject matter.

The main part of the research was carried out while N.P. was employed by Humboldt-UniversitΒ¨at zu Berlin.

β Supported by a Postdoctoral grant of the Fondation Sciences MathΒ΄ematiques de Paris.

β‘Supported by a Ph.D. scholarship of the DFG Research Training Group 1845.

1

we can ο¬x one set of typical price paths, the set of paths which have rough paths associated

to them, and set up an analytical theory of integration on that set. Since we can make

inο¬nite proο¬t by investing in the paths which we excluded, we may consider them as βtoo

good to be trueβ, and there is no problem in restricting the set of paths with which we work.

This is similar in spirit to the usual assumption in ο¬nancial mathematics that there exists

an equivalent local martingale measure, because models without equivalent local martingale

measure allow for free lunches with vanishing risk [DS94].

We follow Gubinelliβs [Gub04] controlled path approach to rough path integration, which

simultaneously extends

β’the Riemann-Stieltjes integral of Sagainst functions of bounded variation, formally

deο¬ned by Rt

0FsdSs=FtStβF0S0βRt

0SsdFs;

β’the Young integral [You36]: typical price paths have ο¬nite p-variation for every p > 2,

and therefore for every Fof ο¬nite q-variation for q < 2 (so that 1/p + 1/q > 1), the

integral RFdSis deο¬ned as limit of Riemann sums;

β’FΒ¨ollmerβs [FΒ¨ol81] βcalcul dβItΛo sans probabilitΒ΄esβ: FΒ¨ollmer shows that if the quadratic

variation hSiexists along a sequence of partitions, then for every gradient βF, the inte-

gral RβF(S)dSexists as limit of Riemann sums along that same sequence of partitions.

That this last integral is a special case of the controlled rough path integral is, to the

best of our knowledge, proved for the ο¬rst time in this paper.

The rough path integral is usually deο¬ned as a limit of compensated Riemann sums,

which have no obvious ο¬nancial interpretation. Here we show that if the integral RSdSis

given as limit of Riemann sums along a given sequence of partitions, then the rough path

integral RFdSfor Fcontrolled by Scan be deο¬ned as limit of Riemann sums along the same

sequence of partitions, similarly as in FΒ¨ollmerβs pathwise ItΛo calculus. We also show that

FΒ¨ollmerβs condition on the existence of the quadratic variation is equivalent to the existence

and regularity of the symmetric part of the matrix (RSdS). Since only the symmetric part

counts when integrating gradients, FΒ¨ollmerβs integral is really a special case of the rough path

integral.

Model free ο¬nance and the need for pathwise integrals

Before we go into detail, let us give some context by motivating why we need a pathwise

stochastic integral in model free ο¬nance.

One of the basic problems in mathematical ο¬nance consists in calculating fair prices for

ο¬nancial derivatives. It is part of the folklore that the minimal superhedging price of a given

derivative is equal to its maximal risk-neutral expectation. More precisely, let (St:tβ[0, T ])

be an adapted process on a ο¬ltered probability space (β¦,F,(Ft)tβ[0,T ],P), which models the

discounted price of a ο¬nancial asset. Let Xbe a nonnegative random variable on (β¦,F),

which models the payoο¬ of a given derivative. Then the minimal superhedging price of Xis

deο¬ned as

p(X) := inf nΞ» > 0 : βadmissible strategy Hs.t. Ξ»+ZT

0

HsdSsβ₯XPβa.s.o,(1)

and we have

p(X) = sup{EQ[X] : Qis equivalent to Pand Sis a Qβlocal martingale}.

2

Of course, in practice the probability measure Pthat describes the statistical behavior of

the asset price process is not known with absolute certainty. Consider for example the Black-

Scholes model, where the discounted price process is described by a geometric Brownian

motion with drift,

dSΟ

t=SΟ

t(ΟdWt+bdt),

where Wis a one-dimensional standard Brownian motion, Ο > 0, and bβR. It is a very

active ο¬eld of research to derive statistical estimates for the volatility Ο, but even the best

statistical method will only give us a conο¬dence interval Οβ[Ο1, Ο2], and not the exact value

of Οβ unless the full, continuous time path of Sis observed, which is obviously not feasible

in practice.

This motivated [ALP95] and [Lyo95b] to study the option pricing problem under volatility

uncertainty, i.e. to obtain results that hold simultaneously under all PΟfor Οβ[Ο1, Ο2],

where PΟdenotes the measure on the path space under which the coordinate process has

the dynamics of a geometric Brownian motion with volatility Ο. The ο¬rst problem that one

encounters here is how to deο¬ne the minimal superhedging price. It is natural to replace the

βa.s.β assumption in (1) by βa.s. under every PΟ,Οβ[Ο1, Ο2]β. But then the stochastic

integral RHdShas to be constructed simultaneously under all the measures PΟ, which is not

a trivial task because PΟand PΛΟare mutually singular for Ο6= ΛΟ. Luckily it turns out that

here we can essentially still use ItΛoβs integral, because while we have to deal with uncountably

many singular probability measures at once, the price process is a semimartingale under every

single one of them.

On the other hand, model free mathematical ο¬nance no longer assumes any semimartingale

structure for the price process. Instead, some basic market data is assumed to be known (for

example European call and put prices), and the aim is to calculate all prices for a given

derivative that are compatible with this data.

In [BHLP13] it is assumed that S= (St)t=0,...,T is a discrete time process, and that the

prices for all European call options with payoο¬ (StβK)+for 0 β€tβ€Tand KβRare

known. Based on this assumption, they derive all arbitrage free prices for a given path-

dependent derivative X(S0,...,ST), and they develop a duality theory between subhedging

prices and martingale expectations under relatively mild continuity assumptions on X. Since

time is discrete, no problems arise in the deο¬nition of the stochastic integrals RFdS, which

are just ο¬nite sums.

In continuous time however, it is not immediately clear how to deο¬ne stochastic integrals

without a probability measure. In [DS13], this problem is resolved by only considering strate-

gies of bounded variation, so that the integrals can be deο¬ned in a pathwise sense, for example

by formally applying integration by parts. They develop a duality theory for path-dependent

functionals which are Lipschitz continuous if the path space is equipped with the supremum

norm. Observe that this excludes any derivative which depends on the volatility.

In [DOR13], FΒ¨ollmerβs pathwise ItΛo calculus [FΒ¨ol81] is used to deο¬ne pathwise stochastic

integrals, and these integrals are used to derive arbitrage consistent prices for weighted vari-

ance swaps. Here, the given data is a ο¬nite number of European call and put prices. The

use of FΒ¨ollmerβs integral, which relies on the pathwise existence of the quadratic variation, is

justiο¬ed by referring to Vovk [Vov12], who proved that typical price paths admit a nontrivial

quadratic variation.

However, FΒ¨ollmerβs integral essentially only applies to one-dimensional integrators. If we

are considering more than one ο¬nancial asset, then we can only integrate gradients RβF(S)dS

3

with FΒ¨ollmerβs integral. A multidimensional extension is given by the rough path integral.

Here we prove that each typical price path has a rough path associated to it, so that the rough

path integral is a natural tool to use in model free ο¬nance. Also, while with FΒ¨ollmerβs integral

it is only possible to integrate functions of S, the rough path integral allows in principal to

integrate path-dependent functionals, and it gives a uniο¬ed framework for both FΒ¨ollmerβs and

Youngβs integrals.

Plan of the paper

In Section 2 we brieο¬y recall Vovkβs game-theoretic approach to mathematical ο¬nance. Sec-

tion 3 is devoted to the construction of a model free ItΛo integral and to the construction of

rough paths associated to typical price paths. Section 4 presents some basic results from

rough path theory, and we prove that the rough path integral is given as a limit of Riemann

sums rather than compensated Riemann sums, which are usually used to deο¬ne it. In Sec-

tion 5 we compare FΒ¨ollmerβs pathwise ItΛo integral with the rough path integral and prove that

the latter is really an extension of the former. Appendix A presents a pathwise version of the

Hoeο¬ding inequality which is due to Vovk. In Appendix B, we show that a result of Davie

which also allows to calculate rough path integrals as limit of Riemann sums, is a special case

of our results in Section 4.

Notation and conventions

Throughout the paper, we ο¬x T > 0 and we write β¦ := C([0, T ],Rd) for the space of d-

dimensional continuous paths. The coordinate process on β¦ is denoted by St(Ο) = Ο(t),

tβ[0, T ]. For iβ {1,...,d}, we also write Si

t(Ο) = Οi(t), where Ο= (Ο1,...,Οd). The

ο¬ltration (Ft)tβ[0,T ]is deο¬ned as Ft:= Ο(Ss:sβ€t), and we set F:= FT. Stopping times Ο

and the associated Ο-algebras FΟare deο¬ned as usually.

Unless explicitly stated otherwise, inequalities of the type Ftβ₯Gt, where Fand G

processes on β¦, are supposed to hold for all Οββ¦, and not modulo null sets, as it is usually

assumed in stochastic analysis.

The indicator function of a set Ais denoted by 1A.

Apartition Οof [0, T ] is a ο¬nite set of time points, Ο={0 = t0< t1<Β·Β·Β·< tm=T}.

Occasionally, we will identify Οwith the set of intervals {[t0, t1],[t1, t2],...,[tmβ1, tm]}, and

write expressions like P[s,t]βΟ.

For f: [0, T ]βRnand t1, t2β[0, T ], denote ft1,t2:= ft2βft1and deο¬ne the p-variation

of frestricted to [s, t]β[0, T ] as

kfkpβvar,[s,t]:= sup ξξmβ1

X

k=0 |ftk,tk+1 |pξ1/p

:s=t0<Β·Β·Β·< tm=t, m βNξ, p > 0,(2)

(possibly taking the value +β). We set kfkpβvar := kfkpβvar,[0,T ]. We write βT={(s, t) :

0β€sβ€tβ€T}for the simplex and deο¬ne the p-variation of a function g: βTβRnin the

same manner, replacing ftk,tk+1 in (2) by g(tk, tk+1).

For Ξ± > 0, the space CΞ±consists of those functions that are βΞ±βtimes continuously

diο¬erentiable, with (Ξ±β βΞ±β)-HΒ¨older continuous partial derivatives of order βΞ±β. The space

CΞ±

bconsists of those functions in CΞ±that are bounded, together with their partial derivatives,

4

and we deο¬ne the norm kΒ·kCΞ±

bby setting

kfkCΞ±

b:=

βΞ±β

X

k=0 kDkfkβ+kDβΞ±βfkΞ±ββΞ±β,

where kΒ·kΞ²denotes the Ξ²-HΒ¨older norm for Ξ²β(0,1), and kΒ·kβdenotes the supremum norm.

For x, y βRd, we write xy := Pd

i=1 xiyifor the usual inner product. However, often

we will encounter terms of the form RSdSor SsSs,t for s, t β[0, T ], where we recall that S

denotes the coordinate process on β¦. Those expressions are to be understood as the matrix

(RSidSj)1β€i,jβ€d, and similarly for SsSs,t. The interpretation will be usually clear from the

context, otherwise we will make a remark to clarify things.

We use the notation a.bif there exists a constant c > 0, independent of the variables

under consideration, such that aβ€cΒ·b, and we write aβbif a.band b.a. If we want to

emphasize the dependence of con the variable x, then we write a(x).xb(x).

We make the convention that 0/0 := 0 Β· β := 0 and inf β
:= β.

2 Superhedging and typical price paths

In a recent series of papers, Vovk [Vov08, Vov11, Vov12] has introduced a model free, hedging

based approach to mathematical ο¬nance that uses arbitrage considerations to examine which

properties are satisο¬ed by βtypical price pathsβ. This is achieved with the help of an outer

measure given by the cheapest superhedging price.

Recall that T > 0 and β¦ = C([0, T ],Rd) is the space of continuous paths, with coordinate

process S, natural ο¬ltration (Ft)tβ[0,T ], and F=FT. A process H: β¦ Γ[0, T ]βRdis called a

simple strategy if there exist stopping times 0 = Ο0< Ο1< . . . , and FΟn-measurable bounded

functions Fn: β¦ βRd, such that for every Οββ¦ we have Οn(Ο) = βfor all but ο¬nitely many

n, and such that

Ht(Ο) =

β

X

n=0

Fn(Ο)1(Οn(Ο),Οn+1(Ο)](t).

In that case, the integral

(HΒ·S)t(Ο) :=

β

X

n=0

Fn(Ο)(SΟn+1β§t(Ο)βSΟnβ§t(Ο)) =

β

X

n=0

Fn(Ο)SΟnβ§t,Οn+1β§t(Ο)

is well deο¬ned for all Οββ¦, tβ[0, T ]. Here Fn(Ο)SΟnβ§t,Οn+1β§t(Ο) denotes the usual inner

product on Rd. For Ξ» > 0, a simple strategy His called Ξ»-admissible if (HΒ·S)t(Ο)β₯ βΞ»for

all Οββ¦, tβ[0, T ]. The set of Ξ»-admissible simple strategies is denoted by HΞ».

Deο¬nition 1. The outer measure of Aββ¦ is deο¬ned as the cheapest superhedging price for

1A, that is

P(A) := inf nΞ» > 0 : β(Hn)nβNβ HΞ»s.t. lim inf

nββ (Ξ»+ (HnΒ·S)T(Ο)) β₯1A(Ο)βΟββ¦o.

A set of paths Aββ¦ is called a null set if it has outer measure zero.

5

The outer measure Pis very similar to the one used by Vovk [Vov12], but not quite the

same. For a discussion see Section 2.1 below. By deο¬nition, every ItΛo stochastic integral

is the limit of stochastic integrals against simple functions. Therefore, our deο¬nition of the

cheapest superhedging price is essentially the same as in the classical setting, see (1), with

one important diο¬erence: we require superhedging for all Οββ¦, and not just almost surely.

Remark 2 ([Vov12], p. 564).An equivalent deο¬nition of Pwould be

e

P(A) := infξΞ» > 0 : β(Hn)nβNβ HΞ»s.t. lim inf

nββ sup

tβ[0,T ]

(Ξ»+ (HnΒ·S)t(Ο)) β₯1A(Ο)βΟββ¦ξ.

Clearly e

Pβ€P. To see the opposite inequality, let e

P(A)< Ξ». Let (Hn)nβNβ HΞ»be a sequence

of simple strategies such that lim inf nββ suptβ[0,T](Ξ»+ (HnΒ·S)t)β₯1A, and let Ξ΅ > 0. Deο¬ne

Οn:= inf{tβ[0, T ] : Ξ»+Ξ΅+ (HnΒ·S)tβ₯1}. Then the stopped strategy Gn

t(Ο) := Hn

t(Ο)1t<Οn(Ο)

is in HΞ»β HΞ»+Ξ΅and

lim inf

nββ (Ξ»+Ξ΅+ (GnΒ·S)T(Ο)) β₯lim inf

nββ 1{Ξ»+Ξ΅+suptβ[0,T ](HnΒ·S)tβ₯1}(Ο)β₯1A(Ο).

Therefore P(A)β€Ξ»+Ξ΅, and since Ξ΅ > 0was arbitrary Pβ€e

P, and thus P=e

P.

Lemma 3 ([Vov12], Lemma 4.1).Pis in fact an outer measure, i.e. a nonnegative function

deο¬ned on the subsets of β¦such that

-P(β
) = 0;

-P(A)β€P(B)if AβB;

- if (An)nβNis a sequence of subsets of β¦, then P(SnAn)β€PnP(An).

Proof. Monotonicity and P(β
) = 0 are obvious. So let (An) be a sequence of subsets of β¦.

Let Ξ΅ > 0, nβN, and let (Hn,m)mβNbe a sequence of (P(An) + Ξ΅2βnβ1)-admissible simple

strategies such that lim inf mββ(P(An) + Ξ΅2βnβ1+ (Hn,m Β·S)T)β₯1An. Deο¬ne for mβNthe

(PnP(An) + Ξ΅)-admissible simple strategy Gm:= Pm

n=0 Hn,m. Then by Fatouβs lemma

lim inf

mββ ξβ

X

n=0

P(An) + Ξ΅+ (GmΒ·S)Tξβ₯

k

X

n=0 ξP(An) + Ξ΅2βnβ1+ lim inf

mββ (Hn,m Β·S)Tξ

β₯1Sk

n=0 An

for all kβN. Since the left hand side does not depend on k, we can replace 1Sk

n=0 Anby

1SnAnand the proof is complete.

Maybe the most important property of Pis that there exists an arbitrage interpretation

for sets with outer measure zero:

Lemma 4. A set Aββ¦is a null set if and only if there exists a sequence of 1-admissible

simple strategies (Hn)nβ H1such that

lim inf

nββ (1 + (HnΒ·S)T)β₯ β Β· 1A(Ο),(3)

where we recall that by convention 0Β· β = 0.

6

Proof. If such a sequence exists, then we can scale it down by an arbitrary factor Ξ΅ > 0 to

obtain a sequence of strategies in HΞ΅that superhedge 1Aand therefore P(A) = 0.

If conversely P(A) = 0, then for every nβNthere exists a sequence of simple strategies

(Hn,m)mβNβ H2βnβ1such that 2βnβ1+ lim inf mββ(Hn,m Β·Ο)Tβ₯1A(Ο) for all Οββ¦. Deο¬ne

Gm:= Pm

n=0 Hn,m, so that Gmβ H1. For every kβNwe obtain

lim inf

mββ (1 + (GmΒ·S)T)β₯

k

X

n=0

(2βnβ1+ lim inf

mββ (Hn,m Β·S)T)β₯k1A.

Since the left hand side does not depend on k, the sequence (Gm) satisο¬es (3).

In other words, if a set Ahas outer measure 0, then we can make inο¬nite proο¬t by

investing in the paths from A, without ever risking to lose more than the initial capital 1.

This motivates the following deο¬nition:

Deο¬nition 5. We say that a property (P) holds for typical price paths if the set Awhere (P)

is violated is a null set.

Before we continue, let us present some results which link our outer content with classi-

cal mathematical ο¬nance. First, observe that Pis an outer measure which simultaneously

dominates all local martingale measures on β¦:

Propostion 6 ([Vov12], Lemma 6.3).Let Pbe a probability measure on (β¦,F), such that the

coordinate process Sis a P-local martingale, and let Aβ F. Then P(A)β€P(A).

Proof. Let Ξ» > 0 and let (Hn)β HΞ»be such that lim inf n(Ξ»+ (HnΒ·S)T)β₯1A. Then

P(A)β€EP[lim inf

n(Ξ»+ (HnΒ·S)T)] β€lim inf

nEP[Ξ»+ (HnΒ·S)T]β€Ξ»,

where in the last step we used that Ξ»+ (HnΒ·S) is a nonnegative P-local martingale and thus

aP-supermartingale.

Recall the fundamental theorem of asset pricing by Delbaen and Schachermayer [DS94]:

If Pis a probability measure on (β¦,F) under which Sis a semimartingale, then there ex-

ists an equivalent measure Qsuch that Sis a Q-local martingale if and only if Sadmits

no free lunch with vanishing risk (NFLVR). It was observed already by [DS94] that (NFLVR)

is equivalent to the two conditions no arbitrage (NA) (intuitively: no proο¬t without risk) and

no arbitrage opportunities of the ο¬rst kind (NA1) (intuitively: no very large proο¬t with a

small risk). The (NA) property has no real translation to the model free setting. But it turns

out that (3) is essentially a model free version of the (NA1) property:

The process Sis said to satisfy (NA1) under Pif {1 + (HΒ·S)T:Hβ H1}is bounded in

P-probability, i.e. if

lim

nββ sup

HβH1

P(1 + (HΒ·S)Tβ₯n) = 0.

Thus, we can interpret a null set Aββ¦ as a model free arbitrage opportunity of the ο¬rst kind.

(NA1) is the minimal property under which there exists a non-degenerate utility maximization

problem (see [IP11]), and therefore it is no restriction to only work with models satisfying

(NA1). Similarly, it is no restriction to only work on a ο¬xed set of typical price paths, rather

than the full space β¦.

Not surprisingly, we can relate our model free notion of (NA1) with the classical (NA1)

property:

7

Propostion 7. Let Aβ F be a null set, and let Pbe a probability measure on (β¦,F)such

that the coordinate process satisο¬es (NA1). Then P(A) = 0.

Proof. Let (Hn)nβNβ H1be such that lim inf n(HnΒ·S)Tβ₯ β Β· 1A. Then for every c > 0

P(A) = PξAβ©ξlim inf

nββ (HnΒ·S)T> cξξβ€lim

nββ PξAβ©ξ\

kβ₯n{(HkΒ·S)T> c}ξξ

β€sup

HβH1

P({(HΒ·S)T> c}).

By assumption, the right hand side converges to 0 as cβ β and thus P(A) = 0.

Remark 8. The proof shows that the measurability assumption on Acan be relaxed: If

P(A) = 0, then Ais contained in a measurable set of the form {lim infnββ(HnΒ·S)T=β},

and this set has P-measure zero. Hence, Ais contained in the P-completion of Fand gets

assigned mass 0by the unique extension of Pto the completion.

Remark 9. Proposition 7 is in fact a direct consequence of Proposition 6, because if S

satisο¬es (NA1) under P, then there exists a dominating measure Qβ«P, such that Sis a

Q-local martingale. See Ruf [Ruf13] for the case of continuous S, and [IP11] for the general

case.

2.1 Relation to Vovkβs outer measure

Our deο¬nition of the outer measure Pis not exactly the same as Vovkβs [Vov12]. We ο¬nd the

deο¬nition given above more intuitive and also it seems to be easier to work with. However,

since we rely on some of the results established by Vovk, let us compare the two notions.

For Ξ» > 0, Vovk deο¬nes the set of processes

SΞ»:= ξβ

X

k=0

Hk:Hkβ HΞ»k, Ξ»k>0,

β

X

k=0

Ξ»k=Ξ»ξ.

For every G=Pkβ₯0Hkβ SΞ», every Οββ¦ and every tβ[0, T ], the integral

(GΒ·S)t(Ο) := X

kβ₯0

(HkΒ·S)t(Ο) = X

kβ₯0

(Ξ»k+ (HkΒ·S)t(Ο)) βΞ»

is well deο¬ned and takes values in [βΞ», β]. Vovk then deο¬nes for Aββ¦ the cheapest

superhedging price as

Q(A) := inf ξΞ» > 0 : βGβ SΞ»s.t. Ξ»+ (GΒ·S)Tβ₯1Aξ.

Vovkβs deο¬nition corresponds to the usual construction of an outer measure from an outer

content (i.e. an outer measure which is only ο¬nitely subadditive and not countably subad-

ditive); see [Fol99], Chapter 1.4, or [Tao11], Chapter 1.7. Here, the outer content is given

by the cheapest superhedging price using only simple strategies. It is easy to see that Pis

dominated by Q:

Lemma 10. Let Aββ¦. Then P(A)β€Q(A).

8

Proof. Let G=PkHk, with Hkβ HΞ»kand PkΞ»k=Ξ», and assume that Ξ»+ (GΒ·S)Tβ₯1A.

Then (Pn

k=0 Hk)nβNdeο¬nes a sequence of simple strategies in HΞ», such that

lim inf

nββ ξΞ»+ξξ n

X

k=0

HkξΒ·SξTξ=Ξ»+ (GΒ·S)Tβ₯1A.

So if Q(A)< Ξ», then also P(A)β€Ξ», and therefore P(A)β€Q(A).

Corollary 11. For every p > 2, the set Ap:= {Οββ¦ : kS(Ο)kpβvar =β} has outer measure

zero, that is P(Ap) = 0.

Proof. Theorem 1 of Vovk [Vov08] states that Q(Ap) = 0, so P(Ap) = 0 by Lemma 10.

It is a remarkable result of [Vov12] that if β¦ = C([0,β),R) (i.e. if the asset price process

is one-dimensional), and if Aββ¦ is βinvariant under time changesβ and such that S0(Ο) = 0

for all ΟβA, then Aβ F and Q(A) = P(A), where Pdenotes the Wiener measure. This can

be interpreted as a pathwise Dambis Dubins-Schwarz theorem.

3 Construction of the pathwise ItΛo integral

The present section is devoted to the pathwise construction of an ItΛo type integral for typical

price paths. The main ingredient in the construction of the integral is a (weak) type of model

free ItΛo isometry, which allows us to estimate the integral against a step function in terms of

the amplitude of the step function and the quadratic variation of the price path. Then we

can extend the integral to c`adl`ag integrands by a continuity argument and we get an explicit

rate of convergence. The rate of convergence turns out to be all we need to prove that the

integral RSdSis suο¬ciently regular to obtain a rough path (S, RSdS).

Since we are in an unusual setting, let us spell out the following standard deο¬nitions:

Deο¬nition 12. A process H: β¦ Γ[0, T ]βRdis called adapted if the random variable

Ο7β Ht(Ο) is Ft-measurable for all tβ[0, T ].

The process His said to be c`adl`ag if the sample path t7β Ht(Ο) is c`adl`ag for all Οββ¦.

For proving our weak ItΛo isometry, we will need an appropriate sequence of stopping times:

Let Ο= (Ο1,...,Οd)βC([0, T ],Rd) and nβN. For each i= 1,...,d, deο¬ne inductively

Οn,i

0(Ο) := 0, Οn,i

k+1(Ο) := inf ξtβ₯Οn,i

k:|Οi(t)βΟi(Οn,i

k)| β₯ 2βnξ, k βN.

Note that we are working with continuous paths and we are considering entrance times into

closed sets. Therefore, the (Οn,i) are indeed stopping times, despite the fact that our ο¬ltration

(Ft) is neither complete nor right-continuous. Denote by Οn,i the partition corresponding to

(Οn,i

k)kβN, that is Οn,i := {Οn,i

k:kβN}. To obtain an increasing sequence of partitions, we

take the union of the (Οn,i). More precisely, for nβNwe deο¬ne Οn

0:= 0 and then

Οn

k+1(Ο) := min ξt > Οn

k(Ο) : tβ

d

[

i=1

Οn,i(Ο)ξ, k βN,(4)

and we write Οn:= {Οn

k:kβN}for the corresponding partition. We will rely on the following

result, which is due to Vovk:

9

Lemma 13 ([Vov11], Theorem 4.1).For typical price paths Οββ¦, the quadratic variation

along (Οn,i(Ο))nβNexists. That is,

Vn,i

t(Ο) :=

β

X

k=0 ξΟi(Οn,i

k+1 β§t)βΟi(Οn,i

kβ§t)ξ2, t β[0, T ], n βN,

converges uniformly to a function hSii(Ο)βC([0, T ],R)for all iβ {1,...,d}.

For later reference, let us estimate Nn

t:= max{kβN:Οn

kβ€t}, the number of stopping

times Οn

k6= 0 in Οnwith values in [0, t]:

Lemma 14. For all Οββ¦,nβN, and tβ[0, T ], we have

2β2nNn

t(Ο)β€

d

X

i=1

Vn,i

t(Ο) =: Vn

t(Ο).

Proof. For iβ {1, ..., d}deο¬ne Nn,i

t:= max{kβN:Οn,i

kβ€t}. Since Οiis continuous, we

have |Οi(Οn,i

k+1)βΟi(Οn,i

k)|= 2βnas long as Οn,i

k+1 β€T. Therefore, we obtain

Nn

t(Ο)β€

d

X

i=1

Nn,i

t(Ο) =

d

X

i=1

Nn,i

t(Ο)β1

X

k=0

1

2β2nξΟ(Οn,i

k+1)βΟ(Οn,i

k)ξ2β€22n

d

X

i=1

Vn,i

t(Ο).

Since we want to extend the integral from step functions to c`adl`ag integrands via a con-

tinuity argument, let us ο¬rst specify what we mean by step functions. They are essentially

just simple strategies, except that they do not have to be bounded:

A process H: β¦ Γ[0, T ]βRdis called a step function if there exist stopping times

0 = Ο0< Ο1< . . . , and FΟn-measurable functions Fn: β¦ βRd, such that for every Οββ¦ we

have Οn(Ο) = βfor all but ο¬nitely many n, and such that

Ht(Ο) =

β

X

n=0

Fn(Ο)1[Οn(Ο),Οn+1(Ο))(t).

Note that for notational convenience, we are now considering the interval [Οn(Ο), Οn+1(Ο))

which is closed on the right hand side. This allows us to deο¬ne the integral

(HΒ·S)t:=

β

X

n=0

FnSΟnβ§t,Οn+1β§t=

β

X

n=0

HΟnSΟnβ§t,Οn+1β§t, t β[0, T ].

The following lemma will be the main building block in our construction of the integral.

Lemma 15 (Model free ItΛo isometry).Let a > 0and let Hbe a step function such that

kH(Ο)kββ€afor all Οββ¦. Then for all b, c > 0we have

Pξ{k(HΒ·S)kββ₯abβc} β© {hSiTβ€c}ξβ€2 exp(βb2/(2d)),

where the set {hSiTβ€c}should be read as {hSiT= limnVn

Texists and satisο¬es hSiTβ€c}.

10

Proof. Assume Ht=Pβ

n=0 Fn1[Οn,Οn+1)(t). Let nβNand deο¬ne Οn

0:= 0 and then for kβN

Οn

k+1 := min ξt > Οn

k:tβΟnβͺ {Οm:mβN}ξ,

where we recall that Οn={Οn

k:kβN}is the n-th generation dyadic partition generated

by S. For tβ[0, T ], we have (HΒ·S)t=PkHΟn

kSΟn

kβ§t,Οn

k+1β§t. Since kH(Ο)kββ€a, and by

deο¬nition of Οn(Ο), we get

sup

tβ[0,T ]ξξHΟn

kSΟn

kβ§t,Οn

k+1β§tξξβ€aβd2βn.

Hence, the pathwise Hoeο¬ding inequality, Lemma 35 in Appendix A, yields for every Ξ»βR

the existence of a 1-admissible simple strategy GΞ»,n β H1such that

1 + (GΞ»,n Β·S)tβ₯exp ξΞ»(HΒ·S)tβΞ»2

2(N(Οn)

t+ 1)2β2na2dξ=: EΞ»,n

t

for all tβ[0, T ], where

N(Οn)

t:= max{k:Οn

kβ€t} β€ Nn

t+N(Ο)

t:= Nn

t+ max{k:Οkβ€t}.

By Lemma 14, we have Nn

tβ€22nVn

t, so that

EΞ»,n

tβ₯exp ξΞ»(HΒ·S)tβΞ»2

2Vn

Ta2dβΞ»2

2(N(Ο)

T+ 1)2β2na2dξ.

If now k(HΒ·S)kββ₯abβcand hSiTβ€c, then

lim inf

nββ sup

tβ[0,T ]

EΞ»,n

t+EβΞ»,n

t

2β₯1

2exp ξΞ»abβcβΞ»2

2ca2dξ.

The argument inside the exponential is maximized for Ξ»=b/(aβcd), in which case we obtain

1/2 exp(b2/(2d)). The statement now follows from Remark 2.

Of course, we did not really establish an isometry, but only an upper bound for the integral.

But this estimate is the key ingredient which allows us to construct the pathwise stochastic

integral for more general integrands than step functions, just like the ItΛo isometry is the key

ingredient in the construction of the ItΛo integral. The term βmodel free ItΛo isometryβ alludes

to that analogy.

Theorem 16. Let Hbe an adapted, c`adl`ag process with values in Rd. Then there exists a

map RHdS: β¦ ββ¦, such that if (cm)mβNis a sequence of strictly positive numbers, and if

(Hm)mβNis a sequence of step functions with kHm(Ο)βH(Ο)kββ€cmfor all Οββ¦and all

mβN, then for typical price paths Οthere exists a constant C(Ο)>0such that

ξ

ξ

ξ(HmΒ·S)(Ο)βZHdS(Ο)ξ

ξ

ξββ€C(Ο)cmplog m(5)

for all mβN. In particular, if (cmβlog m)converges to 0, then (HmΒ·S)(Ο)converges to

RHdS(Ο)for typical price paths Ο.

We usually write Rt

0HsdSs:= RHdS(t), and we call the function RHdSthe pathwise

ItΛo integral of Hwith respect to S.

11

Proof. For the construction of RHdS, we consider dyadic approximations of H: Deο¬ne for

nβNthe stopping times Οn

0:= 0 and then inductively

Οn

k+1 := inf{tβ₯Οn

k:|HtβHΟn

k| β₯ 2βn}.

Set also Gn

t:= Pβ

k=0 HΟn

k1[Οn

k,Οn

k+1)(t), so that kGnβHkββ€2βn, and thus kGnβGn+1kββ€

2βn+1. We claim that

Pξlim sup

nββ k(GnΒ·S)β(Gn+1 Β·S)kβ

2βnβlog n=βξ= 0.

Since P(hSiT=β) = 0 and by countable subadditivity of P, it suο¬ces to show

Pξξlim sup

nββ k(GnΒ·S)β(Gn+1 Β·S)kβ

2βnβlog n=βξβ© {hSiTβ€c}ξ= 0

for every c > 0. But we obtain from Lemma 15 that

P \

mβN[

nβ₯mξnk(GnΒ·S)β(Gn+1 Β·S)kββ₯2βn+1p4dlog nβcoβ© {hSiTβ€c}ξ!

β€

β

X

n=n0

Pξnk(GnΒ·S)β(Gn+1 Β·S)kββ₯2βn+1p4dlog nβcoβ© {hSiTβ€c}ξ

β€

β

X

n=n0

exp ξβ4dlog n

2dξ=

β

X

n=n0

1

n2

for all n0βN. Since the right hand side converges to zero as n0tends to β, we conclude that

for typical price paths Οthere exists C(Ο)β₯0 such that k(GnΒ·S)(Ο)β(Gn+1 Β·S)(Ο)kβ<

C(Ο)2βnβlog nfor all n, and in particular ((GnΒ·S)(Ο))nis a Cauchy sequence. We deο¬ne

ZHdS(Ο)(t) := (limnββ(GnΒ·S)t(Ο),if ((GnΒ·S)(Ο))nconverges uniformly,

0,otherwise.

Let now (Hm)mβNbe a sequence of step functions that approximates Huniformly, such that

kHmβHkββ€cm, where (cm) is a sequence of strictly positive numbers, and let c > 0. Then

Pξξk(HmΒ·S)βZHdSkββ₯3cmp4dlog mβcξβ©{hSiTβ€c}ξ

β€Pξnk(HmΒ·S)β(GnΒ·S)kββ₯2cmp4dlog mβcoβ© {hSiTβ€c}ξ

+Pξξk(GnΒ·S)βZHdSkββ₯cmp4dlog mβcξβ© {hSiTβ€c}ξ

for every nβN. As we have seen above, the second term on the right hand side converges to

zero as nβ β. And if nis large enough so that 2βnβ€cm, then kHmβGnkββ€2cm, so in

particular the ο¬rst term on the right hand side can be estimated by

Pξnk(HmΒ·Ο)β(GnΒ·Ο)kββ₯2cmp4dlog mβcoβ© {VT(Ο)β€c}ξβ€1

m2.

Since this is summable in m, we conclude as before that for typical price paths Οthere exists

C(Ο)>0 such that k(HmΒ·S)(Ο)βRHdS(Ο)kββ€C(Ο)cmβlog mfor all m.

12

Remark 17. The pathwise ItΛo integral is inspired by Karandikar [Kar95]. Just as Karandikar,

we obtain a map on path space, such that for every measure under which the coordinate process

is a local martingale, the map almost surely coincides with the ItΛo integral (recall Proposi-

tion 6). The model free ItΛo isometry and the speed of convergence (5) however seem to be

new. The arbitrage interpretation for the non-existence of the integral is new.

It might seem as if the pathwise ItΛo integral was already suο¬cient for applications. How-

ever, the trading strategies which we constructed in the existence proof of the integral de-

pended on the integrand, and therefore also the null set where the integral does not exist

depends on the integrand. A short moment of contemplation convinces us that unless we

restrict the space of integrands, there cannot exist a βuniversal null setβ outside of which

all integrals can be constructed. Already for the set of deterministic c`adl`ag integrands there

exists no such universal null set. To obtain a set on which we can set up a theory of integra-

tion that works for all paths in the set, we should use an analytic rather than βprobabilisticβ

construction of the integral. Such an analytic construction is given by Lyonsβ rough path

integral, which does not work for all c`adl`ag integrands but instead only for those integrands

which βlook like the integratorβ.

In order to apply the rough path machinery, we will need to show that the integral process

RSdS:= (RSidSj)1β€i,jβ€dis suο¬ciently regular. Fortunately, this is a direct consequence of

the speed of convergence (5):

Corollary 18. For (s, t)ββT,Οββ¦, and i, j β {1,...,d}deο¬ne

Ai,j

s,t(Ο) := Zt

s

Si

rdSj

r(Ο)βSi

s(Ο)Sj

s,t(Ο) := Zt

0

Si

rdSj

r(Ο)βZs

0

Si

rdSj

r(Ο)βSi

s(Ο)Sj

s,t(Ο).

Then for typical price paths, A= (Ai,j )1β€i,jβ€dhas ο¬nite p/2-variation for all p > 2.

Proof. Deο¬ne the dyadic stopping times (Οn

k)n,kβNby Οn

0:= 0 and

Οn

k+1 := inf{tβ₯Οn

k:|StβSΟn

k|= 2βn},

and set Sn

t:= PkSΟn

k1[Οn

k,Οn

k+1)(t), so that kSnβSkββ€2βn. Accorcing to (5), for typical

price paths Οthere exists C(Ο)>0 such that

ξ

ξ

ξ(SnΒ·S)(Ο)βZSdS(Ο)ξ

ξ

ξββ€C(Ο)2βnplog n.

Fix such a typical price path Ο, which is also of ο¬nite q-variation for every q > 2 (recall from

Corollary 11 that this is satisο¬ed by typical price paths). Let us show that for such Ο, the

process Ais of ο¬nite p/2-variation for every p > 2.

We have for (s, t)ββT, omitting the argument Οof the processes under consideration,

|As,t| β€ ξξξZt

s

SrdSrβ(SnΒ·S)s,tξξξ+|(SnΒ·S)s,t βSsSs,t|

β€C(Ο)2βnplog n+|(SnΒ·S)s,t βSsSs,t|.Ξ΅C(Ο)2βn(1βΞ΅)+|(SnΒ·S)s,t βSsSs,t|

for every nβN,Ξ΅ > 0. The second term on the right hand side can be estimated, using a

standard argument based on Youngβs maximal inequality (see [LCL07], Theorem 1.16), by

max{2βnvq(s, t)1/q,(#{k:Οn

kβ[s, t]})1β2/q vq(s, t)2/q +vq(s, t)2/q },(6)

13

where vq(s, t) := kSkq

qβvar,[s,t]. For the convenience of the reader, we sketch the argument:

If there exists no kfor which Οn

kβ[s, t], then |(SnΒ·S)s,t βSsSs,t| β€ 2βnvq(s, t)1/q , using

that |Ss,t| β€ vq(s, t)1/q . This corresponds to the ο¬rst term in the maximum in (6).

Otherwise, note that at the price of adding vq(s, t)2/q to the right hand side, we may

suppose that s=Οn

k0for some k0. Let now Οn

k0,...,Οn

k0+Nβ1be those (Οn

k)kwhich are in [s, t).

Without loss of generality we may suppose Nβ₯2, because otherwise (SnΒ·S)s,t =SsSs,t .

Abusing notation, we write Οn

k0+N=t. The idea is now to successively delete points (Οn

k0+β)

from the partition, in order to pass from (SnΒ·S) to SsSs,t. By super-additivity of vq, there

must exist ββ {1,...,N β1}, for which

vq(Οn

k0+ββ1, Ο n

k0+β+1)β€2

Nβ1vq(s, t).

Deleting Οn

k0+βfrom the partition and subtracting the resulting integral from (SnΒ·S)s,t, we

get

|SΟn

k0+ββ1SΟn

k0+ββ1,Οn

k0+β+SΟn

k0+βSΟn

k0+β,Οn

k0+β+1 βSΟn

k0+ββ1SΟn

k0+ββ1,Οn

k0+β+1 |

=|SΟn

k0+ββ1,Οn

k0+βSΟn

k0+β,Οn

k0+β+1 | β€ vq(Οn

k0+ββ1, Ο n

k0+β+1)2/q β€ξ2

Nβ1vq(s, t)ξ2/q.

Successively deleting all the points except Οn

k0=sand Οn

k0+N=tfrom the partition gives

|(SnΒ·S)s,t βSsSs,t| β€

N

X

k=2 ξ2

kβ1vq(s, t)ξ2/q .N1β2/qvq(s, t)2/q ,

and therefore (6). Now it is easy to see that #{k:Οn

kβ[s, t]} β€ 2nq vq(s, t) (compare also the

proof of Lemma 14), and thus

|As,t|.Ξ΅C(Ο)2βn(1βΞ΅)+ max{2βnvq(s, t)1/q ,(2nq vq(s, t))1β2/qvq(s, t)2/q +vq(s, t)2/q }

=C(Ο)2βn(1βΞ΅)+ max{2βnvq(s, t)1/q,2βn(2βq)vq(s, t) + vq(s, t)2/q }.(7)

This holds for all nβN,Ξ΅ > 0, q > 2. Let us suppose for the moment that vq(s, t)β€1 and let

Ξ± > 0 to be determined later. Then there exists nβNfor which 2βnβ1< vq(s, t)1/Ξ±(1βΞ΅)β€

2βn. Using this nin (7), we get

|As,t|Ξ±.Ξ΅,Ο,Ξ± vq(s, t) + max nvq(s, t)1/(1βΞ΅)vq(s, t)Ξ±/q , vq(s, t)(2βq)/(1βΞ΅)+Ξ±+vq(s, t)2Ξ±/q o

=vq(s, t) + max ξvq(s, t)

q+Ξ±(1βΞ΅)

q(1βΞ΅), vq(s, t)2βq+Ξ±(1βΞ΅)

1βΞ΅+vq(s, t)2Ξ±/qξ.

We would like all the exponents in the maximum on the right hand side to be larger or equal

to 1. For the ο¬rst term, this is satisο¬ed as long as Ξ΅ < 1. For the third term, we need Ξ±β₯q/2.

For the second term, we need Ξ±β₯(qβ1βΞ΅)/(1 βΞ΅). Since Ξ΅ > 0 can be chosen arbitrarily

close to 0, it suο¬ces if Ξ± > q β1. Now, since q > 2 can be chosen arbitrarily close to 2, we

see that Ξ±can be chosen arbitrarily close to 1. In particular, we may take Ξ±=p/2 for any

p > 2, and we obtain

|As,t|p/2.Ο,Ξ΄ vq(s, t)(1 + vq(s, t)Ξ΄)β€vq(s, t)(1 + vq(0, T )Ξ΄)

for a suitable Ξ΄ > 0.

14

It remains to treat the case vq(s, t)>1, for which we simply estimate

|As,t|p/2.pξ

ξ

ξZΒ·

0

SrdSrξ

ξ

ξ

p/2

β+kSkp

ββ€ξξ

ξ

ξZΒ·

0

SrdSrξ

ξ

ξ

p/2

β+kSkp

βξvq(s, t).

So for every interval [s, t] we can estimate |As,t|p/2.Ο,p vq(s, t). The claim now follows from

the super-additivity of vq.

In fact, we only used two properties of the function vqin the proof: it is nonnegative, and

if 0 β€sβ€uβ€t, then vq(s, u) + vq(u, t)β€vq(s, t). We call control function any function

c: βTβ[0,β) which satisο¬es these two properties and is moreover continuous and such that

c(t, t) = 0 for all tβ[0, T ]. Observe that if f: [0, T ]βRdsatisο¬es |fs,t|pβ€c(s, t) for all

(s, t)ββT, then the p-variation of fis bounded from above by c(0, T ).

Remark 19. Corollary 18 states that for typical price paths Ο,(S(Ο),RSdS(Ο)) is a p-rough

path for every p > 2. See Section 4 below for details on rough paths theory. To the best of

our knowledge, this is one of the ο¬rst times that a non-geometric rough path is constructed

in a non-probabilistic setting, and certainly we are not aware of any other work where rough

paths are constructed using ο¬nancial arguments.

We also point out that, thanks to Proposition 6, we gave a simple, model free, and pathwise

proof for the fact that a local martingale together with its ItΛo integral deο¬nes a rough path.

While this seems intuitively clear, the usual proofs are somewhat involved, see [CL05] or

Chapter 14 of [FV10].

The following result will allow us to obtain the rough path integral as a limit of Riemann

sums, rather than compensated Riemann sums which are usually used to deο¬ne it.

Corollary 20. Let (cn)nβNbe a sequence of positive numbers such that (cΞ΅

nβlog n)converges

to 0 for every Ξ΅ > 0. For nβNdeο¬ne Οn

0:= 0 and Οn

k+1 := inf{tβ₯Οn

k:|StβSΟn

k|=cn},

kβN, and set Sn

t=PkSΟn

k1[Οn

k,Οn

k+1)(t). Then for typical price paths, ((SnΒ·S)) converges

uniformly to RSdS. Moreover, for p > 2and for typical price paths there exists a control

function c=c(p, Ο)such that

sup

nsup

k<β

|(SnΒ·S)Οn

k,Οn

β(Ο)βSΟn

k(Ο)SΟn

k,Οn

β(Ο)|p/2

c(Οn

k, Ο n

β)β€1.

Proof. The uniform convergence of ((SnΒ·S)) to RSdSfollows from Theorem 16.

For the second claim, ο¬x nβNand k < β such that Οn

ββ€T. Then

|(SnΒ·S)Οn

k,Οn

ββSΟn

kSΟn

k,Οn

β|.ξ

ξ

ξ(SnΒ·S)βZΒ·

0

SsdSsξ

ξ

ξβ+ξξξAΟn

k,Οn

βξξξ

.Οcnplog n+vp/2(Οn

k, Ο n

β)2/p .Ξ΅c1βΞ΅

n+vp/2(Οn

k, Ο n

β)2/p,(8)

where Ξ΅ > 0 and the last estimate holds by our assumption on the sequence (cn), and where

vp/2(s, t) := kAkp/2

p/2βvar,[s,t]for (s, t)ββT. Of course, this inequality only holds for typical

price paths and not for all Οββ¦.

On the other side, the same argument as in the proof of Corollary 18 (using Youngβs

maximal inequality and successively deleting points from the partition) shows that

|(SnΒ·S)Οn

k,Οn

ββSΟn

kSΟn

k,Οn

β|.c2βq

nvq(Οn

k, Ο n

β),(9)

15

where vq(s, t) := kSkq

qβvar,[s,t]for (s, t)ββT.

Let us deο¬ne the control function Λc:= vq+vp/2. Take Ξ± > 0 to be determined below. If

cn>Λc(s, t)1/Ξ±(1βΞ΅), then we use (9) and the fact that 2 βq < 0, to obtain

|(SnΒ·S)Οn

k,Οn

ββSΟn

kSΟn

k,Οn

β|Ξ±.(Λc(Οn

k, Ο n

β)) 2βq

(1βΞ΅)vq(Οn

k, Ο n

β)Ξ±β€Λc(Οn

k, Ο n

β)

2βq+Ξ±(1βΞ΅)

(1βΞ΅).

The exponent is larger or equal to 1 as long as Ξ±β₯(qβ1βΞ΅)/(1 βΞ΅). Since qand Ξ΅can be

chosen arbitrarily close to 2 and 0 respectively, we can take Ξ±=p/2, and get

|(SnΒ·S)Οn

k,Οn

ββSΟn

kSΟn

k,Οn

β|p/2.Λc(Οn

k, Ο n

β)(1 + Λc(0, T )Ξ΄)

for a suitable Ξ΄ > 0.

On the other side, if cnβ€Λc(s, t)1/Ξ±(1βΞ΅), then we use (8) to obtain

|(SnΒ·S)Οn

k,Οn

ββSΟn

kSΟn

k,Οn

β|Ξ±.Λc(Οn

k, Ο n

β) + Λc(Οn

k, Ο n

β)2Ξ±/p,

so that also in this case we may take Ξ±=p/2, and thus we have in both cases

|(SnΒ·S)Οn

k,Οn

ββSΟn

kSΟn

k,Οn

β|p/2β€c(Οn

k, Ο n

β),

where cis a suitable (Ο-dependent) multiple of Λc.

4 Rough path integration for typical price paths

4.1 The Lyons-Gubinelli rough path integral

We have now collected all the ingredients needed to set up the rough path integral for typical

price paths. We follow more or less the lecture notes [FH13], to which we refer for a gentle

introduction to rough paths. More advanced monographs on rough paths are [LQ02, LCL07,

FV10]. The main diο¬erence to [FH13] in the derivation below is that we use p-variation

to describe the regularity, and not HΒ¨older continuity, because it is not true that all typical

price paths are HΒ¨older continuous. Also, we make an eο¬ort to give reasonably sharp results,

whereas in [FH13] the focus lies more on the pedagogical presentation of the material. We

want to point out that we are merely presenting well known results in this subsection.

Recall from Theorem 16 and Corollary 18, that for typical price paths Ο, the pro-

cess S(Ο) is of ο¬nite p-variation for every p > 2, and the integral process RSdS(Ο) =

(RSidSj(Ο))1β€i,jβ€dis a continuous function of ο¬nite p/2-variation, in the sense that there

exists a control function c=c(Ο) such that

ξξξZt

s

SrdSr(Ο)βSs(Ο)Ss,t(Ο)ξξξp/2β€c(s, t)(Ο)

for all (s, t)ββT. From now on, we ο¬x one path Swhich satisο¬es these two conditions.

Let us write S= (S, A), where

A: βTβRdΓd, A(s, t) = Zt

s

SrdSrβSsSs,t,

denotes the area of S. This name stems from the fact that if Sis smooth and two-dimensional,

then the antisymmetric part of A(s, t) corresponds to the algebraic area enclosed by the curve

16

(Sr)rβ[s,t]. It is a deep insight of Lyons [Lyo98], proving a conjecture of FΒ¨ollmer, that the area

is exactly the additional information which is needed to solve diο¬erential equations driven by

Sin a pathwise continuous manner, and to construct stochastic integrals as continuous maps.

Actually, [Lyo98] solves a much more general problem and proves that if the driving signal is

of ο¬nite q-variation for some q > 1, then it has to be equipped with the iterated integrals up

to order βqβ β 1 to obtain a continuous integral map. The for us relevant case qβ(2,3) was

already treated in [Lyo95a].

We say that Sis of ο¬nite p-variation if there exists a control function csuch that

|Ss,t|p+|A(s, t)|p/2β€c(s, t) (10)

for all 0 β€sβ€tβ€T. In that case, we deο¬ne

kSkpβvar := kSkpβvar +kAkp/2βvar.

From now on we ο¬x p > 2 and we assume that Sis of ο¬nite p-variation. We call Sap-rough

path or simply a rough path. Gubinelli [Gub04] observed that for every rough path, there is a

naturally associated Banach space of integrands, the space of controlled paths. Heuristically,

a path Fis controlled by S, if it locally βlooks like Sβ, modulo a smooth remainder. The

precise deο¬nition is:

Deο¬nition 21. Let qbe such that 2/p + 1/q > 1. Let F: [0, T ]βRnand Fβ²: [0, T ]βRnΓd.

We say that the pair (F, F β²) is controlled by Sif the derivative Fβ²has ο¬nite q-variation, and

the remainder RF: βTβRndeο¬ned by

RF(s, t) = Fs,t βFβ²

sSs,t,

has ο¬nite r-variation, where 1/r = 1/p + 1/q. In this case, we write (F, F β²)βCq

S=Cq

S(Rn),

and deο¬ne

k(F, F β²)kCq

S:= kFβ²kqβvar +kRFkrβvar.

If it is equipped with the norm |F0|+|Fβ²

0|+k(F, F β²)kCq

S, then the space of controlled paths

Cq

Sis a Banach space.

Naturally, the function Fβ²should be interpreted as the derivative of Fwith respect to S.

The reason for considering couples (F, F β²) and not just functions Fis that the smoothness

requirement on the remainder RFusually does not determine Fβ²uniquely for a given path F.

For example, if Fand Sboth have ο¬nite r-variation rather than just ο¬nite p-variation, then

for every Fβ²of ο¬nite q-variation we have (F, F β²)βCq

S.

Note that we do not require For Fβ²to be continuous. We will point out below why this

does not pose any problems.

To obtain a more βquantitativeβ feeling for the condition on q, recall that according to

our results from Section 3, for typical price paths we may choose p > 2 arbitrarily close to 2.

Then 2/p + 1/q > 0 as long as q > 0, so that the derivative Fβ²may essentially be as irregular

as we like. The remainder RFhas to be of ο¬nite r-variation for 1/r = 1/p + 1/q, so in other

words it should be of ο¬nite r-variation for some r < 2 and thus slightly more regular than a

typical price path.

17

Example 22. Let ΟβC2

band deο¬ne Fs:= Ο(Ss) and Fβ²

s:= Οβ²(Ss). Then (F, F β²)βCp

S:

Clearly Fβ²has ο¬nite p-variation. For the remainder, we have

|RF(s, t)|p/2=|Ο(St)βΟ(Ss)βΟβ²(Ss)Ss,t|p/2β€ξ1

2kΟβ²β²kβ|Ss,t|2ξp/2= 2βp/2kΟβ²β²kp/2

β|Ss,t|p.

Since Sis of ο¬nite p-variation, RFis of ο¬nite p/2-variation. Now 1/(p/2) = 1/p + 1/p, and

thus (F, F β²)βCp

S.

As the image of the continuous path Sis compact, it is not actually necessary to assume

that Οis bounded. We may always consider a C2function Οof compact support, such that

Οagrees with Οon the image of S.

It is instructive to examine under which regularity conditions on Οwe obtain a controlled

path if Sis a typical price path. As we argued above, Οβ²(S) should be of ο¬nite q-variation

for some q > 0, which is satisο¬ed as long as Οβ²is Ξ΅-HΒ¨older continuous for some Ξ΅ > 0. The

remainder RF(s, t) = Ο(St)βΟ(Ss)βΟβ²(Ss)Ss,t should be of ο¬nite r-variation for some r < 2.

A simple calculation shows that this is satisο¬ed as long as ΟβC1+Ξ΅for some Ξ΅ > 0, so that

for such Οwe obtain a controlled path.

The example also shows that in general RF(s, t) is not a path increment of the form

RF(s, t) = GtβGsfor some function Gdeο¬ned on [0, T ], but really a function of two variables.

Example 23. Let Gbe a path of ο¬nite r-variation for some rwith 1/p + 1/r > 1. Setting

(F, F β²) = (G, 0), we obtain a controlled path in Cq

S, where 1/q = 1/r β1/p. In combination

with Theorem 24 below, this example shows in particular that the controlled rough path

integral extends the Young integral and the Riemann-Stieltjes integral.

The basic idea of rough path integration is that if we already know how to deο¬ne RSdS,

and if Flooks like Son small scales, then we should be able to deο¬ne RFdSas well. The

precise result is given by the following theorem:

Theorem 24 (Theorem 4.9 in [FH13], see also [Gub04], Theorem 1).Let qbe such that

2/p + 1/q > 1. Let (F, F β²)βCq

S. Then there exists a unique function RFdSβC([0, T ],Rn)

which satisο¬es

ξξξZt

s

FudSuβFsSs,t βFβ²

sA(s, t)ξξξ.kSkpβvar,[s,t]kRFkrβvar,[s,t]+kAkp/2βvar,[s,t]kFβ²kqβvar,[s,t]

for all (s, t)ββT. The integral is given as limit of the compensated Riemann sums

Zt

0

FudSu= lim

mββ X

[s1,s2]βΟmξFs1Ss1,s2+Fβ²

s1A(s1, s2)ξ,(11)

where (Οm)is any sequence of partitions of [0, t]with mesh size going to 0.

The map (F, F β²)7β (G, Gβ²) := (RFudSu, F )is continuous from Cq

Sto Cp

Sand satisο¬es

k(G, Gβ²)kCp

S.kFkp+ (kFβ²kβ+kFβ²kqβvar)kAkp/2βvar +kSkpβvarkRFkrβvar.

Remark 25. To the best of our knowledge, there is no publication in which the controlled

path approach to rough paths is formulated using p-variation regularity. Instead, the references

on the subject all work with HΒ¨older continuity. But in the p-variation setting, all the proofs

work exactly as in the HΒ¨older setting, and it is a simple exercise to translate the proof of

18

Theorem 4.9 in [FH13] (which is based on Youngβs maximal inequality that we encountered

above) to obtain Theorem 24.

There is only one small pitfall: We did not require For Fβ²to be continuous. The rough

path integral for discontinuous functions is somewhat tricky, see [Wil01]. But here we do

not run into any problems, because the integrand S= (S, A)is continuous. The convergence

proof based on Youngβs maximal inequality works as long as integrand and integrator have no

common discontinuities, see the Theorem on p. 264 of [You36].

If now ΟβC1+Ξ΅

bfor some Ξ΅ > 0, then using a Taylor expansion one can show that there

exist p > 2 and q > 0 with 2/p + 1/q > 0, such that (F, F β²)7β (Ο(F), Οβ²(F)Fβ²) is a locally

bounded map from Cp

Sto Cq

S. Combining this with the fact that the rough path integral is a

bounded map from Cq

Sto Cp

S, it is not hard to prove the existence of solutions to the rough

diο¬erential equation

dXt=Ο(Xt)dSt, X0=x0,(12)

tβ[0, T ], where XβCp

S,RΟ(Xt)dStdenotes the rough path integral, and Sis a typical price

path. Similarly, if ΟβC2+Ξ΅

b, then there exist p > 2 and q > 0 with 2/p + 1/q > 0, such that

the map (F, F β²)7β (Ο(F), Οβ²(F)Fβ²) is a locally Lipschitz continuous from Cp

Sto Cq

S, and this

yields the uniqueness of the solution to (12) β at least among the functions in the Banach

space Cp

S. See Section 5.3 of [Gub04] for details.

A remark is in order about the stringent regularity requirements on Ο. In the classical ItΛo

theory of SDEs, the function Οis only required to be Lipschitz continuous. But to solve a

Stratonovich SDE, we need better regularity of Ο. This is natural, because the Stratonovich

SDE can be rewritten as an ItΛo SDE with a Stratonovich correction term: the equations

dXt=Ο(Xt)β¦dWtand

dXt=Ο(Xt)dWt+1

2Οβ²(Xt)Ο(Xt)dt

are equivalent (where Wis a standard Brownian motion, dWtdenotes ItΛo integration, and

β¦dWtdenotes Stratonovich integration). To solve the second equation, we need Οβ²Οto be

Lipschitz continuous, which is always satisο¬ed if ΟβC2

b. But rough path theory cannot

distinguish between ItΛo and Stratonovich integrals: If we deο¬ne the area of Wusing ItΛo

(respectively Stratonovich) integration, then the rough path solution of the equation will

coincide with the ItΛo (respectively Stratonovich) solution. So in the rough path setting,

the function Οshould satisfy at least the same requirements as in the Stratonovich setting.

The regularity requirements on Οare essentially sharp, see [Dav07], but the boundedness

assumption can be relaxed, see [Lej12]. See also Section 10.5 of [FV10] for a slight relaxation

of the regularity requirements in the Brownian case.

Of course, the most interesting result of rough path theory is that the solution to a rough

diο¬erential equation depends continuously on the driving signal. This is a consequence of the

following observation:

Propostion 26 (Proposition 9.1 of [FH13]).Let p > 2and q > 0with 2/p + 1/q > 0. Let

S= (S, A)and Λ

S= ( Λ

S, Λ

A)be two rough paths of ο¬nite p-variation, let (F, F β²)βCq

Sand

(Λ

F , Λ

Fβ²)βCq

Λ

S, and let (s, t)ββT. Then for every M > 0there exists CM>0such that

ξ

ξ

ξZΒ·

0

FsdSsβZΒ·

0

Λ

FsdΛ

Ssξ

ξ

ξpβvar β€CMξ|F0βΛ

F0|+|Fβ²

0βΛ

Fβ²

0|+kFβ²βΛ

Fβ²kqβvar

+kRFβRΛ

Fkrβvar +kSβΛ

Skpβvar +kAβΛ

Akp/2βvarξ,

19

as long as

max{|Fβ²

0|+k(F, F β²)kCq

S,|Λ

Fβ²

0|+k(Λ

F , Λ

Fβ²)kCq

Λ

S

,kSkpβvar,kAkp/2βvar,kΛ

Skpβvar,kΛ

Akp/2βvar} β€ M.

In other words, the rough path integral depends on integrand and integrator in a locally

Lipschitz continuous way, and therefore it is no surprise that the solutions to diο¬erential

equations driven by rough paths depend continuously on the signal.

4.2 The rough path integral as limit of Riemann sums

When trying to apply the rough path integral in ο¬nancial mathematics, we encounter a small

philosophical problem. As we have seen in Theorem 24, the rough path integral RFdSis

given as limit of the compensated Riemann sums

Zt

0

FsdSs= lim

mββ X

[r1,r2]βΟmξFr1Sr1,r2+Fβ²

r1A(r1, r2)ξ,

where (Οm) is any sequence of partitions of [0, t] with mesh size going to 0. The term Fr1Sr1,r2

has an obvious ο¬nancial interpretation. This is the proο¬t that we make by buying Fr1units of

the traded asset at time r1and by selling them at time r2. However, for the βcompensatorβ

Fβ²

r1A(r1, r2) there seems to be no ο¬nancial interpretation, and therefore it is not clear whether

the rough path integral can be understood as proο¬t obtained by investing in S.

However, we observed in Section 3 that along suitable stopping times (Οn

k)n,k, we have

Zt

0

SsdSs= lim

nββ X

k

SΟn

kSΟn

kβ§t,Οn

k+1β§t.

By the philosophy of controlled paths, we expect that also for Fwhich looks like Son small

scales we should obtain

Zt

0

FsdSs= lim

nββ X

k

FΟn

kSΟn

kβ§t,Οn

k+1β§t,

without having to introduce the compensator Fβ²

Οn

kA(Οn

kβ§t, Ο n

k+1 β§t) in the Riemann sum. With

the results we have at hand, this statement is actually relatively easy to prove. Nonetheless,

it seems not to have been observed before.

To set the stage, we ο¬rst present a special case for which we can give an elementary proof.

For this purpose, we use again the dyadic stopping times Οn

0:= 0 and Οn

k+1 := inf{tβ₯Οn

k:

|StβSΟn

k| β₯ 2βn}for k, n βN. We write Sn

t:= Pk1[Οn

k,Οn

k+1)(t)SΟn

k. Recall from Theorem 16

that for typical price paths.

ξ

ξ

ξ(SnΒ·S)βZΒ·

0

SsdSsξ

ξ

ξβ.2βnplog n. (13)

Lemma 27. Let Ξ΅ > 0, let FβC1+Ξ΅

b(Rd,Rd), and suppose Sfulο¬lls (13). Then the Riemann

sums β

X

k=0

F(SΟn

k)SΟn

kβ§t,Οn

k+1β§t, t β[0, T ],

converge uniformly in C([0, T ],R).

20

Proof. Let us write Fn

t:= F(Sn

t). Using a ο¬rst order Taylor expansion, we obtain for nβN

and tβ[0, T ] that

(Fn+1 Β·S)tβ(FnΒ·S)t=

β

X

β=0

β

X

k=0,

[Οn+1

k,Οn+1

k+1 ]β[Οn

β,Οn

β+1]

F(S)Οn

β,Οn+1

kSΟn+1

kβ§t,Οn+1

k+1 β§t(14)

=

β

X

β=0

β

X

k=0,

[Οn+1

k,Οn+1

k+1 ]β[Οn

β,Οn

β+1]

(Fβ²(SΟn

β)SΟn

β,Οn+1

k+RΟn

β,Οn+1

k)SΟn+1

kβ§t,Οn+1

k+1 β§t

with a remainder Rthat satisο¬es |RΟn

β,Οn+1

k|.kFkC1+Ξ΅

b2β(1+Ξ΅)nfor all n, β, k.

It is a simple observation, which we will prove in Lemma 28 below, that the uniform

convergence of (SnΒ·S) to RSdSimplies the existence of the quadratic variation of Salong

(Οn

k)n,k. In particular, Lemma 14 yields max{k:Οn

kβ€t}.22nVn

tfor a uniformly bounded

sequence of increasing functions (Vn). Choose now m=m(n) := βn/3β. Applying HΒ¨olderβs

inequality gives us

ξξ(Fn+1 Β·S)tβ(FnΒ·S)tξξ

β€

β

X

j=0

β

X

β=0,

[Οn

β,Οn

β+1]β[Οm

j,Οm

j+1]

β

X

k=0,

[Οn+1

k,Οn+1

k+1 ]β[Οn

β,Οn

β+1]ξξFβ²(S)Οm

j,Οn

βSΟn

β,Οn+1

kSΟn+1

kβ§t,Οn+1

k+1 β§tξξ

+ξξξξ

β

X

β=0

β

X

k=0,

[Οn+1

k,Οn+1

k+1 ]β[Οn

β,Οn

β+1]

RΟn

β,Οn+1

kSΟn+1

kβ§t,Οn+1

k+1 β§tξξξξ

+

β

X

j=0 ξξFβ²(SΟm

j)ξξξξξξ

β

X

β=0,

[Οn

β,Οn

β+1]β[Οm

j,Οm

j+1]

β

X

k=0,

[Οn+1

k,Οn+1

k+1 ]β[Οn

β,Οn

β+1]

SΟn

β,Οn+1

kSΟn+1

kβ§t,Οn+1

k+1 β§tξξξξ

.(22nVn

T)kFkC1+Ξ΅

b2βmΞ΅2β2n+ (22nVn

T)kFkC1+Ξ΅

b2β(1+Ξ΅)n2βn

+

β

X

j=0 ξξFβ²(SΟm

j)ξξξξ(SnΒ·S)Οm

jβ§t,Οm

j+1β§tβ(Sn+1 Β·S)Οm

jβ§t,Οm

j+1β§tξξ

.sup

NβN

(VN

T)kFkC1+Ξ΅

b(2βnΞ΅/3+ 2βnΞ΅) + kFβ²kβsup

NβN

(VN

T)22m2βnlog n.

Since mβ€n/3, the right hand side is summable in n, which gives the uniform convergence.

For the remainder of this section, we work under the following assumption:

Assumption (rie).Assume Οn={0 = tn

0< tn

1<Β·Β·Β·< tn

Nn=T},nβN, is a given sequence

of partitions such that sup{|Stn

k,tn

k+1 |:k= 0,...,Nnβ1}converges to 0, and set

Sn

t:=

Nnβ1

X

k=0

Stn

k1[tn

k,tn

k+1)(t).

21

We assume that the Riemann sums (SnΒ·S) converge uniformly to a continuous function

RSdS. We also assume that pβ(2,3) and that there exists a control function cfor which

sup

(s,t)ββT

|Ss,t|p

c(s, t)+ sup

nsup

0β€k<ββ€Nn

|(SnΒ·S)tn

k,tn

ββStn

kStn

k,tn

β|p/2

c(tn

k, tn

β)β€1.(15)

Our general proof that the rough path integral is given as limit of Riemann sums is

somewhat indirect. We translate everything from ItΛo type integrals to related Stratonovich

type integrals, for which the convergence follows from the continuity of the rough path integral,

Proposition 26. Then we translate everything back to our ItΛo type integrals. To go from ItΛo

to Stratonovich integral, we ο¬rst need the quadratic variation:

Lemma 28. Under Assumption (rie), let 1β€i, j β€d, and deο¬ne

hSi, Sjit:= Si

tSj

tβSi

0Sj

0βZt

0

Si

rdSj

rβZt

0

Sj

rdSi

r.

Then hSi, Sjiis a continuous function and

hSi, Sjit= lim

nββhSi, Sjin

t= lim

nββ

Nnβ1

X

k=0

(Si

tn

k+1β§tβSi

tn

kβ§t)(Sj

tn

k+1β§tβSj

tn

kβ§t).(16)

The sequence (hSi, Sjin)nis of uniformly bounded total variation, and in particular hSi, Sji

is of bounded variation. We write hSi=hS, Si= (hSi, Sji)1β€i,jβ€d, and call hSithe quadratic

variation of S.

Proof. The function hSi, Sjiis continuous by deο¬nition. The speciο¬c form (16) of hSi, S ji

follows from two simple observations:

Si

tSj

tβSi

0Sj

0=

Nnβ1

X

k=0 ξSi

tn

k+1β§tSj

tn

k+1β§tβSi

tn

kβ§tSj

tn

kβ§tξ

for every nβN, and

Si

tn

k+1β§tSj

tn

k+1β§tβSi

tn

kβ§tSj

tn

kβ§t=Si

tn

kβ§tSj

tn

kβ§t,tn

k+1β§t+Sj

tn

kβ§tSi

tn

kβ§t,tn

k+1β§t+Si

tn

kβ§t,tn

k+1β§tSj

tn

kβ§t,tn

k+1β§t,

so that the convergence in (16) is a consequence of the convergence of (SnΒ·S) to RSdS.

To see that hSi, Sjiis of bounded variation, note that

Si

tn

kβ§t,tn

k+1β§tSj

tn

kβ§t,tn

k+1β§t=1

4ξξ(Si+Sj)tn

kβ§t,tn

k+1β§tξ2βξ(SiβSj)tn

kβ§t,tn

k+1β§tξ2ξ

(read hSi, Sji= 1/4(hSi+SjiβhSiβSji)). In other words, the n-th approximation of hSi, Sji

is the diο¬erence of two increasing functions, and its total variation is bounded from above by

Nnβ1

X

k=0 ξξ(Si+Sj)tn

k,tn

k+1 ξ2+ξ(SiβSj)tn

k,tn

k+1 ξ2ξ.sup

m

Nmβ1

X

k=0 ξ(Si

tm

k,tm

k+1 )2+ (Sj

tm

k,tm

k+1 )2ξ.

Since the right hand side is ο¬nite, also the limit hSi, Sjiis of bounded variation.

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