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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 finance
Nicolas Perkowski
CEREMADE & CNRS UMR 7534
Universit´e Paris-Dauphine
perkowski@ceremade.dauphine.fr
David J. Pr¨omel
Humboldt-Universit¨at zu Berlin
Institut f¨ur Mathematik
proemel@math.hu-berlin.de
November 26, 2013
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 ollmer’s pathwise Itˆo
integral. Moreover, we construct a “model free Itˆo integral” in the spirit of Karandikar.
1 Introduction
In this paper, we use Vovk’s [Vov12] game-theoretic approach to mathematical finance to set
up an integration theory for “typical price paths”. Vovk’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
profit by investing in those paths which violate the property.
We slightly adapt the definition of Vovk’s outer measure, to obtain an object which in
our opinion has a slightly more natural financial interpretation, and with which it is easier to
work. For this outer measure, we prove a sort of “model free Itˆo isometry”, based on which we
construct pathwise Itˆo 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 Itˆo 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 Itˆo 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
We would like to thank Mathias Beiglb¨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-Universit¨at zu Berlin.
Supported by a Postdoctoral grant of the Fondation Sciences Math´ematiques de Paris.
Supported by a Ph.D. scholarship of the DFG Research Training Group 1845.
1
we can fix 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
infinite profit by investing in the paths which we excluded, we may consider them as “too
good to be true”, 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 financial 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 Gubinelli’s [Gub04] controlled path approach to rough path integration, which
simultaneously extends
the Riemann-Stieltjes integral of Sagainst functions of bounded variation, formally
defined by Rt
0FsdSs=FtStF0S0Rt
0SsdFs;
the Young integral [You36]: typical price paths have finite p-variation for every p > 2,
and therefore for every Fof finite q-variation for q < 2 (so that 1/p + 1/q > 1), the
integral RFdSis defined as limit of Riemann sums;
ollmer’s [F¨ol81] “calcul d’Itˆo sans probabilit´es”: ollmer shows that if the quadratic
variation hSiexists along a sequence of partitions, then for every gradient F, the inte-
gral RF(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 first time in this paper.
The rough path integral is usually defined as a limit of compensated Riemann sums,
which have no obvious financial 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 defined as limit of Riemann sums along the same
sequence of partitions, similarly as in F¨ollmer’s pathwise Itˆo calculus. We also show that
ollmer’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, F¨ollmer’s integral is really a special case of the rough path
integral.
Model free finance 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 finance.
One of the basic problems in mathematical finance consists in calculating fair prices for
financial 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:t[0, T ])
be an adapted process on a filtered probability space (Ω,F,(Ft)t[0,T ],P), which models the
discounted price of a financial asset. Let Xbe a nonnegative random variable on (Ω,F),
which models the payoff of a given derivative. Then the minimal superhedging price of Xis
defined as
p(X) := inf nλ > 0 : admissible strategy Hs.t. λ+ZT
0
HsdSsXPa.s.o,(1)
and we have
p(X) = sup{EQ[X] : Qis equivalent to Pand Sis a Qlocal 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,
dSσ
t=Sσ
t(σdWt+bdt),
where Wis a one-dimensional standard Brownian motion, σ > 0, and bR. It is a very
active field of research to derive statistical estimates for the volatility σ, but even the best
statistical method will only give us a confidence interval σ[σ1, σ2], and not the exact value
of σ– 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 Pσfor σ[σ1, σ2],
where Pσdenotes the measure on the path space under which the coordinate process has
the dynamics of a geometric Brownian motion with volatility σ. The first problem that one
encounters here is how to define the minimal superhedging price. It is natural to replace the
“a.s.” assumption in (1) by “a.s. under every Pσ,σ[σ1, σ2]”. But then the stochastic
integral RHdShas to be constructed simultaneously under all the measures Pσ, which is not
a trivial task because Pσand P˜σare mutually singular for σ6= ˜σ. Luckily it turns out that
here we can essentially still use Itˆo’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 finance 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 payoff (StK)+for 0 tTand KRare
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 definition of the stochastic integrals RFdS, which
are just finite sums.
In continuous time however, it is not immediately clear how to define 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 defined 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], F¨ollmer’s pathwise Itˆo calculus [F¨ol81] is used to define pathwise stochastic
integrals, and these integrals are used to derive arbitrage consistent prices for weighted vari-
ance swaps. Here, the given data is a finite number of European call and put prices. The
use of F¨ollmer’s integral, which relies on the pathwise existence of the quadratic variation, is
justified by referring to Vovk [Vov12], who proved that typical price paths admit a nontrivial
quadratic variation.
However, F¨ollmer’s integral essentially only applies to one-dimensional integrators. If we
are considering more than one financial asset, then we can only integrate gradients RF(S)dS
3
with F¨ollmer’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 finance. Also, while with F¨ollmer’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 unified framework for both F¨ollmer’s and
Young’s integrals.
Plan of the paper
In Section 2 we briefly recall Vovk’s game-theoretic approach to mathematical finance. Sec-
tion 3 is devoted to the construction of a model free Itˆo 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 define it. In Sec-
tion 5 we compare F¨ollmer’s pathwise Itˆo 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
Hoeffding 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 fix T > 0 and we write Ω := C([0, T ],Rd) for the space of d-
dimensional continuous paths. The coordinate process on Ω is denoted by St(ω) = ω(t),
t[0, T ]. For i∈ {1,...,d}, we also write Si
t(ω) = ωi(t), where ω= (ω1,...d). The
filtration (Ft)t[0,T ]is defined as Ft:= σ(Ss:st), and we set F:= FT. Stopping times τ
and the associated σ-algebras Fτare defined as usually.
Unless explicitly stated otherwise, inequalities of the type FtGt, where Fand G
processes on Ω, are supposed to hold for all ωΩ, and not modulo null sets, as it is usually
assumed in stochastic analysis.
The indicator function of a set Ais denoted by 1A.
Apartition πof [0, T ] is a finite set of time points, π={0 = t0< t1<···< tm=T}.
Occasionally, we will identify πwith the set of intervals {[t0, t1],[t1, t2],...,[tm1, tm]}, and
write expressions like P[s,t]π.
For f: [0, T ]Rnand t1, t2[0, T ], denote ft1,t2:= ft2ft1and define the p-variation
of frestricted to [s, t][0, T ] as
kfkpvar,[s,t]:= sup m1
X
k=0 |ftk,tk+1 |p1/p
:s=t0<···< tm=t, m N, p > 0,(2)
(possibly taking the value +). We set kfkpvar := kfkpvar,[0,T ]. We write ∆T={(s, t) :
0stT}for the simplex and define the p-variation of a function g: ∆TRnin the
same manner, replacing ftk,tk+1 in (2) by g(tk, tk+1).
For α > 0, the space Cαconsists of those functions that are αtimes continuously
differentiable, with (α− ⌊α)-H¨older continuous partial derivatives of order α. The space
Cα
bconsists of those functions in Cαthat are bounded, together with their partial derivatives,
4
and we define the norm k·kCα
bby setting
kfkCα
b:=
α
X
k=0 kDkfk+kDαfkα−⌊α,
where k·kβdenotes the β-H¨older norm for β(0,1), and k·kdenotes the supremum norm.
For x, y 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 [0, T ], where we recall that S
denotes the coordinate process on Ω. Those expressions are to be understood as the matrix
(RSidSj)1i,jd, 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 ac·b, and we write abif 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 · ∞ := 0 and inf := .
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 finance that uses arbitrage considerations to examine which
properties are satisfied by “typical price paths”. This is achieved with the help of an outer
measure given by the cheapest superhedging price.
Recall that T > 0 and Ω = C([0, T ],Rd) is the space of continuous paths, with coordinate
process S, natural filtration (Ft)t[0,T ], and F=FT. A process H: Ω ×[0, T ]Rdis called a
simple strategy if there exist stopping times 0 = τ0< τ1< . . . , and Fτn-measurable bounded
functions Fn: Ω Rd, such that for every ωΩ we have τn(ω) = for all but finitely many
n, and such that
Ht(ω) =
X
n=0
Fn(ω)1(τn(ω)n+1(ω)](t).
In that case, the integral
(H·S)t(ω) :=
X
n=0
Fn(ω)(Sτn+1t(ω)Sτnt(ω)) =
X
n=0
Fn(ω)Sτnt,τn+1t(ω)
is well defined for all ωΩ, t[0, T ]. Here Fn(ω)Sτnt,τn+1t(ω) denotes the usual inner
product on Rd. For λ > 0, a simple strategy His called λ-admissible if (H·S)t(ω)≥ −λfor
all ωΩ, t[0, T ]. The set of λ-admissible simple strategies is denoted by Hλ.
Definition 1. The outer measure of AΩ is defined as the cheapest superhedging price for
1A, that is
P(A) := inf nλ > 0 : (Hn)nN⊆ Hλs.t. lim inf
n→∞ (λ+ (Hn·S)T(ω)) 1A(ω)ωo.
A set of paths AΩ 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 definition, every Itˆo stochastic integral
is the limit of stochastic integrals against simple functions. Therefore, our definition of the
cheapest superhedging price is essentially the same as in the classical setting, see (1), with
one important difference: we require superhedging for all ωΩ, and not just almost surely.
Remark 2 ([Vov12], p. 564).An equivalent definition of Pwould be
e
P(A) := infλ > 0 : (Hn)nN⊆ Hλs.t. lim inf
n→∞ sup
t[0,T ]
(λ+ (Hn·S)t(ω)) 1A(ω)ω.
Clearly e
PP. To see the opposite inequality, let e
P(A)< λ. Let (Hn)nN⊂ Hλbe a sequence
of simple strategies such that lim inf n→∞ supt[0,T](λ+ (Hn·S)t)1A, and let ε > 0. Define
τn:= inf{t[0, T ] : λ+ε+ (Hn·S)t1}. Then the stopped strategy Gn
t(ω) := Hn
t(ω)1t<τn(ω)
is in Hλ⊆ Hλ+εand
lim inf
n→∞ (λ+ε+ (Gn·S)T(ω)) lim inf
n→∞ 1{λ+ε+supt[0,T ](Hn·S)t1}(ω)1A(ω).
Therefore P(A)λ+ε, and since ε > 0was arbitrary Pe
P, and thus P=e
P.
Lemma 3 ([Vov12], Lemma 4.1).Pis in fact an outer measure, i.e. a nonnegative function
defined on the subsets of such that
-P() = 0;
-P(A)P(B)if AB;
- if (An)nNis a sequence of subsets of , then P(SnAn)PnP(An).
Proof. Monotonicity and P() = 0 are obvious. So let (An) be a sequence of subsets of Ω.
Let ε > 0, nN, and let (Hn,m)mNbe a sequence of (P(An) + ε2n1)-admissible simple
strategies such that lim inf m→∞(P(An) + ε2n1+ (Hn,m ·S)T)1An. Define for mNthe
(PnP(An) + ε)-admissible simple strategy Gm:= Pm
n=0 Hn,m. Then by Fatou’s lemma
lim inf
m→∞
X
n=0
P(An) + ε+ (Gm·S)T
k
X
n=0 P(An) + ε2n1+ lim inf
m→∞ (Hn,m ·S)T
1Sk
n=0 An
for all kN. 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 Ais a null set if and only if there exists a sequence of 1-admissible
simple strategies (Hn)n⊂ H1such that
lim inf
n→∞ (1 + (Hn·S)T)≥ ∞ · 1A(ω),(3)
where we recall that by convention 0· ∞ = 0.
6
Proof. If such a sequence exists, then we can scale it down by an arbitrary factor ε > 0 to
obtain a sequence of strategies in Hεthat superhedge 1Aand therefore P(A) = 0.
If conversely P(A) = 0, then for every nNthere exists a sequence of simple strategies
(Hn,m)mN⊂ H2n1such that 2n1+ lim inf m→∞(Hn,m ·ω)T1A(ω) for all ωΩ. Define
Gm:= Pm
n=0 Hn,m, so that Gm∈ H1. For every kNwe obtain
lim inf
m→∞ (1 + (Gm·S)T)
k
X
n=0
(2n1+ lim inf
m→∞ (Hn,m ·S)T)k1A.
Since the left hand side does not depend on k, the sequence (Gm) satisfies (3).
In other words, if a set Ahas outer measure 0, then we can make infinite profit by
investing in the paths from A, without ever risking to lose more than the initial capital 1.
This motivates the following definition:
Definition 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 finance. First, observe that Pis an outer measure which simultaneously
dominates all local martingale measures on Ω:
Propostion 6 ([Vov12], Lemma 6.3).Let Pbe a probability measure on (Ω,F), such that the
coordinate process Sis a P-local martingale, and let A∈ F. Then P(A)P(A).
Proof. Let λ > 0 and let (Hn)⊆ Hλbe such that lim inf n(λ+ (Hn·S)T)1A. Then
P(A)EP[lim inf
n(λ+ (Hn·S)T)] lim inf
nEP[λ+ (Hn·S)T]λ,
where in the last step we used that λ+ (Hn·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 (Ω,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 profit without risk) and
no arbitrage opportunities of the first kind (NA1) (intuitively: no very large profit 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 + (H·S)T:H∈ H1}is bounded in
P-probability, i.e. if
lim
n→∞ sup
H∈H1
P(1 + (H·S)Tn) = 0.
Thus, we can interpret a null set AΩ as a model free arbitrage opportunity of the first 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 fixed set of typical price paths, rather
than the full space Ω.
Not surprisingly, we can relate our model free notion of (NA1) with the classical (NA1)
property:
7
Propostion 7. Let A∈ F be a null set, and let Pbe a probability measure on (Ω,F)such
that the coordinate process satisfies (NA1). Then P(A) = 0.
Proof. Let (Hn)nN⊆ H1be such that lim inf n(Hn·S)T≥ ∞ · 1A. Then for every c > 0
P(A) = PAlim inf
n→∞ (Hn·S)T> clim
n→∞ PA\
kn{(Hk·S)T> c}
sup
H∈H1
P({(H·S)T> c}).
By assumption, the right hand side converges to 0 as c→ ∞ 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 infn→∞(Hn·S)T=∞},
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
satisfies (NA1) under P, then there exists a dominating measure QP, 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 Vovk’s outer measure
Our definition of the outer measure Pis not exactly the same as Vovk’s [Vov12]. We find the
definition 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 λ > 0, Vovk defines the set of processes
Sλ:=
X
k=0
Hk:Hk∈ Hλk, λk>0,
X
k=0
λk=λ.
For every G=Pk0Hk∈ Sλ, every ωΩ and every t[0, T ], the integral
(G·S)t(ω) := X
k0
(Hk·S)t(ω) = X
k0
(λk+ (Hk·S)t(ω)) λ
is well defined and takes values in [λ, ]. Vovk then defines for AΩ the cheapest
superhedging price as
Q(A) := inf λ > 0 : G∈ Sλs.t. λ+ (G·S)T1A.
Vovk’s definition corresponds to the usual construction of an outer measure from an outer
content (i.e. an outer measure which is only finitely 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 A. Then P(A)Q(A).
8
Proof. Let G=PkHk, with Hk∈ Hλkand Pkλk=λ, and assume that λ+ (G·S)T1A.
Then (Pn
k=0 Hk)nNdefines a sequence of simple strategies in Hλ, such that
lim inf
n→∞ λ+ n
X
k=0
Hk·ST=λ+ (G·S)T1A.
So if Q(A)< λ, then also P(A)λ, and therefore P(A)Q(A).
Corollary 11. For every p > 2, the set Ap:= {ωΩ : kS(ω)kpvar =∞} 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 Ω = C([0,),R) (i.e. if the asset price process
is one-dimensional), and if AΩ is “invariant under time changes” and such that S0(ω) = 0
for all ωA, then A∈ 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 Itˆo integral
The present section is devoted to the pathwise construction of an Itˆo type integral for typical
price paths. The main ingredient in the construction of the integral is a (weak) type of model
free Itˆo 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 sufficiently regular to obtain a rough path (S, RSdS).
Since we are in an unusual setting, let us spell out the following standard definitions:
Definition 12. A process H: ×[0, T ]Rdis called adapted if the random variable
ω7→ Ht(ω) is Ft-measurable for all t[0, T ].
The process His said to be c`adl`ag if the sample path t7→ Ht(ω) is c`adl`ag for all ωΩ.
For proving our weak Itˆo isometry, we will need an appropriate sequence of stopping times:
Let ω= (ω1,...,ωd)C([0, T ],Rd) and nN. For each i= 1,...,d, define inductively
σn,i
0(ω) := 0, σn,i
k+1(ω) := inf tσn,i
k:|ωi(t)ωi(σn,i
k)| ≥ 2n, k N.
Note that we are working with continuous paths and we are considering entrance times into
closed sets. Therefore, the (σn,i) are indeed stopping times, despite the fact that our filtration
(Ft) is neither complete nor right-continuous. Denote by πn,i the partition corresponding to
(σn,i
k)kN, that is πn,i := {σn,i
k:kN}. To obtain an increasing sequence of partitions, we
take the union of the (πn,i). More precisely, for nNwe define σn
0:= 0 and then
σn
k+1(ω) := min t > σn
k(ω) : t
d
[
i=1
πn,i(ω), k N,(4)
and we write πn:= {σn
k:kN}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 ω, the quadratic variation
along (πn,i(ω))nNexists. That is,
Vn,i
t(ω) :=
X
k=0 ωi(σn,i
k+1 t)ωi(σn,i
kt)2, t [0, T ], n N,
converges uniformly to a function hSii(ω)C([0, T ],R)for all i∈ {1,...,d}.
For later reference, let us estimate Nn
t:= max{kN:σn
kt}, the number of stopping
times σn
k6= 0 in πnwith values in [0, t]:
Lemma 14. For all ω,nN, and t[0, T ], we have
22nNn
t(ω)
d
X
i=1
Vn,i
t(ω) =: Vn
t(ω).
Proof. For i∈ {1, ..., d}define Nn,i
t:= max{kN:σn,i
kt}. Since ωiis continuous, we
have |ωi(σn,i
k+1)ωi(σn,i
k)|= 2nas long as σn,i
k+1 T. Therefore, we obtain
Nn
t(ω)
d
X
i=1
Nn,i
t(ω) =
d
X
i=1
Nn,i
t(ω)1
X
k=0
1
22nω(σn,i
k+1)ω(σn,i
k)222n
d
X
i=1
Vn,i
t(ω).
Since we want to extend the integral from step functions to c`adl`ag integrands via a con-
tinuity argument, let us first 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: ×[0, T ]Rdis called a step function if there exist stopping times
0 = τ0< τ1< . . . , and Fτn-measurable functions Fn: Ω Rd, such that for every ωΩ we
have τn(ω) = for all but finitely many n, and such that
Ht(ω) =
X
n=0
Fn(ω)1[τn(ω)n+1(ω))(t).
Note that for notational convenience, we are now considering the interval [τn(ω), τn+1(ω))
which is closed on the right hand side. This allows us to define the integral
(H·S)t:=
X
n=0
FnSτnt,τn+1t=
X
n=0
HτnSτnt,τn+1t, t [0, T ].
The following lemma will be the main building block in our construction of the integral.
Lemma 15 (Model free Itˆo isometry).Let a > 0and let Hbe a step function such that
kH(ω)kafor all ω. Then for all b, c > 0we have
P{k(H·S)kabc} ∩ {hSiTc}2 exp(b2/(2d)),
where the set {hSiTc}should be read as {hSiT= limnVn
Texists and satisfies hSiTc}.
10
Proof. Assume Ht=P
n=0 Fn1[τnn+1)(t). Let nNand define ρn
0:= 0 and then for kN
ρn
k+1 := min t > ρn
k:tπn∪ {τm:mN},
where we recall that πn={σn
k:kN}is the n-th generation dyadic partition generated
by S. For t[0, T ], we have (H·S)t=PkHρn
kSρn
kt,ρn
k+1t. Since kH(ω)ka, and by
definition of πn(ω), we get
sup
t[0,T ]Hρn
kSρn
kt,ρn
k+1tad2n.
Hence, the pathwise Hoeffding inequality, Lemma 35 in Appendix A, yields for every λR
the existence of a 1-admissible simple strategy Gλ,n ∈ H1such that
1 + (Gλ,n ·S)texp λ(H·S)tλ2
2(N(ρn)
t+ 1)22na2d=: Eλ,n
t
for all t[0, T ], where
N(ρn)
t:= max{k:ρn
kt} ≤ Nn
t+N(τ)
t:= Nn
t+ max{k:τkt}.
By Lemma 14, we have Nn
t22nVn
t, so that
Eλ,n
texp λ(H·S)tλ2
2Vn
Ta2dλ2
2(N(τ)
T+ 1)22na2d.
If now k(H·S)kabcand hSiTc, then
lim inf
n→∞ sup
t[0,T ]
Eλ,n
t+Eλ,n
t
21
2exp λabcλ2
2ca2d.
The argument inside the exponential is maximized for λ=b/(acd), 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 Itˆo isometry is the key
ingredient in the construction of the Itˆo integral. The term “model free Itˆo isometry” alludes
to that analogy.
Theorem 16. Let Hbe an adapted, c`adl`ag process with values in Rd. Then there exists a
map RHdS: Ω , such that if (cm)mNis a sequence of strictly positive numbers, and if
(Hm)mNis a sequence of step functions with kHm(ω)H(ω)kcmfor all ωand all
mN, then for typical price paths ωthere exists a constant C(ω)>0such that
(Hm·S)(ω)ZHdS(ω)
C(ω)cmplog m(5)
for all mN. In particular, if (cmlog m)converges to 0, then (Hm·S)(ω)converges to
RHdS(ω)for typical price paths ω.
We usually write Rt
0HsdSs:= RHdS(t), and we call the function RHdSthe pathwise
Itˆo integral of Hwith respect to S.
11
Proof. For the construction of RHdS, we consider dyadic approximations of H: Define for
nNthe stopping times τn
0:= 0 and then inductively
τn
k+1 := inf{tτn
k:|HtHτn
k| ≥ 2n}.
Set also Gn
t:= P
k=0 Hτn
k1[τn
kn
k+1)(t), so that kGnHk2n, and thus kGnGn+1k
2n+1. We claim that
Plim sup
n→∞ k(Gn·S)(Gn+1 ·S)k
2nlog n== 0.
Since P(hSiT=) = 0 and by countable subadditivity of P, it suffices to show
Plim sup
n→∞ k(Gn·S)(Gn+1 ·S)k
2nlog n=∩ {hSiTc}= 0
for every c > 0. But we obtain from Lemma 15 that
P \
mN[
nmnk(Gn·S)(Gn+1 ·S)k2n+1p4dlog nco∩ {hSiTc}!
X
n=n0
Pnk(Gn·S)(Gn+1 ·S)k2n+1p4dlog nco∩ {hSiTc}
X
n=n0
exp 4dlog n
2d=
X
n=n0
1
n2
for all n0N. Since the right hand side converges to zero as n0tends to , we conclude that
for typical price paths ωthere exists C(ω)0 such that k(Gn·S)(ω)(Gn+1 ·S)(ω)k<
C(ω)2nlog nfor all n, and in particular ((Gn·S)(ω))nis a Cauchy sequence. We define
ZHdS(ω)(t) := (limn→∞(Gn·S)t(ω),if ((Gn·S)(ω))nconverges uniformly,
0,otherwise.
Let now (Hm)mNbe a sequence of step functions that approximates Huniformly, such that
kHmHkcm, where (cm) is a sequence of strictly positive numbers, and let c > 0. Then
Pk(Hm·S)ZHdSk3cmp4dlog mc{hSiTc}
Pnk(Hm·S)(Gn·S)k2cmp4dlog mco∩ {hSiTc}
+Pk(Gn·S)ZHdSkcmp4dlog mc∩ {hSiTc}
for every nN. As we have seen above, the second term on the right hand side converges to
zero as n→ ∞. And if nis large enough so that 2ncm, then kHmGnk2cm, so in
particular the first term on the right hand side can be estimated by
Pnk(Hm·ω)(Gn·ω)k2cmp4dlog mco∩ {VT(ω)c}1
m2.
Since this is summable in m, we conclude as before that for typical price paths ωthere exists
C(ω)>0 such that k(Hm·S)(ω)RHdS(ω)kC(ω)cmlog mfor all m.
12
Remark 17. The pathwise Itˆo 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 Itˆo integral (recall Proposi-
tion 6). The model free Itˆo 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 Itˆo integral was already sufficient 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 “universal null set” 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 “probabilistic”
construction of the integral. Such an analytic construction is given by Lyons’ rough path
integral, which does not work for all c`adl`ag integrands but instead only for those integrands
which “look like the integrator”.
In order to apply the rough path machinery, we will need to show that the integral process
RSdS:= (RSidSj)1i,jdis sufficiently regular. Fortunately, this is a direct consequence of
the speed of convergence (5):
Corollary 18. For (s, t)T,ω, and i, j ∈ {1,...,d}define
Ai,j
s,t(ω) := Zt
s
Si
rdSj
r(ω)Si
s(ω)Sj
s,t(ω) := Zt
0
Si
rdSj
r(ω)Zs
0
Si
rdSj
r(ω)Si
s(ω)Sj
s,t(ω).
Then for typical price paths, A= (Ai,j )1i,jdhas finite p/2-variation for all p > 2.
Proof. Define the dyadic stopping times (τn
k)n,kNby τn
0:= 0 and
τn
k+1 := inf{tτn
k:|StSτn
k|= 2n},
and set Sn
t:= PkSτn
k1[τn
kn
k+1)(t), so that kSnSk2n. Accorcing to (5), for typical
price paths ωthere exists C(ω)>0 such that
(Sn·S)(ω)ZSdS(ω)
C(ω)2nplog n.
Fix such a typical price path ω, which is also of finite q-variation for every q > 2 (recall from
Corollary 11 that this is satisfied by typical price paths). Let us show that for such ω, the
process Ais of finite p/2-variation for every p > 2.
We have for (s, t)T, omitting the argument ωof the processes under consideration,
|As,t| ≤ Zt
s
SrdSr(Sn·S)s,t+|(Sn·S)s,t SsSs,t|
C(ω)2nplog n+|(Sn·S)s,t SsSs,t|.εC(ω)2n(1ε)+|(Sn·S)s,t SsSs,t|
for every nN,ε > 0. The second term on the right hand side can be estimated, using a
standard argument based on Young’s maximal inequality (see [LCL07], Theorem 1.16), by
max{2nvq(s, t)1/q,(#{k:τn
k[s, t]})12/q vq(s, t)2/q +vq(s, t)2/q },(6)
13
where vq(s, t) := kSkq
qvar,[s,t]. For the convenience of the reader, we sketch the argument:
If there exists no kfor which τn
k[s, t], then |(Sn·S)s,t SsSs,t| ≤ 2nvq(s, t)1/q , using
that |Ss,t| ≤ vq(s, t)1/q . This corresponds to the first 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=τn
k0for some k0. Let now τn
k0,...,τn
k0+N1be those (τn
k)kwhich are in [s, t).
Without loss of generality we may suppose N2, because otherwise (Sn·S)s,t =SsSs,t .
Abusing notation, we write τn
k0+N=t. The idea is now to successively delete points (τn
k0+)
from the partition, in order to pass from (Sn·S) to SsSs,t. By super-additivity of vq, there
must exist ∈ {1,...,N 1}, for which
vq(τn
k0+1, τ n
k0++1)2
N1vq(s, t).
Deleting τn
k0+from the partition and subtracting the resulting integral from (Sn·S)s,t, we
get
|Sτn
k0+1Sτn
k0+1n
k0++Sτn
k0+Sτn
k0+n
k0++1 Sτn
k0+1Sτn
k0+1n
k0++1 |
=|Sτn
k0+1n
k0+Sτn
k0+n
k0++1 | ≤ vq(τn
k0+1, τ n
k0++1)2/q 2
N1vq(s, t)2/q.
Successively deleting all the points except τn
k0=sand τn
k0+N=tfrom the partition gives
|(Sn·S)s,t SsSs,t| ≤
N
X
k=2 2
k1vq(s, t)2/q .N12/qvq(s, t)2/q ,
and therefore (6). Now it is easy to see that #{k:τn
k[s, t]} ≤ 2nq vq(s, t) (compare also the
proof of Lemma 14), and thus
|As,t|.εC(ω)2n(1ε)+ max{2nvq(s, t)1/q ,(2nq vq(s, t))12/qvq(s, t)2/q +vq(s, t)2/q }
=C(ω)2n(1ε)+ max{2nvq(s, t)1/q,2n(2q)vq(s, t) + vq(s, t)2/q }.(7)
This holds for all nN,ε > 0, q > 2. Let us suppose for the moment that vq(s, t)1 and let
α > 0 to be determined later. Then there exists nNfor which 2n1< vq(s, t)1(1ε)
2n. Using this nin (7), we get
|As,t|α.ε,ωvq(s, t) + max nvq(s, t)1/(1ε)vq(s, t)α/q , vq(s, t)(2q)/(1ε)+α+vq(s, t)2α/q o
=vq(s, t) + max vq(s, t)
q+α(1ε)
q(1ε), vq(s, t)2q+α(1ε)
1ε+vq(s, t)2α/q.
We would like all the exponents in the maximum on the right hand side to be larger or equal
to 1. For the first term, this is satisfied as long as ε < 1. For the third term, we need αq/2.
For the second term, we need α(q1ε)/(1 ε). Since ε > 0 can be chosen arbitrarily
close to 0, it suffices if α > q 1. Now, since q > 2 can be chosen arbitrarily close to 2, we
see that αcan be chosen arbitrarily close to 1. In particular, we may take α=p/2 for any
p > 2, and we obtain
|As,t|p/2.ω,δ vq(s, t)(1 + vq(s, t)δ)vq(s, t)(1 + vq(0, T )δ)
for a suitable δ > 0.
14
It remains to treat the case vq(s, t)>1, for which we simply estimate
|As,t|p/2.p
Z·
0
SrdSr
p/2
+kSkp
Z·
0
SrdSr
p/2
+kSkp
vq(s, t).
So for every interval [s, t] we can estimate |As,t|p/2.ω,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 sut, then vq(s, u) + vq(u, t)vq(s, t). We call control function any function
c: ∆T[0,) which satisfies these two properties and is moreover continuous and such that
c(t, t) = 0 for all t[0, T ]. Observe that if f: [0, T ]Rdsatisfies |fs,t|pc(s, t) for all
(s, t)T, then the p-variation of fis bounded from above by c(0, T ).
Remark 19. Corollary 18 states that for typical price paths ω,(S(ω),RSdS(ω)) 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 first 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 financial 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 Itˆo integral defines 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 define it.
Corollary 20. Let (cn)nNbe a sequence of positive numbers such that (cε
nlog n)converges
to 0 for every ε > 0. For nNdefine τn
0:= 0 and τn
k+1 := inf{tτn
k:|StSτn
k|=cn},
kN, and set Sn
t=PkSτn
k1[τn
kn
k+1)(t). Then for typical price paths, ((Sn·S)) converges
uniformly to RSdS. Moreover, for p > 2and for typical price paths there exists a control
function c=c(p, ω)such that
sup
nsup
k<ℓ
|(Sn·S)τn
kn
(ω)Sτn
k(ω)Sτn
kn
(ω)|p/2
c(τn
k, τ n
)1.
Proof. The uniform convergence of ((Sn·S)) to RSdSfollows from Theorem 16.
For the second claim, fix nNand k < ℓ such that τn
T. Then
|(Sn·S)τn
kn
Sτn
kSτn
kn
|.
(Sn·S)Z·
0
SsdSs
+Aτn
kn
.ωcnplog n+vp/2(τn
k, τ n
)2/p .εc1ε
n+vp/2(τn
k, τ n
)2/p,(8)
where ε > 0 and the last estimate holds by our assumption on the sequence (cn), and where
vp/2(s, t) := kAkp/2
p/2var,[s,t]for (s, t)T. Of course, this inequality only holds for typical
price paths and not for all ωΩ.
On the other side, the same argument as in the proof of Corollary 18 (using Young’s
maximal inequality and successively deleting points from the partition) shows that
|(Sn·S)τn
kn
Sτn
kSτn
kn
|.c2q
nvq(τn
k, τ n
),(9)
15
where vq(s, t) := kSkq
qvar,[s,t]for (s, t)T.
Let us define the control function ˜c:= vq+vp/2. Take α > 0 to be determined below. If
cn>˜c(s, t)1(1ε), then we use (9) and the fact that 2 q < 0, to obtain
|(Sn·S)τn
kn
Sτn
kSτn
kn
|α.c(τn
k, τ n
)) 2q
(1ε)vq(τn
k, τ n
)α˜c(τn
k, τ n
)
2q+α(1ε)
(1ε).
The exponent is larger or equal to 1 as long as α(q1ε)/(1 ε). Since qand εcan be
chosen arbitrarily close to 2 and 0 respectively, we can take α=p/2, and get
|(Sn·S)τn
kn
Sτn
kSτn
kn
|p/2.˜c(τn
k, τ n
)(1 + ˜c(0, T )δ)
for a suitable δ > 0.
On the other side, if cn˜c(s, t)1(1ε), then we use (8) to obtain
|(Sn·S)τn
kn
Sτn
kSτn
kn
|α.˜c(τn
k, τ n
) + ˜c(τn
k, τ n
)2α/p,
so that also in this case we may take α=p/2, and thus we have in both cases
|(Sn·S)τn
kn
Sτn
kSτn
kn
|p/2c(τn
k, τ n
),
where cis a suitable (ω-dependent) multiple of ˜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 difference to [FH13] in the derivation below is that we use p-variation
to describe the regularity, and not H¨older continuity, because it is not true that all typical
price paths are H¨older continuous. Also, we make an effort 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 ω, the pro-
cess S(ω) is of finite p-variation for every p > 2, and the integral process RSdS(ω) =
(RSidSj(ω))1i,jdis a continuous function of finite p/2-variation, in the sense that there
exists a control function c=c(ω) such that
Zt
s
SrdSr(ω)Ss(ω)Ss,t(ω)p/2c(s, t)(ω)
for all (s, t)T. From now on, we fix one path Swhich satisfies these two conditions.
Let us write S= (S, A), where
A: ∆TRd×d, A(s, t) = Zt
s
SrdSrSsSs,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)r[s,t]. It is a deep insight of Lyons [Lyo98], proving a conjecture of F¨ollmer, that the area
is exactly the additional information which is needed to solve differential 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 finite q-variation for some q > 1, then it has to be equipped with the iterated integrals up
to order q⌋ − 1 to obtain a continuous integral map. The for us relevant case q(2,3) was
already treated in [Lyo95a].
We say that Sis of finite p-variation if there exists a control function csuch that
|Ss,t|p+|A(s, t)|p/2c(s, t) (10)
for all 0 stT. In that case, we define
kSkpvar := kSkpvar +kAkp/2var.
From now on we fix p > 2 and we assume that Sis of finite 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 “looks like S”, modulo a smooth remainder. The
precise definition is:
Definition 21. Let qbe such that 2/p + 1/q > 1. Let F: [0, T ]Rnand F: [0, T ]Rn×d.
We say that the pair (F, F ) is controlled by Sif the derivative Fhas finite q-variation, and
the remainder RF: ∆TRndefined by
RF(s, t) = Fs,t F
sSs,t,
has finite r-variation, where 1/r = 1/p + 1/q. In this case, we write (F, F )Cq
S=Cq
S(Rn),
and define
k(F, F )kCq
S:= kFkqvar +kRFkrvar.
If it is equipped with the norm |F0|+|F
0|+k(F, F )kCq
S, then the space of controlled paths
Cq
Sis a Banach space.
Naturally, the function Fshould be interpreted as the derivative of Fwith respect to S.
The reason for considering couples (F, F ) and not just functions Fis that the smoothness
requirement on the remainder RFusually does not determine Funiquely for a given path F.
For example, if Fand Sboth have finite r-variation rather than just finite p-variation, then
for every Fof finite q-variation we have (F, F )Cq
S.
Note that we do not require For Fto be continuous. We will point out below why this
does not pose any problems.
To obtain a more “quantitative” 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 Fmay essentially be as irregular
as we like. The remainder RFhas to be of finite r-variation for 1/r = 1/p + 1/q, so in other
words it should be of finite r-variation for some r < 2 and thus slightly more regular than a
typical price path.
17
Example 22. Let ϕC2
band define Fs:= ϕ(Ss) and F
s:= ϕ(Ss). Then (F, F )Cp
S:
Clearly Fhas finite p-variation. For the remainder, we have
|RF(s, t)|p/2=|ϕ(St)ϕ(Ss)ϕ(Ss)Ss,t|p/21
2kϕ′′k|Ss,t|2p/2= 2p/2kϕ′′kp/2
|Ss,t|p.
Since Sis of finite p-variation, RFis of finite p/2-variation. Now 1/(p/2) = 1/p + 1/p, and
thus (F, F )Cp
S.
As the image of the continuous path Sis compact, it is not actually necessary to assume
that ϕis bounded. We may always consider a C2function ψof compact support, such that
ψagrees with ϕon the image of S.
It is instructive to examine under which regularity conditions on ϕwe obtain a controlled
path if Sis a typical price path. As we argued above, ϕ(S) should be of finite q-variation
for some q > 0, which is satisfied as long as ϕis ε-H¨older continuous for some ε > 0. The
remainder RF(s, t) = ϕ(St)ϕ(Ss)ϕ(Ss)Ss,t should be of finite r-variation for some r < 2.
A simple calculation shows that this is satisfied as long as ϕC1+εfor some ε > 0, so that
for such ϕwe 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) = GtGsfor some function Gdefined on [0, T ], but really a function of two variables.
Example 23. Let Gbe a path of finite r-variation for some rwith 1/p + 1/r > 1. Setting
(F, F ) = (G, 0), we obtain a controlled path in Cq
S, where 1/q = 1/r 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 define RSdS,
and if Flooks like Son small scales, then we should be able to define 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 )Cq
S. Then there exists a unique function RFdSC([0, T ],Rn)
which satisfies
Zt
s
FudSuFsSs,t F
sA(s, t).kSkpvar,[s,t]kRFkrvar,[s,t]+kAkp/2var,[s,t]kFkqvar,[s,t]
for all (s, t)T. The integral is given as limit of the compensated Riemann sums
Zt
0
FudSu= lim
m→∞ X
[s1,s2]πmFs1Ss1,s2+F
s1A(s1, s2),(11)
where (πm)is any sequence of partitions of [0, t]with mesh size going to 0.
The map (F, F )7→ (G, G) := (RFudSu, F )is continuous from Cq
Sto Cp
Sand satisfies
k(G, G)kCp
S.kFkp+ (kFk+kFkqvar)kAkp/2var +kSkpvarkRFkrvar.
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 H¨older continuity. But in the p-variation setting, all the proofs
work exactly as in the H¨older setting, and it is a simple exercise to translate the proof of
18
Theorem 4.9 in [FH13] (which is based on Young’s maximal inequality that we encountered
above) to obtain Theorem 24.
There is only one small pitfall: We did not require For Fto 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 Young’s maximal inequality works as long as integrand and integrator have no
common discontinuities, see the Theorem on p. 264 of [You36].
If now ϕC1+ε
bfor some ε > 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 )7→ (ϕ(F), ϕ(F)F) 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
differential equation
dXt=ϕ(Xt)dSt, X0=x0,(12)
t[0, T ], where XCp
S,Rϕ(Xt)dStdenotes the rough path integral, and Sis a typical price
path. Similarly, if ϕC2+ε
b, then there exist p > 2 and q > 0 with 2/p + 1/q > 0, such that
the map (F, F )7→ (ϕ(F), ϕ(F)F) is a locally Lipschitz continuous from Cp
Sto Cq
S, and this
yields the uniqueness of the solution to (12) – 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 ϕ. In the classical Itˆo
theory of SDEs, the function ϕis only required to be Lipschitz continuous. But to solve a
Stratonovich SDE, we need better regularity of ϕ. This is natural, because the Stratonovich
SDE can be rewritten as an Itˆo SDE with a Stratonovich correction term: the equations
dXt=ϕ(Xt)dWtand
dXt=ϕ(Xt)dWt+1
2ϕ(Xt)ϕ(Xt)dt
are equivalent (where Wis a standard Brownian motion, dWtdenotes Itˆo integration, and
dWtdenotes Stratonovich integration). To solve the second equation, we need ϕϕto be
Lipschitz continuous, which is always satisfied if ϕC2
b. But rough path theory cannot
distinguish between Itˆo and Stratonovich integrals: If we define the area of Wusing Itˆo
(respectively Stratonovich) integration, then the rough path solution of the equation will
coincide with the Itˆo (respectively Stratonovich) solution. So in the rough path setting,
the function ϕshould satisfy at least the same requirements as in the Stratonovich setting.
The regularity requirements on ϕare 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
differential 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 ˜
S= ( ˜
S, ˜
A)be two rough paths of finite p-variation, let (F, F )Cq
Sand
(˜
F , ˜
F)Cq
˜
S, and let (s, t)T. Then for every M > 0there exists CM>0such that
Z·
0
FsdSsZ·
0
˜
Fsd˜
Ss
pvar CM|F0˜
F0|+|F
0˜
F
0|+kF˜
Fkqvar
+kRFR˜
Fkrvar +kS˜
Skpvar +kA˜
Akp/2var,
19
as long as
max{|F
0|+k(F, F )kCq
S,|˜
F
0|+k(˜
F , ˜
F)kCq
˜
S
,kSkpvar,kAkp/2var,k˜
Skpvar,k˜
Akp/2var} ≤ 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 differential
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 financial 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
m→∞ X
[r1,r2]πmFr1Sr1,r2+F
r1A(r1, r2),
where (πm) is any sequence of partitions of [0, t] with mesh size going to 0. The term Fr1Sr1,r2
has an obvious financial interpretation. This is the profit that we make by buying Fr1units of
the traded asset at time r1and by selling them at time r2. However, for the “compensator”
F
r1A(r1, r2) there seems to be no financial interpretation, and therefore it is not clear whether
the rough path integral can be understood as profit obtained by investing in S.
However, we observed in Section 3 that along suitable stopping times (τn
k)n,k, we have
Zt
0
SsdSs= lim
n→∞ X
k
Sτn
kSτn
kt,τn
k+1t.
By the philosophy of controlled paths, we expect that also for Fwhich looks like Son small
scales we should obtain
Zt
0
FsdSs= lim
n→∞ X
k
Fτn
kSτn
kt,τn
k+1t,
without having to introduce the compensator F
τn
kA(τn
kt, τ n
k+1 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 first present a special case for which we can give an elementary proof.
For this purpose, we use again the dyadic stopping times τn
0:= 0 and τn
k+1 := inf{tτn
k:
|StSτn
k| ≥ 2n}for k, n N. We write Sn
t:= Pk1[τn
kn
k+1)(t)Sτn
k. Recall from Theorem 16
that for typical price paths.
(Sn·S)Z·
0
SsdSs
.2nplog n. (13)
Lemma 27. Let ε > 0, let FC1+ε
b(Rd,Rd), and suppose Sfulfills (13). Then the Riemann
sums
X
k=0
F(Sτn
k)Sτn
kt,τn
k+1t, t [0, T ],
converge uniformly in C([0, T ],R).
20
Proof. Let us write Fn
t:= F(Sn
t). Using a first order Taylor expansion, we obtain for nN
and t[0, T ] that
(Fn+1 ·S)t(Fn·S)t=
X
=0
X
k=0,
[τn+1
kn+1
k+1 ][τn
n
+1]
F(S)τn
n+1
kSτn+1
kt,τn+1
k+1 t(14)
=
X
=0
X
k=0,
[τn+1
kn+1
k+1 ][τn
n
+1]
(F(Sτn
)Sτn
n+1
k+Rτn
n+1
k)Sτn+1
kt,τn+1
k+1 t
with a remainder Rthat satisfies |Rτn
n+1
k|.kFkC1+ε
b2(1+ε)nfor all n, ℓ, k.
It is a simple observation, which we will prove in Lemma 28 below, that the uniform
convergence of (Sn·S) to RSdSimplies the existence of the quadratic variation of Salong
(τn
k)n,k. In particular, Lemma 14 yields max{k:τn
kt}.22nVn
tfor a uniformly bounded
sequence of increasing functions (Vn). Choose now m=m(n) := n/3. Applying H¨older’s
inequality gives us
(Fn+1 ·S)t(Fn·S)t
X
j=0
X
=0,
[τn
n
+1][τm
jm
j+1]
X
k=0,
[τn+1
kn+1
k+1 ][τn
n
+1]F(S)τm
jn
Sτn
n+1
kSτn+1
kt,τn+1
k+1 t
+
X
=0
X
k=0,
[τn+1
kn+1
k+1 ][τn
n
+1]
Rτn
n+1
kSτn+1
kt,τn+1
k+1 t
+
X
j=0 F(Sτm
j)
X
=0,
[τn
n
+1][τm
jm
j+1]
X
k=0,
[τn+1
kn+1
k+1 ][τn
n
+1]
Sτn
n+1
kSτn+1
kt,τn+1
k+1 t
.(22nVn
T)kFkC1+ε
b222n+ (22nVn
T)kFkC1+ε
b2(1+ε)n2n
+
X
j=0 F(Sτm
j)(Sn·S)τm
jt,τm
j+1t(Sn+1 ·S)τm
jt,τm
j+1t
.sup
NN
(VN
T)kFkC1+ε
b(2nε/3+ 2) + kFksup
NN
(VN
T)22m2nlog n.
Since mn/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 πn={0 = tn
0< tn
1<···< tn
Nn=T},nN, is a given sequence
of partitions such that sup{|Stn
k,tn
k+1 |:k= 0,...,Nn1}converges to 0, and set
Sn
t:=
Nn1
X
k=0
Stn
k1[tn
k,tn
k+1)(t).
21
We assume that the Riemann sums (Sn·S) converge uniformly to a continuous function
RSdS. We also assume that p(2,3) and that there exists a control function cfor which
sup
(s,t)T
|Ss,t|p
c(s, t)+ sup
nsup
0k<ℓNn
|(Sn·S)tn
k,tn
Stn
kStn
k,tn
|p/2
c(tn
k, tn
)1.(15)
Our general proof that the rough path integral is given as limit of Riemann sums is
somewhat indirect. We translate everything from Itˆo 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 Itˆo type integrals. To go from Itˆo
to Stratonovich integral, we first need the quadratic variation:
Lemma 28. Under Assumption (rie), let 1i, j d, and define
hSi, Sjit:= Si
tSj
tSi
0Sj
0Zt
0
Si
rdSj
rZt
0
Sj
rdSi
r.
Then hSi, Sjiis a continuous function and
hSi, Sjit= lim
n→∞hSi, Sjin
t= lim
n→∞
Nn1
X
k=0
(Si
tn
k+1tSi
tn
kt)(Sj
tn
k+1tSj
tn
kt).(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)1i,jd, and call hSithe quadratic
variation of S.
Proof. The function hSi, Sjiis continuous by definition. The specific form (16) of hSi, S ji
follows from two simple observations:
Si
tSj
tSi
0Sj
0=
Nn1
X
k=0 Si
tn
k+1tSj
tn
k+1tSi
tn
ktSj
tn
kt
for every nN, and
Si
tn
k+1tSj
tn
k+1tSi
tn
ktSj
tn
kt=Si
tn
ktSj
tn
kt,tn
k+1t+Sj
tn
ktSi
tn
kt,tn
k+1t+Si
tn
kt,tn
k+1tSj
tn
kt,tn
k+1t,
so that the convergence in (16) is a consequence of the convergence of (Sn·S) to RSdS.
To see that hSi, Sjiis of bounded variation, note that
Si
tn
kt,tn
k+1tSj
tn
kt,tn
k+1t=1
4(Si+Sj)tn
kt,tn
k+1t2(SiSj)tn
kt,tn
k+1t2
(read hSi, Sji= 1/4(hSi+SjihSiSji)). In other words, the n-th approximation of hSi, Sji
is the difference of two increasing functions, and its total variation is bounded from above by
Nn1
X
k=0 (Si+Sj)tn
k,tn
k+1 2+(SiSj)tn
k,tn
k+1 2.sup
m
Nm1
X
k=0 (Si
tm
k,tm
k+1 )2+ (Sj
tm
k,tm
k+1 )2.
Since the right hand side is finite, also the limit hSi, Sjiis of bounded variation.
22