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Pathwise stochastic integrals for model free finance

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Abstract

We show that Lyons' rough path integral is a natural tool to use in model free financial mathematics by proving that it is possible to make an arbitrarily large profit 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 define it. This proves that the rough path integral is really an extension of F\"ollmer's pathwise It\^o integral. Moreover, we construct a "model free It\^o integral" in the spirit of Karandikar.
arXiv:1311.6187v1 [math.PR] 24 Nov 2013
Pathwise stochastic integrals for model free ο¬nance β
Nicolas Perkowskiβ
UniversitΒ΄e Paris-Dauphine
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¨ollmerβs pathwise ItΛo
integral. Moreover, we construct a βmodel free ItΛo integralβ in the spirit of Karandikar.
1 Introduction
In this paper, we use Vovkβs [Vov12] game-theoretic approach to mathematical ο¬nance to set
up an integration theory for βtypical price pathsβ. Vovkβs 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 Vovkβs 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 βmodel free ItΛo isometryβ, based on which we
construct pathwise ItΛo 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 ItΛo 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 ItΛo 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
βWe 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 βtoo
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
We follow Gubinelliβs [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=FtStβF0S0βRt
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¨ollmerβs [FΒ¨ol81] βcalcul dβItΛo sans probabilitΒ΄esβ: FΒ¨ollmer shows that if the quadratic
variation hSiexists along a sequence of partitions, then for every gradient βF, the inte-
gral RβF(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¨ollmerβs pathwise ItΛo calculus. We also show that
FΒ¨ollmerβs 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¨ollmerβs 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 : βadmissible strategy Hs.t. Ξ»+ZT
0
HsdSsβ₯XPβa.s.o,(1)
and we have
p(X) = sup{EQ[X] : Qis equivalent to Pand Sis a Qβlocal 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(ΟdWt+bdt),
where Wis a one-dimensional standard Brownian motion, Ο > 0, and bβR. 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 Οβ[Ο1, Ο2], and not the exact value
of Οβ 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 PΟfor Οβ[Ο1, Ο2],
where PΟdenotes 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
βa.s.β assumption in (1) by βa.s. under every PΟ,Οβ[Ο1, Ο2]β. But then the stochastic
integral RHdShas to be constructed simultaneously under all the measures PΟ, which is not
a trivial task because PΟand PΛΟare mutually singular for Ο6= ΛΟ. Luckily it turns out that
here we can essentially still use ItΛoβs 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¬ (StβK)+for 0 β€tβ€Tand KβRare
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¨ollmerβs pathwise ItΛo 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¨ollmerβs 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
However, FΒ¨ollmerβs integral essentially only applies to one-dimensional integrators. If we
are considering more than one ο¬nancial asset, then we can only integrate gradients RβF(S)dS
3
with FΒ¨ollmerβs 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¨ollmerβs 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¨ollmerβs and
Youngβs integrals.
Plan of the paper
In Section 2 we brieο¬y recall Vovkβs game-theoretic approach to mathematical ο¬nance. Sec-
tion 3 is devoted to the construction of a model free ItΛo 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¨ollmerβs pathwise ItΛo 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) = Οi(t), where Ο= (Ο1,...,Οd). 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 FΟare 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 Οββ¦, and not modulo null sets, as it is usually
assumed in stochastic analysis.
The indicator function of a set Ais denoted by 1A.
Apartition Οof [0, T ] is a ο¬nite set of time points, Ο={0 = t0< t1<Β·Β·Β·< tm=T}.
Occasionally, we will identify Οwith the set of intervals {[t0, t1],[t1, t2],...,[tmβ1, tm]}, and
write expressions like P[s,t]βΟ.
For f: [0, T ]βRnand t1, t2β[0, T ], denote ft1,t2:= ft2βft1and deο¬ne the p-variation
of frestricted to [s, t]β[0, T ] as
kfkpβvar,[s,t]:= sup ξξmβ1
X
k=0 |ftk,tk+1 |pξ1/p
:s=t0<Β·Β·Β·< tm=t, m βNξ, p > 0,(2)
(possibly taking the value +β). We set kfkpβvar := kfkpβvar,[0,T ]. We write βT={(s, t) :
0β€sβ€tβ€T}for the simplex and deο¬ne the p-variation of a function g: βTβRnin 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 βΞ±βtimes continuously
diο¬erentiable, with (Ξ±β βΞ±β)-HΒ¨older continuous partial derivatives of order βΞ±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:=
βΞ±β
X
k=0 kDkfkβ+kDβΞ±βfkΞ±ββΞ±β,
where kΒ·kΞ²denotes the Ξ²-HΒ¨older norm for Ξ²β(0,1), and kΒ·kβdenotes the supremum norm.
For x, y βRd, 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 aβbif 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 βtypical 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 ]βRdis called a
simple strategy if there exist stopping times 0 = Ο0< Ο1< . . . , and FΟn-measurable bounded
functions Fn: β¦ βRd, such that for every Οββ¦ we have Οn(Ο) = βfor all but ο¬nitely many
n, and such that
Ht(Ο) =
β
X
n=0
Fn(Ο)1(Οn(Ο),Οn+1(Ο)](t).
In that case, the integral
(HΒ·S)t(Ο) :=
β
X
n=0
Fn(Ο)(SΟn+1β§t(Ο)βSΟnβ§t(Ο)) =
β
X
n=0
Fn(Ο)SΟnβ§t,Οn+1β§t(Ο)
is well deο¬ned for all Οββ¦, tβ[0, T ]. Here Fn(Ο)SΟnβ§t,Οn+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₯ βΞ»for
all Οββ¦, tβ[0, T ]. The set of Ξ»-admissible simple strategies is denoted by HΞ».
Deο¬nition 1. The outer measure of Aββ¦ is deο¬ned as the cheapest superhedging price for
1A, that is
P(A) := inf nΞ» > 0 : β(Hn)nβNβ HΞ»s.t. lim inf
nββ (Ξ»+ (HnΒ·S)T(Ο)) β₯1A(Ο)βΟββ¦o.
A set of paths Aββ¦ 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 ItΛo 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 Οββ¦, and not just almost surely.
Remark 2 ([Vov12], p. 564).An equivalent deο¬nition of Pwould be
e
P(A) := infξΞ» > 0 : β(Hn)nβNβ HΞ»s.t. lim inf
nββ sup
tβ[0,T ]
(Ξ»+ (HnΒ·S)t(Ο)) β₯1A(Ο)βΟββ¦ξ.
Clearly e
Pβ€P. To see the opposite inequality, let e
P(A)< Ξ». Let (Hn)nβNβ HΞ»be a sequence
of simple strategies such that lim inf nββ suptβ[0,T](Ξ»+ (HnΒ·S)t)β₯1A, and let Ξ΅ > 0. Deο¬ne
Οn:= inf{tβ[0, T ] : Ξ»+Ξ΅+ (HnΒ·S)tβ₯1}. Then the stopped strategy Gn
t(Ο) := Hn
t(Ο)1t<Οn(Ο)
is in HΞ»β HΞ»+Ξ΅and
lim inf
nββ (Ξ»+Ξ΅+ (GnΒ·S)T(Ο)) β₯lim inf
nββ 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 AβB;
- if (An)nβNis 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, nβN, and let (Hn,m)mβNbe a sequence of (P(An) + Ξ΅2βnβ1)-admissible simple
strategies such that lim inf mββ(P(An) + Ξ΅2βnβ1+ (Hn,m Β·S)T)β₯1An. Deο¬ne for mβNthe
(PnP(An) + Ξ΅)-admissible simple strategy Gm:= Pm
n=0 Hn,m. Then by Fatouβs lemma
lim inf
mββ ξβ
X
n=0
P(An) + Ξ΅+ (GmΒ·S)Tξβ₯
k
X
n=0 ξP(An) + Ξ΅2βnβ1+ lim inf
mββ (Hn,m Β·S)Tξ
β₯1Sk
n=0 An
for all kβN. 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 Aββ¦is a null set if and only if there exists a sequence of 1-admissible
simple strategies (Hn)nβ H1such that
lim inf
nββ (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 nβNthere exists a sequence of simple strategies
(Hn,m)mβNβ H2βnβ1such that 2βnβ1+ lim inf mββ(Hn,m Β·Ο)Tβ₯1A(Ο) for all Οββ¦. Deο¬ne
Gm:= Pm
n=0 Hn,m, so that Gmβ H1. For every kβNwe obtain
lim inf
mββ (1 + (GmΒ·S)T)β₯
k
X
n=0
(2βnβ1+ lim inf
mββ (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
nββ sup
HβH1
P(1 + (HΒ·S)Tβ₯n) = 0.
Thus, we can interpret a null set Aββ¦ 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)nβNβ H1be such that lim inf n(HnΒ·S)Tβ₯ β Β· 1A. Then for every c > 0
nββ (HnΒ·S)T> cξξβ€lim
kβ₯n{(HkΒ·S)T> c}ξξ
β€sup
HβH1
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 infnββ(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 Vovkβs outer measure
Our deο¬nition of the outer measure Pis not exactly the same as Vovkβs [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»:= ξβ
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 Οββ¦ 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)) βΞ»
is well deο¬ned and takes values in [βΞ», β]. Vovk then deο¬nes for Aββ¦ the cheapest
superhedging price as
Q(A) := inf ξΞ» > 0 : βGβ SΞ»s.t. Ξ»+ (GΒ·S)Tβ₯1Aξ.
Vovkβs 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 Aββ¦. 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)nβNdeο¬nes a sequence of simple strategies in HΞ», such that
lim inf
nββ ξΞ»+ξξ n
X
k=0
HkξΒ·SξTξ=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:= {Οββ¦ : kS(Ο)kpβvar =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 Aββ¦ is βinvariant under time changesβ and such that S0(Ο) = 0
for all ΟβA, 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 ItΛo integral
The present section is devoted to the pathwise construction of an ItΛo type integral for typical
price paths. The main ingredient in the construction of the integral is a (weak) type of model
free ItΛo 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 ]βRdis called adapted if the random variable
Ο7β 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 Οββ¦.
For proving our weak ItΛo isometry, we will need an appropriate sequence of stopping times:
Let Ο= (Ο1,...,Οd)βC([0, T ],Rd) and nβN. For each i= 1,...,d, deο¬ne inductively
Οn,i
0(Ο) := 0, Οn,i
k+1(Ο) := inf ξtβ₯Οn,i
k:|Οi(t)βΟi(Οn,i
k)| β₯ 2βnξ, k βN.
Note that we are working with continuous paths and we are considering entrance times into
closed sets. Therefore, the (Οn,i) are indeed stopping times, despite the fact that our ο¬ltration
(Ft) is neither complete nor right-continuous. Denote by Οn,i the partition corresponding to
(Οn,i
k)kβN, that is Οn,i := {Οn,i
k:kβN}. To obtain an increasing sequence of partitions, we
take the union of the (Οn,i). More precisely, for nβNwe deο¬ne Οn
0:= 0 and then
Οn
k+1(Ο) := min ξt > Οn
k(Ο) : tβ
d
[
i=1
Οn,i(Ο)ξ, k βN,(4)
and we write Οn:= {Οn
k:kβN}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 Οββ¦, the quadratic variation
along (Οn,i(Ο))nβNexists. That is,
Vn,i
t(Ο) :=
β
X
k=0 ξΟi(Οn,i
k+1 β§t)βΟi(Οn,i
kβ§t)ξ2, t β[0, T ], n βN,
converges uniformly to a function hSii(Ο)βC([0, T ],R)for all iβ {1,...,d}.
For later reference, let us estimate Nn
t:= max{kβN:Οn
kβ€t}, the number of stopping
times Οn
k6= 0 in Οnwith values in [0, t]:
Lemma 14. For all Οββ¦,nβN, and tβ[0, T ], we have
2β2nNn
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{kβN:Οn,i
kβ€t}. Since Οiis continuous, we
have |Οi(Οn,i
k+1)βΟi(Οn,i
k)|= 2βnas long as Οn,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)β1
X
k=0
1
2β2nξΟ(Οn,i
k+1)βΟ(Οn,i
k)ξ2β€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 ]βRdis called a step function if there exist stopping times
0 = Ο0< Ο1< . . . , and FΟn-measurable functions Fn: β¦ βRd, such that for every Οββ¦ we
have Οn(Ο) = βfor all but ο¬nitely many n, and such that
Ht(Ο) =
β
X
n=0
Fn(Ο)1[Οn(Ο),Οn+1(Ο))(t).
Note that for notational convenience, we are now considering the interval [Οn(Ο), Οn+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
FnSΟnβ§t,Οn+1β§t=
β
X
n=0
HΟnSΟnβ§t,Οn+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 ItΛo isometry).Let a > 0and let Hbe a step function such that
kH(Ο)kββ€afor all Οββ¦. Then for all b, c > 0we have
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[Οn,Οn+1)(t). Let nβNand deο¬ne Οn
0:= 0 and then for kβN
Οn
k+1 := min ξt > Οn
k:tβΟnβͺ {Οm:mβN}ξ,
where we recall that Οn={Οn
k:kβN}is the n-th generation dyadic partition generated
by S. For tβ[0, T ], we have (HΒ·S)t=PkHΟn
kSΟn
kβ§t,Οn
k+1β§t. Since kH(Ο)kββ€a, and by
deο¬nition of Οn(Ο), we get
sup
tβ[0,T ]ξξHΟn
kSΟn
kβ§t,Οn
k+1β§tξξβ€aβd2βn.
Hence, the pathwise Hoeο¬ding inequality, Lemma 35 in Appendix A, yields for every Ξ»βR
the existence of a 1-admissible simple strategy GΞ»,n β H1such that
1 + (GΞ»,n Β·S)tβ₯exp ξΞ»(HΒ·S)tβΞ»2
2(N(Οn)
t+ 1)2β2na2dξ=: EΞ»,n
t
for all tβ[0, T ], where
N(Οn)
t:= max{k:Οn
kβ€t} β€ Nn
t+N(Ο)
t:= Nn
t+ max{k:Οkβ€t}.
By Lemma 14, we have Nn
tβ€22nVn
t, so that
EΞ»,n
tβ₯exp ξΞ»(HΒ·S)tβΞ»2
2Vn
Ta2dβΞ»2
2(N(Ο)
T+ 1)2β2na2dξ.
If now k(HΒ·S)kββ₯abβcand hSiTβ€c, then
lim inf
nββ sup
tβ[0,T ]
EΞ»,n
t+EβΞ»,n
t
2β₯1
2exp ξΞ»abβcβΞ»2
2ca2dξ.
The argument inside the exponential is maximized for Ξ»=b/(aβcd), 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 ItΛo isometry is the key
ingredient in the construction of the ItΛo integral. The term βmodel free ItΛo 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¦ ββ¦, such that if (cm)mβNis a sequence of strictly positive numbers, and if
(Hm)mβNis a sequence of step functions with kHm(Ο)βH(Ο)kββ€cmfor all Οββ¦and all
mβN, then for typical price paths Οthere exists a constant C(Ο)>0such that
ξ
ξ
ξ(HmΒ·S)(Ο)βZHdS(Ο)ξ
ξ
ξββ€C(Ο)cmplog m(5)
for all mβN. In particular, if (cmβlog 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
ItΛo integral of Hwith respect to S.
11
Proof. For the construction of RHdS, we consider dyadic approximations of H: Deο¬ne for
nβNthe stopping times Οn
0:= 0 and then inductively
Οn
k+1 := inf{tβ₯Οn
k:|HtβHΟn
k| β₯ 2βn}.
Set also Gn
t:= Pβ
k=0 HΟn
k1[Οn
k,Οn
k+1)(t), so that kGnβHkββ€2βn, and thus kGnβGn+1kββ€
2βn+1. We claim that
Pξlim sup
nββ k(GnΒ·S)β(Gn+1 Β·S)kβ
2βnβlog n=βξ= 0.
Since P(hSiT=β) = 0 and by countable subadditivity of P, it suο¬ces to show
Pξξlim sup
nββ k(GnΒ·S)β(Gn+1 Β·S)kβ
for every c > 0. But we obtain from Lemma 15 that
P \
mβN[
β€
β
X
n=n0
β€
β
X
n=n0
exp ξβ4dlog n
2dξ=
β
X
n=n0
1
n2
for all n0βN. Since the right hand side converges to zero as n0tends to β, we conclude that
for typical price paths Οthere exists C(Ο)β₯0 such that k(GnΒ·S)(Ο)β(Gn+1 Β·S)(Ο)kβ<
C(Ο)2βnβlog nfor all n, and in particular ((GnΒ·S)(Ο))nis a Cauchy sequence. We deο¬ne
ZHdS(Ο)(t) := (limnββ(GnΒ·S)t(Ο),if ((GnΒ·S)(Ο))nconverges uniformly,
0,otherwise.
Let now (Hm)mβNbe a sequence of step functions that approximates Huniformly, such that
kHmβHkββ€cm, where (cm) is a sequence of strictly positive numbers, and let c > 0. Then
for every nβN. 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 2βnβ€cm, then kHmβGnkββ€2cm, so in
particular the ο¬rst term on the right hand side can be estimated by
m2.
Since this is summable in m, we conclude as before that for typical price paths Οthere exists
C(Ο)>0 such that k(HmΒ·S)(Ο)βRHdS(Ο)kββ€C(Ο)cmβlog mfor all m.
12
Remark 17. The pathwise ItΛo 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 ItΛo integral (recall Proposi-
tion 6). The model free ItΛo 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 ItΛo 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 βuniversal 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 βprobabilisticβ
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 βlook 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)ββT,Οββ¦, and i, j β {1,...,d}deο¬ne
Ai,j
s,t(Ο) := Zt
s
Si
rdSj
r(Ο)βSi
s(Ο)Sj
s,t(Ο) := Zt
0
Si
rdSj
r(Ο)βZs
0
Si
rdSj
r(Ο)βSi
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 (Οn
k)n,kβNby Οn
0:= 0 and
Οn
k+1 := inf{tβ₯Οn
k:|StβSΟn
k|= 2βn},
and set Sn
t:= PkSΟn
k1[Οn
k,Οn
k+1)(t), so that kSnβSkββ€2βn. Accorcing to (5), for typical
price paths Οthere exists C(Ο)>0 such that
ξ
ξ
ξ(SnΒ·S)(Ο)βZSdS(Ο)ξ
ξ
ξββ€C(Ο)2βnplog 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)ββT, omitting the argument Οof the processes under consideration,
|As,t| β€ ξξξZt
s
SrdSrβ(SnΒ·S)s,tξξξ+|(SnΒ·S)s,t βSsSs,t|
β€C(Ο)2βnplog n+|(SnΒ·S)s,t βSsSs,t|.Ξ΅C(Ο)2βn(1βΞ΅)+|(SnΒ·S)s,t βSsSs,t|
for every nβN,Ξ΅ > 0. The second term on the right hand side can be estimated, using a
standard argument based on Youngβs maximal inequality (see [LCL07], Theorem 1.16), by
max{2βnvq(s, t)1/q,(#{k:Οn
kβ[s, t]})1β2/q vq(s, t)2/q +vq(s, t)2/q },(6)
13
where vq(s, t) := kSkq
qβvar,[s,t]. For the convenience of the reader, we sketch the argument:
If there exists no kfor which Οn
kβ[s, t], then |(SnΒ·S)s,t βSsSs,t| β€ 2βnvq(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=Οn
k0for some k0. Let now Οn
k0,...,Οn
k0+Nβ1be those (Οn
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 Οn
k0+N=t. The idea is now to successively delete points (Οn
k0+β)
from the partition, in order to pass from (SnΒ·S) to SsSs,t. By super-additivity of vq, there
must exist ββ {1,...,N β1}, for which
vq(Οn
k0+ββ1, Ο n
k0+β+1)β€2
Nβ1vq(s, t).
Deleting Οn
k0+βfrom the partition and subtracting the resulting integral from (SnΒ·S)s,t, we
get
|SΟn
k0+ββ1SΟn
k0+ββ1,Οn
k0+β+SΟn
k0+βSΟn
k0+β,Οn
k0+β+1 βSΟn
k0+ββ1SΟn
k0+ββ1,Οn
k0+β+1 |
=|SΟn
k0+ββ1,Οn
k0+βSΟn
k0+β,Οn
k0+β+1 | β€ vq(Οn
k0+ββ1, Ο n
k0+β+1)2/q β€ξ2
Nβ1vq(s, t)ξ2/q.
Successively deleting all the points except Οn
k0=sand Οn
k0+N=tfrom the partition gives
|(SnΒ·S)s,t βSsSs,t| β€
N
X
k=2 ξ2
kβ1vq(s, t)ξ2/q .N1β2/qvq(s, t)2/q ,
and therefore (6). Now it is easy to see that #{k:Οn
kβ[s, t]} β€ 2nq vq(s, t) (compare also the
proof of Lemma 14), and thus
|As,t|.Ξ΅C(Ο)2βn(1βΞ΅)+ max{2βnvq(s, t)1/q ,(2nq vq(s, t))1β2/qvq(s, t)2/q +vq(s, t)2/q }
=C(Ο)2βn(1βΞ΅)+ max{2βnvq(s, t)1/q,2βn(2βq)vq(s, t) + vq(s, t)2/q }.(7)
This holds for all nβN,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 nβNfor which 2βnβ1< vq(s, t)1/Ξ±(1βΞ΅)β€
2βn. Using this nin (7), we get
|As,t|Ξ±.Ξ΅,Ο,Ξ± vq(s, t) + max nvq(s, t)1/(1βΞ΅)vq(s, t)Ξ±/q , vq(s, t)(2βq)/(1βΞ΅)+Ξ±+vq(s, t)2Ξ±/q o
=vq(s, t) + max ξvq(s, t)
q+Ξ±(1βΞ΅)
q(1βΞ΅), vq(s, t)2βq+Ξ±(1βΞ΅)
1βΞ΅+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₯(qβ1βΞ΅)/(1 βΞ΅). Since Ξ΅ > 0 can be chosen arbitrarily
close to 0, it suο¬ces if Ξ± > q β1. 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
ξZΒ·
0
SrdSrξ
ξ
ξ
p/2
β+kSkp
ββ€ξξ
ξ
ξZΒ·
0
SrdSrξ
ξ
ξ
p/2
β+kSkp
βξvq(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
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: βTβ[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 ]βRdsatisο¬es |fs,t|pβ€c(s, t) for all
(s, t)ββT, 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 ItΛo 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)nβNbe a sequence of positive numbers such that (cΞ΅
nβlog n)converges
to 0 for every Ξ΅ > 0. For nβNdeο¬ne Οn
0:= 0 and Οn
k+1 := inf{tβ₯Οn
k:|StβSΟn
k|=cn},
kβN, and set Sn
t=PkSΟn
k1[Οn
k,Οn
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)Οn
k,Οn
β(Ο)βSΟn
k(Ο)SΟn
k,Οn
β(Ο)|p/2
c(Οn
k, Ο n
β)β€1.
Proof. The uniform convergence of ((SnΒ·S)) to RSdSfollows from Theorem 16.
For the second claim, ο¬x nβNand k < β such that Οn
ββ€T. Then
|(SnΒ·S)Οn
k,Οn
ββSΟn
kSΟn
k,Οn
β|.ξ
ξ
ξ(SnΒ·S)βZΒ·
0
SsdSsξ
ξ
ξβ+ξξξAΟn
k,Οn
βξξξ
.Οcnplog n+vp/2(Οn
k, Ο n
β)2/p .Ξ΅c1βΞ΅
n+vp/2(Οn
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/2βvar,[s,t]for (s, t)ββT. Of course, this inequality only holds for typical
price paths and not for all Οββ¦.
On the other side, the same argument as in the proof of Corollary 18 (using Youngβs
maximal inequality and successively deleting points from the partition) shows that
|(SnΒ·S)Οn
k,Οn
ββSΟn
kSΟn
k,Οn
β|.c2βq
nvq(Οn
k, Ο n
β),(9)
15
where vq(s, t) := kSkq
qβvar,[s,t]for (s, t)ββT.
Let us deο¬ne the control function Λc:= vq+vp/2. Take Ξ± > 0 to be determined below. If
cn>Λc(s, t)1/Ξ±(1βΞ΅), then we use (9) and the fact that 2 βq < 0, to obtain
|(SnΒ·S)Οn
k,Οn
ββSΟn
kSΟn
k,Οn
β|Ξ±.(Λc(Οn
k, Ο n
β)) 2βq
(1βΞ΅)vq(Οn
k, Ο n
β)Ξ±β€Λc(Οn
k, Ο n
β)
2βq+Ξ±(1βΞ΅)
(1βΞ΅).
The exponent is larger or equal to 1 as long as Ξ±β₯(qβ1βΞ΅)/(1 βΞ΅). Since qand Ξ΅can be
chosen arbitrarily close to 2 and 0 respectively, we can take Ξ±=p/2, and get
|(SnΒ·S)Οn
k,Οn
ββSΟn
kSΟn
k,Οn
β|p/2.Λc(Οn
k, Ο n
β)(1 + Λc(0, T )Ξ΄)
for a suitable Ξ΄ > 0.
On the other side, if cnβ€Λc(s, t)1/Ξ±(1βΞ΅), then we use (8) to obtain
|(SnΒ·S)Οn
k,Οn
ββSΟn
kSΟn
k,Οn
β|Ξ±.Λc(Οn
k, Ο n
β) + Λc(Οn
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)Οn
k,Οn
ββSΟn
kSΟn
k,Οn
β|p/2β€c(Οn
k, Ο n
β),
where cis a suitable (Ο-dependent) multiple of Λc.
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
ξξξZt
s
SrdSr(Ο)βSs(Ο)Ss,t(Ο)ξξξp/2β€c(s, t)(Ο)
for all (s, t)ββT. From now on, we ο¬x one path Swhich satisο¬es these two conditions.
Let us write S= (S, A), where
A: βTβRdΓd, A(s, t) = Zt
s
SrdSrβSsSs,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 βqβ β 1 to obtain a continuous integral map. The for us relevant case qβ(2,3) was
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
kSkpβvar := kSkpβvar +kAkp/2βvar.
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 βlooks 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 ]βRnand Fβ²: [0, T ]βRnΓd.
We say that the pair (F, F β²) is controlled by Sif the derivative Fβ²has ο¬nite q-variation, and
the remainder RF: βTβRndeο¬ned by
RF(s, t) = Fs,t βFβ²
sSs,t,
has ο¬nite r-variation, where 1/r = 1/p + 1/q. In this case, we write (F, F β²)βCq
S=Cq
S(Rn),
and deο¬ne
k(F, F β²)kCq
S:= kFβ²kqβvar +kRFkrβvar.
If it is equipped with the norm |F0|+|Fβ²
0|+k(F, F β²)kCq
S, then the space of controlled paths
Cq
Sis a Banach space.
Naturally, the function Fβ²should be interpreted as the derivative of Fwith respect to S.
The reason for considering couples (F, F β²) and not just functions Fis that the smoothness
requirement on the remainder RFusually does not determine Fβ²uniquely 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 Fβ²of ο¬nite q-variation we have (F, F β²)βCq
S.
Note that we do not require For Fβ²to be continuous. We will point out below why this
does not pose any problems.
To obtain a more βquantitativeβ 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 Fβ²may 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 ΟβC2
band deο¬ne Fs:= Ο(Ss) and Fβ²
s:= Οβ²(Ss). Then (F, F β²)βCp
S:
Clearly Fβ²has ο¬nite p-variation. For the remainder, we have
|RF(s, t)|p/2=|Ο(St)βΟ(Ss)βΟβ²(Ss)Ss,t|p/2β€ξ1
2kΟβ²β²kβ|Ss,t|2ξp/2= 2βp/2kΟβ²β²kp/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 β²)βCp
S.
As the image of the continuous path Sis compact, it is not actually necessary to assume
that Οis bounded. We may always consider a C2function Οof compact support, such that
Οagrees with Οon the image of S.
It is instructive to examine under which regularity conditions on Οwe obtain a controlled
path if Sis a typical price path. As we argued above, Οβ²(S) should be of ο¬nite q-variation
for some q > 0, which is satisο¬ed as long as Οβ²is Ξ΅-HΒ¨older continuous for some Ξ΅ > 0. The
remainder RF(s, t) = Ο(St)βΟ(Ss)βΟβ²(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 ΟβC1+Ξ΅for some Ξ΅ > 0, so that
for such Οwe 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) = GtβGsfor 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 β²) = (G, 0), we obtain a controlled path in Cq
S, where 1/q = 1/r β1/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 β²)βCq
S. Then there exists a unique function RFdSβC([0, T ],Rn)
which satisο¬es
ξξξZt
s
FudSuβFsSs,t βFβ²
sA(s, t)ξξξ.kSkpβvar,[s,t]kRFkrβvar,[s,t]+kAkp/2βvar,[s,t]kFβ²kqβvar,[s,t]
for all (s, t)ββT. The integral is given as limit of the compensated Riemann sums
Zt
0
FudSu= lim
mββ X
[s1,s2]βΟmξFs1Ss1,s2+Fβ²
s1A(s1, s2)ξ,(11)
where (Οm)is any sequence of partitions of [0, t]with mesh size going to 0.
The map (F, F β²)7β (G, Gβ²) := (RFudSu, F )is continuous from Cq
Sto Cp
Sand satisο¬es
k(G, Gβ²)kCp
S.kFkp+ (kFβ²kβ+kFβ²kqβvar)kAkp/2βvar +kSkpβvarkRFkrβvar.
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 Youngβs maximal inequality that we encountered
above) to obtain Theorem 24.
There is only one small pitfall: We did not require For Fβ²to 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 Youngβs maximal inequality works as long as integrand and integrator have no
common discontinuities, see the Theorem on p. 264 of [You36].
If now ΟβC1+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 β²)7β (Ο(F), Οβ²(F)Fβ²) 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 XβCp
S,RΟ(Xt)dStdenotes the rough path integral, and Sis a typical price
path. Similarly, if ΟβC2+Ξ΅
b, then there exist p > 2 and q > 0 with 2/p + 1/q > 0, such that
the map (F, F β²)7β (Ο(F), Οβ²(F)Fβ²) 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 ItΛo
theory of SDEs, the function Οis 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 ItΛo SDE with a Stratonovich correction term: the equations
dXt=Ο(Xt)β¦dWtand
dXt=Ο(Xt)dWt+1
2Οβ²(Xt)Ο(Xt)dt
are equivalent (where Wis a standard Brownian motion, dWtdenotes ItΛo integration, and
β¦dWtdenotes Stratonovich integration). To solve the second equation, we need Οβ²Οto be
Lipschitz continuous, which is always satisο¬ed if ΟβC2
b. But rough path theory cannot
distinguish between ItΛo and Stratonovich integrals: If we deο¬ne the area of Wusing ItΛo
(respectively Stratonovich) integration, then the rough path solution of the equation will
coincide with the ItΛo (respectively Stratonovich) solution. So in the rough path setting,
the function Οshould satisfy at least the same requirements as in the Stratonovich setting.
The regularity requirements on Οare 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 β²)βCq
Sand
(Λ
F , Λ
Fβ²)βCq
Λ
S, and let (s, t)ββT. Then for every M > 0there exists CM>0such that
ξ
ξ
ξZΒ·
0
FsdSsβZΒ·
0
Λ
FsdΛ
Ssξ
ξ
ξpβvar β€CMξ|F0βΛ
F0|+|Fβ²
0βΛ
Fβ²
0|+kFβ²βΛ
Fβ²kqβvar
+kRFβRΛ
Fkrβvar +kSβΛ
Skpβvar +kAβΛ
Akp/2βvarξ,
19
as long as
max{|Fβ²
0|+k(F, F β²)kCq
S,|Λ
Fβ²
0|+k(Λ
F , Λ
Fβ²)kCq
Λ
S
,kSkpβvar,kAkp/2βvar,kΛ
Skpβvar,kΛ
Akp/2βvar} 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
mββ X
[r1,r2]βΟmξFr1Sr1,r2+Fβ²
r1A(r1, r2)ξ,
where (Οm) 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 βcompensatorβ
Fβ²
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 (Οn
k)n,k, we have
Zt
0
SsdSs= lim
nββ X
k
SΟn
kSΟn
kβ§t,Οn
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
nββ X
k
FΟn
kSΟn
kβ§t,Οn
k+1β§t,
without having to introduce the compensator Fβ²
Οn
kA(Οn
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 Οn
0:= 0 and Οn
k+1 := inf{tβ₯Οn
k:
|StβSΟn
k| β₯ 2βn}for k, n βN. We write Sn
t:= Pk1[Οn
k,Οn
k+1)(t)SΟn
k. Recall from Theorem 16
that for typical price paths.
ξ
ξ
ξ(SnΒ·S)βZΒ·
0
SsdSsξ
ξ
ξβ.2βnplog n. (13)
Lemma 27. Let Ξ΅ > 0, let FβC1+Ξ΅
b(Rd,Rd), and suppose Sfulο¬lls (13). Then the Riemann
sums β
X
k=0
F(SΟn
k)SΟn
kβ§t,Οn
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 nβN
and tβ[0, T ] that
(Fn+1 Β·S)tβ(FnΒ·S)t=
β
X
β=0
β
X
k=0,
[Οn+1
k,Οn+1
k+1 ]β[Οn
β,Οn
β+1]
F(S)Οn
β,Οn+1
kSΟn+1
kβ§t,Οn+1
k+1 β§t(14)
=
β
X
β=0
β
X
k=0,
[Οn+1
k,Οn+1
k+1 ]β[Οn
β,Οn
β+1]
(Fβ²(SΟn
β)SΟn
β,Οn+1
k+RΟn
β,Οn+1
k)SΟn+1
kβ§t,Οn+1
k+1 β§t
with a remainder Rthat satisο¬es |RΟn
β,Οn+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
(Οn
k)n,k. In particular, Lemma 14 yields max{k:Οn
kβ€t}.22nVn
tfor a uniformly bounded
sequence of increasing functions (Vn). Choose now m=m(n) := βn/3β. Applying HΒ¨olderβs
inequality gives us
ξξ(Fn+1 Β·S)tβ(FnΒ·S)tξξ
β€
β
X
j=0
β
X
β=0,
[Οn
β,Οn
β+1]β[Οm
j,Οm
j+1]
β
X
k=0,
[Οn+1
k,Οn+1
k+1 ]β[Οn
β,Οn
β+1]ξξFβ²(S)Οm
j,Οn
βSΟn
β,Οn+1
kSΟn+1
kβ§t,Οn+1
k+1 β§tξξ
+ξξξξ
β
X
β=0
β
X
k=0,
[Οn+1
k,Οn+1
k+1 ]β[Οn
β,Οn
β+1]
RΟn
β,Οn+1
kSΟn+1
kβ§t,Οn+1
k+1 β§tξξξξ
+
β
X
j=0 ξξFβ²(SΟm
j)ξξξξξξ
β
X
β=0,
[Οn
β,Οn
β+1]β[Οm
j,Οm
j+1]
β
X
k=0,
[Οn+1
k,Οn+1
k+1 ]β[Οn
β,Οn
β+1]
SΟn
β,Οn+1
kSΟn+1
kβ§t,Οn+1
k+1 β§tξξξξ
.(22nVn
T)kFkC1+Ξ΅
b2βmΞ΅2β2n+ (22nVn
T)kFkC1+Ξ΅
b2β(1+Ξ΅)n2βn
+
β
X
j=0 ξξFβ²(SΟm
j)ξξξξ(SnΒ·S)Οm
jβ§t,Οm
j+1β§tβ(Sn+1 Β·S)Οm
jβ§t,Οm
j+1β§tξξ
.sup
NβN
(VN
T)kFkC1+Ξ΅
b(2βnΞ΅/3+ 2βnΞ΅) + kFβ²kβsup
NβN
(VN
T)22m2βnlog 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 Οn={0 = tn
0< tn
1<Β·Β·Β·< tn
Nn=T},nβN, is a given sequence
of partitions such that sup{|Stn
k,tn
k+1 |:k= 0,...,Nnβ1}converges to 0, and set
Sn
t:=
Nnβ1
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)ββT
|Ss,t|p
c(s, t)+ sup
nsup
0β€k<ββ€Nn
|(SnΒ·S)tn
k,tn
ββStn
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 ItΛo 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 ItΛo type integrals. To go from ItΛo
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
tβSi
0Sj
0βZt
0
Si
rdSj
rβZt
0
Sj
rdSi
r.
Then hSi, Sjiis a continuous function and
hSi, Sjit= lim
nββhSi, Sjin
t= lim
nββ
Nnβ1
X
k=0
(Si
tn
k+1β§tβSi
tn
kβ§t)(Sj
tn
k+1β§tβSj
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
tβSi
0Sj
0=
Nnβ1
X
k=0 ξSi
tn
k+1β§tSj
tn
k+1β§tβSi
tn
kβ§tSj
tn
kβ§tξ
for every nβN, and
Si
tn
k+1β§tSj
tn
k+1β§tβSi
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
4ξξ(Si+Sj)tn
kβ§t,tn
k+1β§tξ2βξ(SiβSj)tn
kβ§t,tn
k+1β§tξ2ξ
(read hSi, Sji= 1/4(hSi+SjiβhSiβSji)). 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
Nnβ1
X
k=0 ξξ(Si+Sj)tn
k,tn
k+1 ξ2+ξ(SiβSj)tn
k,tn
k+1 ξ2ξ.sup
m
Nmβ1
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.
22