A C\`adl\`ag Rough Path Foundation for Robust Finance

Using rough path theory, we provide a pathwise foundation for stochastic It\^o integration, which covers most commonly applied trading strategies and mathematical models of financial markets, including those under Knightian uncertainty. To this end, we introduce the so-called Property (RIE) for c\`adl\`ag paths, which is shown to imply the existence of a c\`adl\`ag rough path and of quadratic variation in the sense of F\"ollmer. We prove that the corresponding rough integrals exist as limits of left-point Riemann sums along a suitable sequence of partitions. This allows one to treat integrands of non-gradient type, and gives access to the powerful stability estimates of rough path theory. Additionally, we verify that (path-dependent) functionally generated trading strategies and Cover's universal portfolio are admissible integrands, and that Property (RIE) is satisfied by both (Young) semimartingales and typical price paths.


Introduction
A fundamental pillar of mathematical finance is the theory of stochastic integration initiated by K. Itô in the 1940s.Itô's stochastic integration not only allows for a well-posedness theory for most probabilistic models of financial markets, but also comes with invaluable properties, such as having an integration by parts formula and chain rule, and that of being a continuous operator (with respect to suitable spaces of random variables), which is essential for virtually all applications.However, despite the elegance and success of Itô integration, it also admits some significant drawbacks from both theoretical and practical perspectives.
The construction of the Itô integral requires one to fix a probability measure a priori, and is usually based on a limiting procedure of approximating Riemann sums in probability.While in mathematical finance the Itô integral usually represents the capital gain process from continuous-time trading in a financial market, it lacks a robust pathwise meaning.That is, the stochastic Itô integral does not have a well-defined value on a given "state of the world", e.g. a realized price trajectory of a liquidly traded asset on a stock exchange.This presents a gap between probabilistic models and their financial interpretation.Addressing the pathwise meaning of stochastic integration has led to a stream of literature beginning with the classical works of Bichteler [7] and Willinger and Taqqu [52]; see also Karandikar [30] and Nutz [40].
The requirement of fixing a probability measure to have access to Itô integration becomes an even more severe obstacle when one wants to develop mathematical finance under model risk-also known as Knightian uncertainty.Starting from the seminal works of Avellaneda, Date: May 3, 2023.
Levy and Parás [5] and Lyons [37], there has been an enormous and on-going effort to treat the challenges posed by model risk in mathematical finance, that is, the risk stemming from the possible misspecification of an adopted stochastic model, typically represented by a single fixed probability measure.The majority of the existing robust treatments of financial modelling replace the single probability measure by a family of (potentially singular) probability measures, or even take so-called model-free approaches, whereby no probabilistic structure of the underlying price trajectories is assumed; see for example Hobson [27] for classical lecture notes on robust finance.In particular, the latter model-free approaches often require a purely deterministic integration theory sophisticated enough to handle the irregular sample paths of standard continuous-time financial models and commonly employed functionally generated trading strategies.
In the seminal paper [19], Föllmer provided the first deterministic analogue to stochastic Itô integration which had the desired properties required by financial applications.Indeed, assuming that a càdlàg path S: [0, T ] → R d possesses a suitable notion of quadratic variation along a sequence (P n ) n∈N of partitions of the interval [0, T ], Föllmer proved that the limit where Df denotes the gradient of f , exists for all twice continuously differentiable functions f : R d → R. The resulting pathwise integral t 0 Df (S u ) dS u is often called the Föllmer integral, and has proved to be a valuable tool in various applications in model-free finance; for some recent examples we refer to Föllmer and Schied [21], Davis, Ob lój and Raval [14], Schied, Speiser and Voloshchenko [44] and Cuchiero, Schachermayer and Wong [13].In fact, even classical Riemann-Stieltjes integration has been successfully used as a substitution to Itô integration in model-free finance; see e.g.Dolinsky and Soner [16] or Hou and Ob lój [28].
By now arguably the most general pathwise (stochastic) integration theory is provided by the theory of rough paths, as introduced by Lyons [38], and its recent extension to càdlàg rough paths by Friz and Shekhar [25], Friz and Zhang [26] and Chevyrev and Friz [9].Rough integration can be viewed as a generalization of Young integration which is able to handle paths of lower regularity.While rough integration allows one to treat the sample paths of numerous stochastic processes as integrators and offers powerful pathwise stability estimates, it comes with a pitfall from a financial perspective: the rough integral is defined as a limit of so-called compensated Riemann sums, and thus apparently does not correspond to the canonical financial interpretation as the capital gain process generated by continuous-time trading.Even worse, choosing a rough path without care might lead to an anticipating integral, corresponding e.g. to Stratonovich integration, thus introducing undesired arbitrage when used as a capital process.
We overcome these issues by introducing the so-called Property (RIE) for a càdlàg path S: [0, T ] → R d and a sequence (P n ) n∈N of partitions of the interval [0, T ].This property is very much in the same spirit as Föllmer's assumption of quadratic variation along a sequence of partitions.Indeed, we show that Property (RIE) implies the existence of quadratic variation in the sense of Föllmer, and even the existence of a càdlàg rough path S above S, which, loosely speaking, corresponds to an "Itô" rough path in a probabilistic setting.Assuming Property (RIE), we prove that the corresponding rough integrals exist as limits of left-point Riemann sums along the sequence of partitions (P n ) n∈N .This result restores the canonical financial interpretation for rough integration, and links it to Föllmer integration for càdlàg paths.Property (RIE) was previously introduced by Perkowski and Prömel [41] for continuous paths, though we emphasize that the present more general càdlàg setting requires quite different techniques compared to the continuous setting of [41].
Given the aforementioned results, a càdlàg path which satisfies Property (RIE) permits the path-by-path existence of rough integrals with their desired financial interpretation, and moreover maintains access to their powerful stability results which ensure that the integral is a continuous operator.This appears to be a significant advantage compared to the classical notions of pathwise stochastic integration in [7,52,30,40], which do not come with such stability estimates.In particular, the pathwise stability results of rough path theory allow one to prove a model-free version of the so-called fundamental theorem of derivative trading-see Armstrong, Bellani, Brigo and Cass [4]-and may be of interest when investigating discretization errors of continuous-time trading in model-free finance; see Riga [43].Furthermore, in contrast to Föllmer integration, rough integration allows one to consider general functionally generated integrands g(S t ), where g is a general (sufficiently smooth) function g: R d → R d , and not necessarily the gradient of another vector field f : R d → R. For instance, model-free portfolio theory constitutes a research direction in which it is beneficial to consider non-gradient trading strategies; see Allan, Cuchiero, Liu and Prömel [1].Even more generally, rough integration allows one to treat path-dependent functionally generated options in the sense of Dupire [17], and pathwise versions of Cover's universal portfolio, as discussed in Section 3.
Of course, it remains to verify that Property (RIE) is a reasonable modelling assumption in mathematical finance, in the sense that it is fulfilled almost surely by sample paths of the commonly used probabilistic models of financial markets.Since it seems natural that continuous-time trading takes place when the underlying price process fluctuates, we employ sequences of partitions based on such a "space discretization".For such sequences of partitions, we show that the sample paths of càdlàg semimartingales almost surely satisfy Property (RIE).This result is then extended to so-called Young semimartingales, which are stochastic possesses given by the sum of a càdlàg local martingale and an adapted càdlàg process of finite q-variation for some q < 2. Finally, we prove that Property (RIE) is satisfied by typical price paths in the sense of Vovk [48], which correspond to a model-free version of "no unbounded profit with bounded risk".

Organization of the paper:
In Section 2 we introduce Property (RIE) and verify the properties of the associated rough integration as described above.In Section 3 we exhibit functionally generated trading strategies and generalizations thereof which provide valid integrands for rough integration.In Section 4 we prove that (Young) semimartingales and typical price paths satisfy Property (RIE).

Rough integration under Property (RIE)
In this section we develop pathwise integration under Property (RIE).We set up the essential ingredients from rough path theory in Section 2.2 and show in Section 2.3 that paths satisfying (RIE) serve as suitable integrators in mathematical finance.Finally, in Section 2.4 we connect Property (RIE) with the existence of quadratic variation in the sense of Föllmer.
Throughout this section we fix a finite time interval [0, T ] and the dimension d ∈ N. We also adopt the convention that, given a path A defined on [0, T ], we will write A s,t := A t − A s for the increment of A over the interval [s, t].Note however that whenever A is a two-parameter function defined on ∆ [0,T ] , then the notation A s,t will simply denote the value of A evaluated at the pair of times (s, t) ∈ ∆ [0,T ] .
If A denotes either a path from [0, T ] → E or a two-parameter function from ∆ [0,T ] → E for some normed vector space E, then, for any p ∈ [1, ∞), the p-variation of A over the interval [s, t] is defined by where the supremum is taken over all partitions P([s, t]) of the interval [s, t] ⊆ [0, T ], and in the case when A is a path we write We write D p = D p ([0, T ]; E) for the space of all càdlàg paths A: [0, T ] → E of finite pvariation, and we similarly write D p 2 = D p 2 (∆ [0,T ] ; E) for the space of two-parameter functions A: ∆ [0,T ] → E of finite p-variation which are such that the maps s → A s,t for fixed t, and t → A s,t for fixed s, are both càdlàg.Note that A having finite p-variation is equivalent to the existence of a control function w such that |A s,t | p ≤ w(s, t) for all (s, t) ∈ ∆ [0,T ] .(For instance, one may take w(s, t) = A p p,[s,t] .)2.2.Càdlàg rough path theory and Property (RIE).While rough path theory has by now been well studied in the case of continuous paths, as exhibited in a number of books, notably Friz and Hairer [24], its extension to càdlàg paths appeared only recently, starting with Friz and Shekhar [25].In this section we mainly rely on results regarding forward integration with respect to càdlàg rough paths as presented in Friz and Zhang [26].
In the following we fix p ∈ (2, 3) and q ≥ p such that and define r > 1 by the relation This means in particular that 1 < p/2 ≤ r < p ≤ q < ∞.
Throughout the paper, we will use the symbol to denote inequality up to a multiplicative constant which depends only on the numbers p, q and r as chosen above.
We begin by recalling the definition of a càdlàg rough path, as well as the corresponding notion of controlled paths.In the following we will write A ⊗ B for the tensor product of two vectors A, B ∈ R d , i.e. the d × d-matrix with (i, j)-component given by [ holds for all times 0 ≤ s ≤ u ≤ t ≤ T .We denote the space of càdlàg rough paths by V p .
The unfamiliar reader is encouraged to check that, given càdlàg paths X and Z of bounded variation, setting , with the integral defined as a limit of left-point Riemann sums, gives a p-rough path.Although the integral t s Z s,u ⊗ dX u is not in general well-defined when X and Z are not of bounded variation, given a rough path (X, Z, X), we may think of X as postulating a "candidate" for the value of such integrals.
Remark 2.2.The definition of rough paths we have introduced above looks slightly different to the standard definition, in which one takes X = Z.Our definition is slightly more general, but the corresponding theory works in exactly the same way, and turns out to be more convenient in the context of Property (RIE) as we will see later.
More precisely, later the matrix X s,t will for us represent the (a priori ill-defined) 'integral' t s S s,u ⊗ dS u , which will be defined as the limit as n → ∞ of the Riemann sums ( t s S n s,u ⊗ dS u ) n∈N appearing in Property (RIE) below.In the continuous (i.e.without jumps) setting of Perkowski and Prömel [41], a linear interpolation is used to provide a continuous approximation of S n , leading to a Stratonovich type integral in the limit, which is subsequently converted back into an Itô type integral.Thanks to the recently developed theory of càdlàg rough paths, here we can use a more direct argument which avoids this detour.This means working directly with the integral t s S n s,u ⊗ dS u , which corresponds to taking X = S and Z = S n in Definition 2.1, thus requiring X = Z.
For two rough paths, X = (X, Z, X) and X = ( X, Z, X), we use the seminorm , and the pseudometric In the following we write L(R d ; R d ) for the space of linear maps from We say that a pair (F, where the remainder R F is defined implicitly by the relation We refer to F ′ as the Gubinelli derivative of F (with respect to Z), and denote the space of such controlled paths by V q,r Z .Given a path Z ∈ D p ([0, T ]; R d ), the space of controlled paths V q,r Z becomes a Banach space when equipped with the norm (F, With the concepts of rough paths and controlled paths at hand we are ready to introduce rough integration.The following result is a straightforward extension of [1, Lemma 2.6], and its proof follows almost verbatim. Proposition 2.4.Let X = (X, Z, X) ∈ V p be a càdlàg rough path, and let (F, F ′ ) ∈ V q,r Z and (G, G ′ ) ∈ V q,r X be controlled paths with respect to Z and X, respectively, with remainders R F and R G .Then, for each t ∈ [0, T ], the limit exists along every sequence of partitions P of the interval [0, t] with mesh size |P| tending to zero.We call this limit the rough integral of (F, F ′ ) against (G, G ′ ) (relative to the rough path X), which moreover comes with the estimate for all (s, t) ∈ ∆ [0,T ] where the constant C depends only on p, q and r.
For us, the product of vectors F u G u,v appearing in (2.2) will usually be interpreted as the Euclidean inner product, but in general this product may be interpreted as a matrix of any desired shape and size, consisting of linear combinations of products of the components of the two vectors, and the product F ′ u G ′ u X u,v will be a matrix with the same shape.Remark 2.5.In the special case when G = X (so that G ′ is the identity map and R G = 0), the integral defined in Proposition 2.4 reduces to the more classical notion of the rough integral of the controlled path (F, F ′ ) against the rough path X, given by t 0 Remark 2.6.It follows from the estimate in (2.3), combined with the relation X .Notice that the construction of the rough integral in (2.2) is based on so-called compensated Riemann sums While the classical Riemann sums come with a natural interpretation as capital gain processes in the context of mathematical finance, the interpretation of compensated Riemann sums is by no means obvious.However, one advantage of rough integration is that it provides rather powerful stability estimates, for instance as presented in the next proposition.Proposition 2.7.Let X = (X, Z, X), X = ( X, Z, X) ∈ V p be càdlàg rough paths, and let (F, F ′ ) ∈ V q,r Z and ( F , F ′ ) ∈ V q,r Z be controlled paths with remainders R F and R F respectively.(i) We have the estimate where the constant C depends only on p, q and r.
X and ( G, G′ ) ∈ V q,r X also be controlled paths with remainders R G and R G respectively.Let M > 0 be a constant such that We then have the estimate where the new constant C depends on p, q, r and M .
Proof.We present here only the proof of part (ii), since the proof of part (i) follows almost verbatim.Here, the multiplicative constant implied by the symbol will be allowed to depend on the numbers p, q and r as usual, and additionally on the constant M .Following the proof of [26,Lemma 3.4], in our more general setting one deduces the estimates (2.5) Recalling Remark 2.6, we find, using the controlled path structure of the rough integrals, that (2.6) The result then follows upon substituting the estimates (2.4) and (2.5) into (2.6).
In the spirit of Föllmer's assumption of quadratic variation along a sequence of partitions [19], we introduce the following property.
Property (RIE).Let p ∈ (2, 3) and let for each n ∈ N. We assume that: • the sequence of paths (S n ) n∈N converges uniformly to S as n → ∞, • the Riemann sums k+1 ∧t converge uniformly as n → ∞ to a limit, which we denote by • and that there exists a control function w such that In (2.7), and hereafter, we adopt the convention that 0 0 := 0. The name "RIE" is an abbreviation for "Riemann", as we assume the convergence of the Riemann sums S n u ⊗ dS u , instead of the discrete quadratic variations as in [19].Indeed, Property (RIE) is a stronger assumption than the existence of quadratic variation in the sense of Föllmer, and is even enough to allow us to lift S in a canonical way to a rough path-see Lemma 2.12 below-giving us access to the powerful stability results of rough path theory, such as those exhibited in Proposition 2.7.Moreover, Property (RIE) can be verified for most typical stochastic processes in mathematical finance, as we will see in Section 4.
Remark 2.9.We highlight that, rather than simply being a property of a path, Property (RIE) is a property of a path together with a given sequence of partitions (P n ) n∈N .Indeed, such a path will in general not satisfy (RIE) with respect to a different sequence of partitions.However, in practice there is often a natural choice for the sequence of partitions; see Remark 4.2.For clarity, hereafter, whenever we claim that a path satisfies Property (RIE), we will always make explicit the partition with respect to which the path satisfies (RIE), in the sense of Definition 2.8.
Remark 2.10.In Proposition 2.13 below, it is actually shown that it is sufficient in Property (RIE) to assume that the sequence (S n ) n∈N converges only pointwise to S, since the uniformity of this convergence then immediately follows.
Next we shall verify that Property (RIE) ensures the existence of a càdlàg rough path.For this purpose, we consider a suitable approximating sequence of the so-called 'area process', which is represented by X in Definition 2.1.
Lemma 2.11.Suppose S ∈ D([0, T ]; R d ) satisfies Property (RIE) with respect to p and where t s S n s,u ⊗ dS u is defined as in Property (RIE), then there exists a constant C, depending only on p, such that (2.9) for every n ∈ N.
Otherwise, let k 0 be the smallest k such that t n k ∈ (s, t), and let k 1 be the largest such k.It is easy to see that the triplet (S, S n , A n ) satisfies Chen's relation (2.1), from which it follows that As we have already observed, we have that ,t = 0.By the inequality (2.7), we have We estimate the remaining terms as , so that, putting this all together, we deduce the existence of a constant C > 0 such that Taking an arbitrary partition P of the interval [0, T ], it follows that We thus conclude that (2.9) holds with C = (2 C) Lemma 2.12.Suppose that S ∈ D([0, T ]; R d ) satisfies Property (RIE) with respect to p and Then, the triplet S = (S, S, A) is a càdlàg p-rough path.
Proof.It is straightforward to verify Chen's relation (2.1), i.e. that By Property (RIE), we know that lim n→∞ A n s,t = A s,t , where the convergence is uniform in (s, t), and thus, being a uniform limit of càdlàg functions, A is itself càdlàg.By the lower semi-continuity of the p 2 -variation norm, and the result of Lemma 2.11, we have that It follows that (S, S, A) is a càdlàg p-rough path.
2.3.The rough integral as a limit of Riemann sums.While the rough integral in (2.2) is a powerful tool to study various differential equations, it lacks the natural interpretation as the capital gain process in the context of mathematical finance.The aim of this subsection is to restore this interpretation by showing that the rough integral can be obtained as the limit of left-point Riemann sums provided that the integrator satisfies Property (RIE).As preparation we need the following approximation result.
be a sequence of nested partitions with vanishing mesh size, so that P n ⊂ P n+1 for all n, and |P n |→ 0 as n → ∞ (as in the setting of Property (RIE)).Let F : [0, T ] → R d be a càdlàg path, and define (2.10)

Let
(2.11) be the set of jump times of F .The following are equivalent: Proof.We first show that conditions (i) and (ii) are equivalent.To this end, suppose that J F ⊆ ∪ n≥1 P n and let t ∈ (0, T ].If t ∈ J F , then there exists m ≥ 1 such that t ∈ P n for all n ≥ m.In this case we then have that and since the mesh size |P n |→ 0, it follows that F n t → F t− = F t as n → ∞.Now suppose instead that there exists a t ∈ J F such that t / ∈ ∪ n≥1 P n .We then observe that F n t → F t− = F t , so that F n t F t .This establishes the equivalence of (i) and (ii).Since (iii) clearly implies (ii), it only remains to show that (ii) implies (iii).By [22,Theorem 3.3], it is enough to show that the family of paths {F n : n ≥ 1} is equiregulated in the sense of [22,Definition 3.1].
Step 1.Let t ∈ (0, T ] and ε > 0. Since the left limit F t− exists, there exists δ > 0 with Moreover, since the sequence of partitions is nested, we immediately have that, for all n ≥ m, there exists a k such that If n < m, then there does not exist a k such that t n k ∈ (t − δ, t), which implies that F n is constant on the interval (t − δ, t), and hence that F n s = F n t− .Suppose instead that n ≥ m.Let i = max{k : t n k ≤ s} and j = max{k : t n k < t}.By the definition of u, we see that Thus, we have that By part (i), we know that t ∈ ∪ n≥1 P n .Let If n < m, then, since v < u, there does not exist a k such that Hence, F n is constant on the interval [t, v], so that in particular F n s = F n t .Suppose instead that n ≥ m.By the definition of m, there exists a j such that t n j = t.Let i = max{k : t n k ≤ s}.In particular, we then have that If n < m, then, since v < u, there does not exist a k such that t n k ∈ (t, v].Hence, F n is constant on the interval [t, v], so that in particular F n s = F n t .Suppose instead that n ≥ m.Let i = max{k : t n k ≤ s} and j = max{k : t n k ≤ t}.Since, by the definition of m, there exists at least one k such that t n k ∈ (t − δ, t], and since Thus, we have that |F n s − F n t |< ε for all s ∈ (t, v] and all n ≥ 1.It follows that the family of paths {F n : n ≥ 1} is indeed equiregulated. The next theorem is the main result of this section, stating that the rough integral can be approximated by left-point Riemann sums along a suitable sequence of partitions, in the spirit of Föllmer's pathwise integration.Theorem 2.14.Let q ≥ p such that 2 p + 1 q > 1, and let r > 1 such that 1 r = 1 p + 1 q .Suppose that S ∈ D([0, T ]; R d ) satisfies Property (RIE) with respect to p and (P n ) n∈N .Let (F, F ′ ) ∈ V q,r S and (G, G ′ ) ∈ V q,r S be controlled paths with respect to S, and assume that J F ⊆ ∪ n∈N P n , where J F is the set of jump times of F , as in (2.11).Then the rough integral of (F, F ′ ) against (G, G ′ ) relative to the rough path S = (S, S, A), as defined in (2.2), is given by where the convergence is uniform in t ∈ [0, T ].
Proof.We recall from Lemma 2.12 that S = (S, S, A) is a p-rough path, so that, by Proposition 2.4, the rough integral of (F, F ′ ) against (G, G ′ ) (relative to S) exists.It is also clear that S n := (S, S n , A n ) is a p-rough path, where A n was defined in (2.8).Moreover, by Property (RIE), we immediately have that S n and A n converge uniformly to S and A respectively as n → ∞.
For each n ≥ 1, we let F n be the path defined in (2.10).We consider the pair (F n , F ′ ) as a controlled path with respect to S n , defining the remainder term R n by the usual relation: Since S n converges uniformly to S and, by Proposition 2.13, F n converges uniformly to F , it follows that R n also converges uniformly to the remainder term R corresponding to the S-controlled path (F, F ′ ).We observe that S n p,[0,T ] ≤ S p,[0,T ] and F n p,[0,T ] ≤ F p,[0,T ] , and we have from Lemma 2.11 that where in the last line we used Young's inequality, recalling that 1 r = 1 p + 1 q .Otherwise, let k 0 be the smallest k such that t n k ∈ [s, t], and let k 1 be the largest such k.After a short calculation, we find that . We can deal with the terms R n s,t n k 0 and ,t using the above, and we bound Putting this all together, we have that where the constant C depends only on p, q and r.Taking an arbitrary partition P of the interval [0, T ], we deduce that Since the sequence (S n ) n≥1 has uniformly bounded p-variation, and S n converges uniformly to S as n → ∞, it follows by interpolation that S n converges to S with respect to the p ′ -variation norm, i.e. S n − S p ′ ,[0,T ] → 0 as n → ∞.It follows similarly that A n − A p ′ 2 ,[0,T ] → 0 and R n − R r ′ ,[0,T ] → 0, and hence also that S n ; S p ′ ,[0,T ] → 0 as n → ∞.It thus follows from part (ii) of Proposition 2.7 that (2.12) where the convergence is uniform in t ∈ [0, T ].Note that in (2.12) the integral t 0 F n u dG u is defined relative to the rough path S n = (S, S n , A n ), whilst the limiting rough integral t 0 F u dG u is defined relative to S = (S, S, A).We recall from Proposition 2.4 that the integral of (F n , F ′ ) against (G, G ′ ) relative to S n = (S, S n , A n ) is given by the limit where the limit is taken over any sequence of partitions of the interval [0, t] with vanishing mesh size.Take any refinement P of the partition (P n ∪ {t}) ∩ [0, t] (where as usual P n is the partition given in Property (RIE)), and let [u, v] ∈ P. By the choice of the partition P, there exists a k such that t n k ≤ u < v ≤ t n k+1 , which, recalling (2.8), implies that A n u,v = 0. Since the mesh size of P may be arbitrarily small, it follows that lim To conclude, we then simply recall (2.12), and note that We can actually generalize the result of Theorem 2.14 to a slightly larger class of integrands.
Corollary 2.15.Recall the assumptions of Theorem 2.14, and let γ ∈ D r ([0, T ]; R d ).Then, the rough integral of the controlled path H = F + γ, given by (H, The point here is that the path γ may have jump times which do not belong to the set ∪ n∈N P n . Proof.Since γ has finite r-variation, we immediately have that γ is a controlled path with Gubinelli derivative simply given by γ ′ = 0.By linearity, it is then clear that (H, H ′ ) = (F, F ′ ) + (γ, 0) is indeed a controlled path with respect to S. Since γ ′ = 0, we have from Proposition 2.4 that By linearity, we have that , and the result then follows from Theorem 2.14.
2.4.Link to Föllmer integration.In his seminal paper [19], Föllmer introduced a notion of pathwise integration based on the concept of quadratic variation, and derived a corresponding pathwise Itô formula, which have proved to be useful tools in robust approaches to mathematical finance.
In the following we will write B[0, T ] for the Borel σ-algebra on [0, T ].
Definition 2.16.Let S ∈ D([0, T ]; R) and let , be a sequence of partitions with vanishing mesh size.We say that S has quadratic variation along (P n ) n∈N in the sense of Föllmer if the sequence of measures converges weakly to a measure µ, such that the map t → [S] c t := µ([0, t]) − 0<s≤t |S s−,s | 2 is continuous and increasing.In this case we call the function [S], given by [S] t = µ([0, t]), the quadratic variation of S along (P n ) n∈N .
We say that a path S ∈ D([0, T ]; R d ) has quadratic variation along (P n ) n∈N in the sense of Föllmer if the condition above holds for S i and S i + S j for every (i, j), and in this case we write Assuming that a path S ∈ D([0, T ]; R d ) has quadratic variation along (P n ) n∈N and f ∈ C 2 (R d ; R), Föllmer showed that the limit Df (S u ) dS u is only well-defined for gradients Df and not for general functions, as its existence is given by the corresponding pathwise Itô formula.This result can also be explained via the language rough path theory; see Friz and Hairer [24,Chapter 5.3].
In the following we relate Property (RIE) to the existence of quadratic variation in the sense of Föllmer.To this end, for each i = 1, . . ., d, we introduce and the discrete quadratic variation S i , S j n by Proposition 2.17.Let S ∈ D([0, T ]; R d ) and let be a sequence of nested partitions with vanishing mesh size.The following conditions are equivalent: (i) For every pair (i, j), the Riemann sums t 0 S n,i u dS j u + t 0 S n,j u dS i u converge uniformly to a limit, which we denote by t 0 S i u dS j u + t 0 S j u dS i u .(ii) For every pair (i, j), the discrete quadratic variation S i , S j n converges uniformly to a càdlàg path, which we denote by S i , S j .(iii) The path S has quadratic variation along (P n ) n∈N in the sense of Föllmer.
Moreover, if these conditions hold then the path S i , S j has finite total variation, and, for every (i, j), we have that [S i , S j ] = S i , S j and the equality Proof.We have from which it follows that conditions (i) and (ii) are equivalent, and that (2.14) then also holds.In this case, we also have that so that, as the difference of two non-decreasing functions, S i , S j has finite total variation.For one-dimensional paths S, the equivalence of conditions (ii) and (iii) follows from [51, Propositions 3 and 4].The extension of this to d-dimensional paths S and the equality [S i , S j ] = S i , S j then follow from the polarization identity S i , S j n t = 1 2 S i + S j , S i + S j n t − S i , S i n t − S j , S j n t and the definition of [S i , S j ] in (2.13).
Remark 2.18.As an immediate consequence of Proposition 2.17, we have that if a path S satisfies (RIE) along (P n ) n∈N , then it has quadratic variation along (P n ) n∈N in the sense of Föllmer, thus allowing one to apply all the known results regarding Föllmer integration.
In particular, if a vector field f : R d → R is of class C 3 , then, by Theorem 2.14, the Föllmer integral • 0 Df (S u ) dS u coincides with the rough integral • 0 Df (S u ) dS u .We thus obtain the rough Itô formula: which holds for every t ∈ [0, T ], where [S] = ([S i , S j ]) 1≤i,j≤d denotes the quadratic variation matrix, and ∆S u := lim s→u,s<u S s,u .We note that the formula above is precisely the Itô formula for rough paths derived in Friz and Zhang [26].

Functionally generated trading strategies and their generalizations
Given Property (RIE), we can introduce a model-free framework for continuous-time financial markets with a possibly infinite time horizon.In this section we shall verify that most relevant trading strategies from a practical perspective, such as delta-hedging strategies and functionally generated strategies, are admissible integrands for price paths satisfying Property (RIE).Furthermore, the underlying rough integration allows us to deduce stability estimates for admissible strategies.Definition 3.1.For a fixed p ∈ (2, 3), we say that a path S ∈ D([0, ∞); R d ) is a price path, if there exists a nested sequence of locally finite partitions (P n ) n∈N of the interval [0, ∞), with vanishing mesh size on compacts, such that, for all T > 0, the restriction S| [0,T ] satisfies (RIE) with respect to p and (P n ([0, T ])) n∈N .
We denote the family of all such price paths by Ω p .
Note that the sequence of partitions (P n ) n∈N may depend on the choice of price path S ∈ Ω p , consistent with the stochastic framework where this sequence will naturally be defined in terms of (probabilistic) stopping times.
Having fixed the model-free structure of the underlying price paths, we can introduce the class of admissible strategies and the corresponding capital process.Definition 3.2.Let p ∈ (2, 3) and let S ∈ Ω p be a price path.We say that a path ϕ: [0, ∞) → R d is an admissible strategy (with respect to S), if • there exist q ≥ p and r > 1 with 2/p + 1/q > 1 and 1/r = 1/p + 1/q, such that for every T > 0, there exists a path ϕ S is a controlled path with respect to S in the sense of Definition 2.3, • and J ϕ ⊆ ∪ n∈N P n , where J ϕ is the set of jump times of ϕ in (0, ∞), and (P n ) n∈N is the sequence of partitions associated with the price path S ∈ Ω p .We denote the space of all admissible strategies (with respect to S) by A S .
We define the capital process associated with ϕ and S as the path V ϕ (S): [0, ∞) → R given by where T } is the sequence of partitions specified in Property (RIE).Remark 3.3.In a semimartingale setting, as often used in classical mathematical finance, one usually considers left-continuous trading strategies ϕ.In the present setting this assumption is not necessary, as the corresponding capital process V ϕ (S), which, as we will see below, may be expressed as a rough integral, does not change when replacing ϕ by its left-continuous modification; see e.g.[25,Theorem 31].The reason for this is essentially the left-point Riemann sum construction of the integral.Indeed, suppose that S has a jump at a time t > 0. The contribution to the capital process V ϕ (S) at time t is then given by lim s→t, s<t ϕ s S s,t , which is invariant to the choice of ϕ or its left-continuous modification.Furthermore, we will see that V ϕ (S) coincides with the classical stochastic Itô integral, both the rough and stochastic integrals are defined; see Section 4.4 below.
The condition J ϕ ⊆ ∪ n∈N P n means that one is allowed to use trading strategies whose jump points are included in the underlying sequence of partitions (P n ) n∈N .On the one hand, many frequently used trading strategies, such as delta-hedging, satisfy this condition, and further examples will be discussed later in this section.On the other hand, the sequence of partitions (P n ) n∈N can be fixed a priori to allow for a desired class of trading strategies, e.g.buy and hold strategies along the sequence of dyadic partitions.Proposition 3.4.Let p ∈ (2, 3), let S ∈ Ω p be a price path, and let ϕ ∈ A S be an admissible strategy (in the sense of Definition 3.2).Then, the capital process V ϕ (S) as defined in (3.1) exists as a locally uniform limit, and is actually given by that is, the rough integral of the controlled path (ϕ, ϕ ′ ) ∈ V q,r S against the rough path S defined in Lemma 2.12.
Moreover, given another price path S ∈ Ω p and an admissible strategy φ ∈ A S with respect to S, we have, for every T > 0, that where the constant C depends on p, q and r.
Remark 3.5.Recall from Remark 2.5 that the rough integral in (3.2) is defined by the limit , where the limit is taken over any sequence of partitions of the interval [0, t] with vanishing mesh size.Here, ϕ u and S u,v both take values in R d , and we interpret their multiplication as the Euclidean inner product.The derivative ϕ ′ u takes values in L(R d ; R d ), which we can also identify with L(R d×d ; R).Since A u,v ∈ R d×d , the product ϕ ′ u A u,v also takes values in R. Proof of Proposition 3.4.Let T > 0, and let (P n ) n∈N be a sequence of nested partitions such that p, (P n ) n∈N and S satisfy Property (RIE) on the interval [0, T ].Recall from Property (RIE) the existence of the limit t 0 S u ⊗ dS u for every t ∈ [0, T ].By Lemma 2.12, defining the function A: we have that the triplet S = (S, S, A) is a càdlàg rough path (in the sense of Definition 2.1).Hence, the rough integral in (3.2) is well-defined by Proposition 2.4 (see also Remark 2.5), and satisfies (3.1) as a locally uniform limit by Theorem 2.14.
For the stability estimate we simply note that and apply part (i) of Proposition 2.7.
In the following we show that the most relevant trading strategies from a practical viewpoint belong to the class of admissible strategies in the sense of Definition 3.2.

3.2.
Functionally generated trading strategies.Having fixed the set Ω p of underlying price paths, we start by introducing functionally generated portfolios.For this purpose, for some d A ∈ N, we fix a càdlàg path A: [0, ∞) → R d A of locally bounded variation and assume that the jump times of A belong to the union of the partitions (P n ) n∈N appearing in Property (RIE); that is, we assume that J A ⊆ ∪ n∈N P n , where J A := {t ∈ (0, ∞) : A t−,t = 0}.The path A is supposed to include additional information pertaining to the market which a trader would like to include in their trading decisions.For instance, the components of the path A = (A 1 , . . ., A d A ) could include time t → t, the running maximum t → max u∈[0,t] S i u , or the integral t → t 0 S i u du for some (or all) i = 1, . . ., d.A more detailed discussion on practical choices of the path A can be found in Schied, Speiser and Voloshchenko [44].
For ℓ = d + d A , we denote by C 2 b (R ℓ ; R d ) the space of twice continuously differentiable (in the Fréchet sense) functions f : R ℓ → R d such that f and its derivatives up to order 2 are uniformly bounded; that is For S ∈ Ω p we introduce the set G 2 S of all generalized functionally generated trading strategies ϕ f , which are all strategies of the form S the corresponding capital process is given by Proposition 3.6.Let p ∈ (2, 3) and S ∈ Ω p , and let ϕ f , ϕ f ∈ G 2 S .Then, ϕ f ∈ A S is an admissible strategy, and the capital process (V f t (S)) t∈[0,∞) given in (3.4) is well-defined as a locally uniform limit in t ∈ [0, ∞).Moreover, for every T ∈ [0, ∞), we have the stability estimate where the constant C depends only on p, and the triplet S = (S, S, A) is the càdlàg rough path defined in Lemma 2.12.
Proof.Admissibility: Let ϕ = ϕ f ∈ G 2 S be a functionally generated strategy ϕ = (ϕ 1 , . . ., ϕ d ) of the form in (3.3) is a controlled path with respect to S in the sense of Definition 2.3 (with q = p and r = p/2), where and D S f denotes the derivative of f with respect to its first d components.To see this, we first note that . We moreover have that For a given path F , let J F = {t ∈ (0, T ] : F t−,t = 0} denote the jump times of F .It follows from Property (RIE) and Proposition 2.13 that J S ⊆ ∪ n∈N P n .Since we also assumed that J A ⊆ ∪ n∈N P n , it then follows from (3.3) that J ϕ ⊆ ∪ n∈N P n .Thus, by Theorem 2.14, we have that and that this limit is uniform in t ∈ [0, T ].
Stability estimate: Let ϕ and φ be the strategies generated by f and f respectively, as defined in (3.3).By part (i) of Proposition 2.7, we have the estimate As above, here , with φ′ and R φ defined similarly.We will now aim to estimate each term on the right-hand side of (3.8).

3.3.
Path-dependent functionally generated trading strategies.The functionally generated trading strategies considered in Section 3.2 could depend on the past prices only through a process of locally finite variation.In some contexts it is beneficial to work with trading strategies possessing a more general path-dependent structure; see e.g.Schied, Speiser and Voloshchenko [45,44] for more detailed discussions in this direction.A common way to treat path-dependent and non-anticipating trading strategies is the calculus initiated by Dupire [17] and Cont and Fournié [11].For the sake of brevity we recall here only the essential definitions and refer to Ananova [3, Section 3.1] and [11] for full details.
A functional As usual, the non-anticipative functionals are defined on the space of stopped paths, defined as the equivalence class in [0, T ] × D([0, T ]; R d ) with respect to the following equivalence relation: which turns (Λ d T , d ∞ ) into a complete metric space.We introduce the following spaces: is the space of left-continuous functionals F : Λ d T → R d , i.e. for all (t, S) ∈ Λ d T and ε > 0, there exists ν > 0 such that, for all (t is the space of all boundedness-preserving functionals F : Λ d T → R d , i.e. for every compact subset K ⊂ R d and for every t 0 ∈ [0, T ], there exists C > 0 such that, for all t ∈ [0, t 0 ] and (t, S) is the space of all Lipschitz continuous functionals F : Λ d T → R d , i.e. there exists C > 0 such that, for all (t, S), We define C exists, where (e i ) i=1,...,d is the canonical basis of R d , and , and S ∈ Ω p , then the path-dependent functionally generated trading strategy F (•, S) is an admissible strategy in the sense of Definition 3.2.
Proof.That (F (•, S), ∇ x F (•, S)) is a controlled path with respect to S is an immediate consequence of [2, Lemma 5.12]; see also [3,Lemma 3.7].The admissibility condition regarding the jump times of F (•, S) is ensured by the Lipschitz continuity of F .
Remark 3.8.The standard examples of sufficiently regular path-dependent functionals are functionals which depend on the running maximum or on a notion of the average of the underlying path; see e.g.Ananova [2].Further examples include: Cover's universal portfolio.While functionally generated trading strategies are most prominent in the literature regarding hedging and control problems in mathematical finance, various other trading strategies with desirable properties have been considered.One example coming from portfolio theory is Cover's universal portfolio, as introduced in [12].The basic idea is to invest, not according to one specific trading strategy, but according to a mixture of all admissible strategies.Following Cuchiero, Schachermayer and Wong [13], we introduce here a model-free analogue of Cover's universal portfolio.
Let Z be a Borel measurable subset of C 2 b (R d ; R d ), and suppose that ν is a probability measure on Z.A model-free version of Cover's universal portfolio ϕ ν is then given by (3.13) where ϕ f = f (S) is the portfolio generated by f , for some fixed S ∈ Ω p .
Lemma 3.9.Let S ∈ Ω p and let ν be a probability measure on Z, as above.
then Cover's universal portfolio ϕ ν , as defined in (3.13), is an admissible strategy in the sense of Definition 3.2.
Proof.Let T > 0. We know from Proposition 3.6 that, for each f ∈ Z, the corresponding functionally generated portfolio ϕ = ϕ f is an admissible strategy.Let ϕ ′ and R ϕ denote the corresponding Gubinelli derivative and remainder term.It follows from the inequalities in (3.6) and (3.7) that and hence that Recall that the map (ϕ, is a norm on the Banach space of controlled paths V p, p 2 S .Note moreover that the subset of controlled paths (ϕ, S , and thus is itself a Banach space.It follows from the integrability condition in (3.14) that the integral in (3.13) exists as a well-defined Bochner integral, and defines a controlled path (ϕ ν , (ϕ 3.5.Functionally generated portfolios from stochastic portfolio theory.In this section we briefly discuss admissible strategies appearing in stochastic portfolio theory, as initiated by Fernholz [18]; see also e.g.Strong [47] and Karatzas and Ruf [33].Following the model-free framework for stochastic portfolio theory, as introduced in Schied, Speiser and Voloshchenko [44] and Cuchiero, Schachermayer and Wong [13], we consider a d-dimensional càdlàg path S = (S 1 , . . ., S d ) such that S i t > 0 for all t ≥ 0 and i = 1, . . ., d.The total capitalization Σ is defined by Σ and the relative market weight process µ is given by Here we impose that the market weight process µ = (µ 1 , . . ., µ d ) satisfies Property (RIE) with respect to some p ∈ (2, 3) and a sequence of nested partitions (P n ) n∈N .More precisely, we assume that µ is a price path, in the sense of Definition 3.1.
Given a controlled path θ ∈ V q,r µ , we define its associated value evolution V θ by V θ t := d i=1 θ i t µ i t for t ≥ 0. We also denote where t 0 θ s dµ s is interpreted as a rough integral, as in Remark 2.5.As in classical mathematical finance (see e.g.[33, Proposition 2.3]) we can show that every controlled path θ ∈ V q,r µ induces a self-financing trading strategy ϕ, and that every such strategy ϕ is itself a controlled path in V q,r µ .
Proposition 3.10.Given θ ∈ V q,r µ and a constant C ∈ R, we introduce Then the resulting path ϕ = (ϕ 1 , . . ., ϕ d ) is a controlled path in V q,r µ , and ϕ is self-financing, in the sense that where Moreover, if θ is an admissible strategy in the sense of Definition 3.2, then so is ϕ.
Proof.Since θ ∈ V q,r µ is a controlled path, by (the trivial extension to càdlàg paths of) [1, Lemma A.1], the path V θ = d i=1 θ i µ i is also a controlled path.Recalling Remark 2.6, we also have that the rough integral • 0 θ s dµ s is a controlled path, and hence that ϕ i = θ i − Q θ − C is as well for every i = 1, . . ., d, so that ϕ ∈ V q,r µ .The self-financing property of ϕ can be verified by following the proof of [33,Proposition 2.3].
We know from Proposition 2.13 and the fact that µ satisfies Property (RIE) that the jump times J µ of µ satisfy J µ ⊆ ∪ n∈N P n .It is also clear that the jump times of the integral • 0 θ dµ form a subset of J µ .Thus, under the assumption that θ is an admissible strategy, so that J θ ⊆ ∪ n∈N P n , we also deduce that J ϕ ⊆ ∪ n∈N P n , so that ϕ is itself an admissible strategy.
The following definition introduces notions which can be seen as the rough path counterparts of the functionally generated strategies induced by regular and Lyapunov functions from stochastic portfolio theory; see [33, has locally bounded variation.We say that a regular function G: ∆ d + → R is a Lyapunov function for the market weight process µ if the path Γ G is also nondecreasing.
Example 3.12.Suppose that G: It is then straightforward to see that θ = DG(µ) defines a controlled path, with DG here defined as the classical gradient of G.It also follows from the Itô formula for rough paths (recall Remark 2.18) that where [µ] c is the continuous part of the quadratic variation of µ, and ∆µ s = µ s − µ s− denotes the jump of µ at time s.
If we also assume that G is concave, then we infer from (3.15) that Γ G is nondecreasing.In this case G is a Lyapunov function in the sense of Definition 3.11.Two important examples of such functions G are the Gibbs entropy function, H(x) := d i=1 x i log 1 x i , and the quadratic function, Q (c) (x) := c − d i=1 x 2 i for some c ∈ R; see [33, Section 5.1].Remark 3.13.If µ is realized by a semimartingale model, then any continuous concave (but not necessarily C 3 ) function G can be a candidate for a Lyapunov function in the sense of [33,Definition 3.3].This is essentially because in stochastic portfolio theory one only needs the "supergradients" DG to be measurable, so that DG(µ) is integrable with respect to the semimartingale µ.In contrast, in our purely pathwise setup, measurability of DG alone is not enough to ensure that θ = DG(µ) is controlled by µ, so we require more regularity of the generating function G.This illustrates a difference between stochastic integration in a probabilistic setting and rough integration when only a single deterministic path is considered; for more detailed discussions on this theme, we refer to [1].On the other hand, as shown in the previous example, many important Lyapunov functions from stochastic portfolio theory (such as the Gibbs entropy function) are actually smooth, so they do induce controlled paths (and indeed admissible strategies in the sense of Definition 3.2) for almost all trajectories of µ, and the stability results established in Section 3 remain valid for these functionally generated strategies.It would be interesting to explore further Lyapunov functions for rough paths, but this is beyond the scope of the present paper.
Based on the previous observations, we can also extend the following notions from classical stochastic portfolio theory (e.g.[33]) to our rough path setting.The proof is straightforward and is therefore omitted for brevity.
Corollary 3.14.Suppose that the market weight process µ = (µ 1 , . . ., µ d ) is a price path in the sense of Definition 3.1, and let θ = (θ 1 , . . ., θ d ) be an admissible strategy for µ in the sense of Definition 3.2.Let G be a regular function for µ in the sense of Definition 3.11.Then the following paths are also admissible strategies for µ: (i) the additive generated strategy: ) the portfolio weights associated with ϕ: the multiplicative generated strategy: , where here the function G is also assumed to be positive and bounded away from zero, (iv) the portfolio weights associated with Φ:

Semimartingales and typical price paths satisfy Property (RIE)
In this section we show that many stochastic processes commonly used to model the price evolutions on financial markets satisfy Property (RIE) along a suitable sequence of partitions.In particular, we verify that Property (RIE) holds for semimartingales and for typical price paths in the sense of Vovk [50].
4.1.Semimartingales.Semimartingales, such as geometric Brownian motion and Markov jump-diffusion processes, serve as the most frequently used stochastic processes to model price evolutions on financial markets.For more details on semimartingales and Itô integration we refer to the standard textbook by Protter [42,Chapter II].
Usually the considered class of semimartingales is restricted to those satisfying the condition "no free lunch with vanishing risk" in classical mathematical finance, e.g.Delbaen and Schachermayer [15], or the condition "no unbounded profit with bounded risk" (NUPBR) in stochastic portfolio theory; see e.g.Karatzas and Kardaras [31].Such a restriction is not required here.Property (RIE) is fulfilled by general càdlàg semimartingales with respect to any p ∈ (2, 3) and a suitable (random) sequence of partitions.
For each n ∈ N, we introduce stopping times (τ n k ) k∈N∪{0} such that τ n 0 = 0, and We then define a sequence of partitions (P n X ) n∈N by P n X := {τ m k : m ≤ n, k ∈ N ∪ {0}}.Note that (P n X ) n∈N is a nested sequence of adapted partitions.However, this sequence of partitions will not have vanishing mesh size if the path X has an interval of constancy.To amend this, we proceed as follows.For each n ∈ N and k ∈ N ∪ {0}, we set , that is, the beginning and end points of the first interval of constancy of X within the interval [τ n k , τ n k+1 ].For each i ∈ N, we then define Clearly, for each n, k, we will have that ρ n k,i = ς n k for all but finitely many i ∈ N (provided that ς n k < ∞).Finally, we define n∈N is still nested, and moreover has vanishing mesh size on every compact time interval.Moreover, it is straightforward to see that all the points τ n k , σ n k and ρ n k,i appearing in the partitions (Q n X ) n∈N are stopping times with respect to the (right-continuous) filtration (F t ) t∈[0,∞) .In the next proposition we will show that X satisfies Property (RIE) with respect to any p ∈ (2, 3) and the sequence of partitions (Q n X ) n∈N .Proposition 4.1.Let p ∈ (2, 3), and let X be a d-dimensional càdlàg semimartingale.Then almost every sample path of X is a price path in the sense of Definition 3.1.
Proof.Step 1. Fix a T > 0. Let • 0 X u− ⊗ dX u denote the Itô integral of X with respect to itself.For each n ∈ N, we define the discretized process X n = (X n t ) t∈[0,T ] by (4.1) An application of the Burkholder-Davis-Gundy inequality and the Borel-Cantelli lemma, as in the proof of [34,Proposition 3.4], then yields the existence of a measurable set Ω ′ ⊆ Ω with full measure such that, for every ω ∈ Ω ′ and every ε ∈ (0, 1), there exists a constant Thus, we have that X n (ω) → X(ω) and Step 2. We choose a q 0 ∈ (2, 3) close enough to 2 and an ε ∈ (0, 1) small enough such that We also fix a control function w X,q 0 such that (4.4) |X s,t | q 0 ≤ w X,q 0 (s, t) for all (s, t) ∈ ∆ [0,T ] .This is always possible as X has almost surely finite q-variation for every q > 2, and so without loss of generality we may assume that X(ω) has finite q 0 -variation for every ω ∈ Ω ′ .Note that by (4.3) we have p > q 0 and consequently (by increasing the values of w X,q 0 by a multiplicative constant if necessary) we can also assume that (4.5) sup Let 0 ≤ s < t ≤ T be such that s = τ n k 0 and t = τ n k 0 +N for some n ∈ N, k 0 ∈ N ∪ {0} and N ≥ 1.By the superadditivity of the control function w X,q 0 , there must exist an l ∈ {1, 2, . . ., N − 1} such that Thus, by (4.4), By successively removing in this manner all the intermediate points from the partition {s = τ n k 0 , τ n k 0 +1 , . . ., τ n k 0 +N = t}, we obtain the estimate 2/q 0 N 1−2/q 0 w X,q 0 (s, t) 2/q 0 .Since w X,q 0 (s, t) we have that N ≤ 2 nq 0 w X,q 0 (s, t).Substituting this into the above, we have that where here the discretized process X n is defined relative to the partition {τ n for some m 1 , m 2 ≤ n and k 1 , k 2 ∈ N ∪ {0}.In this case, the number of partition points N above satisfies N ≤ n m=1 2 mq 0 w X,q 0 (s, t) 2 nq 0 w X,q 0 (s, t), and we thus still obtain the same bound in (4.6).If we further allow the pair of times s, t to include the times σ m k for m ≤ n and k ∈ N ∪ {0}, then this at most doubles the total number of partition points N , so we can again obtain the same bound.Since the points ρ u− ⊗ dX u = 0 for every i ∈ N, and it follows that these terms will not contribute anything to the bound above.
Thus, the bound in (4.6) actually holds for all 0 ≤ s < t ≤ T with s, t ∈ Q n X , with X n defined relative to the partition Q n X , as in (4.1).
Now we consider the case that w X,q 0 (s, t) Then, in view of (4.2), we obtain t s for some constant C 2 = C 2 (ε, ω), and hence Remark 4.2.The result of Proposition 4.1 implies that almost every sample path of a semimartingale X satisfies (RIE) with respect to a suitable sequence of partitions (which depends on the choice of sample path), and may therefore be canonically lifted to a rough path.However, we also see from the proof of Proposition 4.1 that the resulting rough path lift (s, t) → t s X s,u ⊗ dX u is nothing but the standard Itô-rough path lift of X.Thus, the rough path lift itself does not actually depend on the choice of sequence of partitions.For (almost) every sample path, the sequence of partitions specified in Property (RIE) is merely a choice of partitions along which the value of the Ito integral may be well approximated by left-point Riemann sums in a pathwise fashion.This is analogous to Föllmer integration, in which the quadratic variation of a semimartingale is well-defined as a stochastic process, but to make sense of the quadratic variation in a pathwise sense requires a suitable choice of sequence of partitions, which depends on the sample path being considered.Indeed, given any continuous path, it is possible to find a sequence of partitions along which the quadratic variation of the path is equal to zero (see Freedman [23, (70) Proposition]).Similarly, it is natural to allow the sequence of partitions specified in Property (RIE) to depend on the path being considered, but in practice there typically exists a natural choice for this sequence which results in the desired rough path.4.2.Generalized semimartingales.It is a well observed fact in the empirical literature, see e.g.Lo [35], that price processes appear regularly in financial markets which are not semimartingales.Motivated by this fact, many researchers have proposed and investigated financial models based on fractional Brownian motions; see for instance Jarrow, Protter and Sayit [29], Cheridito [8] or Bender [6].One example of such models are the so-called mixed Black-Scholes models.In these models the (one-dimensional) price process S = (S t ) t∈[0,∞) is usually given by (4.9) for constants s 0 , σ, η > 0 and ν, µ ∈ R, where W = (W t ) t∈[0,∞) is standard Brownian motion and Y = (Y t ) t∈[0,∞) is a fractional Brownian motion with Hurst index H ∈ (0, 1).Multidimensional versions of the mixed Black-Scholes model (4.9) can be obtained by standard modifications.Notice that, while the price process S as defined in (4.9) is not a semimartingale if H = 1/2, the mixed Black-Scholes model (4.9) is still arbitrage-free when restricting the admissible trading strategies to classes of trading strategies which, roughly speaking, exclude continuous rebalancing of the positions in the underlying market, cf.[29,8,6].In particular, the mixed Black-Scholes model (4.9) is free of simple arbitrage opportunities if H > 1/2, as proven in [6, Section 4.1].
In order to demonstrate that Property (RIE) is satisfied by various financial models based on fractional Brownian motion, we consider the following class of generalized semimartingales.On a filtered probability space (Ω, F, (F t ) t∈[0,∞) , P) with a complete right-continuous filtration (F t ) t∈[0,∞) , let Z be a d-dimensional process admitting the decomposition where X is a semimartingale, and Y is a càdlàg adapted process with finite q-variation for some q ∈ [1, 2).Processes Z of this form are sometimes called Young semimartingales, and belong to the class of càdlàg Dirichlet processes in the sense of Föllmer [20].We introduce stopping times (τ n k ), such that, for each n ∈ N, τ n 0 = 0, and Proposition 4.4.Let Z = X + Y be a d-dimensional process such that X is a càdlàg semimartingale, and Y is a càdlàg process with finite q-variation for some q ∈ [1, 2).Then, for any p ∈ (2, 3) such that 1/p + 1/q > 1, almost every sample path of Z is a price path in the sense of Definition 3.1. Proof.
Step 1.It is sufficient to prove that almost all sample paths of Z satisfy Property (RIE) along (Q n Z ([0, T ])) n∈N for an arbitrary T > 0. To this end, for each n ∈ N we set and define X n and Y n in the same way with respect to the partition Q n Z ([0, T ]).Let X = M + A be a decomposition of the semimartingale X such that M is a locally square integrable martingale and A is of bounded variation.By setting B := A + Y , we can write Z = M + B, where B has finite q-variation.We define the Itô integral of Z with respect to itself by where the first integral on the right-hand side is an Itô integral and the second one is interpreted as a Young integral, which exists since Z has finite p-variation and 1/p + 1/q > 1; see e.g.Friz and Zhang [26].Then, since Z n •− − Z •− ∞,[0,T ] ≤ 2 1−n , by the Burkholder-Davis-Gundy inequality and the Borel-Cantelli lemma, we deduce that for almost all ω and for every ε ∈ (0, 1), there exists a constant C = C(ω, ε) such that, for all n ∈ N, for some constant C = C(p, q).Since Z n •− p 0 ,[0,t] ≤ Z p 0 ,[0,T ] holds for every n and every p 0 ∈ (2, p), a routine interpolation argument shows that, for each n ∈ N, Step Towards this aim, we first construct a control function c X such that the above bound holds for X n .Since X n •− − X •− ∞,[0,T ] ≤ 2 1−n , we still have the bound in (4.2).We also choose q 0 ∈ (2, 3) and ε ∈ (0, 1) as in (4.3), and let w X,q 0 and w Y,q be control functions dominating the q 0 -variation and q-variation for X and Y respectively.From the definition of the stopping times τ m k ∈ P n Z and the fact that q < 2 < q 0 , it is easy to check that, for all s < t with s, t ∈ P n Z , the number N of partition points in P n Z between s and t can be bounded by 2 mq 0 w X,q 0 (s, t) + 2 mq w Y,q (s, t) 2 nq 0 w q 0 ,q (s, t), where w q 0 ,q (s, t) := w X,q 0 (s, t) + w Y,q (s, t).By the same argument as in Step 2 of the proof of Proposition 4.1, we deduce that, for all n ∈ N and all s < t with s, t ∈ Q n Z , t s X n u− ⊗ dX u − X s ⊗ X s,t 2 n(q 0 −2) w q 0 ,q (s, t), which, as in Step 3 of the proof of Proposition 4. ≤ C p,q w Y,q (s, t) p/2q w X,p (s, t) 1/2 , Let us recall that typical price paths are of finite p-variation for every p > 2 (see Vovk [49,Theorem 1]) and Vovk's model-free framework allows for setting up a model-free Itô integration, see e.g.Lochowski, Perkowski and Prömel [36].The proof of Lemma 4.7 works verbatim as that of Proposition 4.1, keeping in mind [34,Proposition 3.10] and [36,Corollary 4.9], and is therefore omitted for brevity.4.4.Consistency of rough and stochastic integration.In a probabilistic framework when the underlying process is a semimartingale, one can employ either rough or stochastic Itô integration.In this subsection we briefly demonstrate that, under Property (RIE), these two integrals actually coincide almost surely whenever both are defined.
Let us again fix a filtered probability space (Ω, F, (F t ) t∈[0,∞) , P), and assume that the filtration (F t ) t∈[0,∞) satisfies the usual conditions.Proposition 4.8.Let X = (X t ) t∈[0,∞) be a d-dimensional càdlàg semimartingale.Let Y be an adapted càdlàg process such that, for almost every ω ∈ Ω, the path Y (ω) ∈ A X(ω) is an admissible strategy (in the sense of Definition 3.2).Then the rough and Itô integrals of Y against X coincide almost surely.That is, for almost every ω ∈ Ω, where X(ω) is the canonical rough path lift of X(ω), as defined in Lemma 2.12.
Proof.Fix T > 0. By Proposition 4.1, we know that, for any p ∈ (2, 3), almost every sample path of X is a price path, and satisfies Property (RIE) along a nested sequence of adapted partitions P n X = {0 = t n 0 < t n 1 < • • • < t n Nn = T }, n ∈ N, of the interval [0, T ] with vanishing mesh size.Since Y (ω) ∈ A X(ω) , there exists a càdlàg process Y ′ , such that (Y (ω), Y ′ (ω)) ∈ V q,r X(ω) is a controlled path on [0, T ], for almost every ω ∈ Ω and some suitable numbers q, r.

0
u ) dS u := lim n→∞ [s,t]∈P n Df (S s )S s,t exists and the resulting integral T Df (S u ) dS u satisfies a pathwise Itô formula, see [19, TH ÉOR ÈME].Let us remark that the Föllmer integral T 0

3. 1 .
Price paths and admissible strategies.For a path S: [0, ∞) → R d , we denote by S| [0,T ] the restriction of S to the interval [0, T ].
d and a left-continuous and locally bounded function ψ: Λ d T → R d×d ; see Chiu and Cont [10, Example 4.18].3.4.

Remark 4 . 3 .
Proposition 4.1 holds true even if the semimartingale X takes values in an (infinite dimensional) Hilbert space E, as long as the norm on E ⊗E is admissible in the sense of Lyons, Caruana and Lévy[39, Definition 1.25].In particular, an extension of Proposition 4.1 to so-called piecewise semimartingales, which were introduced in Strong [46, Definition 2.2] as generalized semimartingales with an image dimension evolving randomly in time, appears to be straightforward to implement.As discussed in Karatzas and Kim [32, Remark 6.2 and Section 7], piecewise semimartingales provide a realistic framework to model so-called open markets, which are financial markets with an evolving number of traded assets.
and setP n Z := {τ m k : m ≤ n, k ∈ N ∪ {0}}.As in the previous section, since we insist that the sequence of partitions in Property (RIE) has vanishing mesh size, we also defineQ n Z := P n Z ∪ {σ m k : m ≤ n, k ∈ N ∪ {0}} ∪ {ρ m k,i : m ≤ n, k ∈ N ∪ {0}, i ∈ N},where the times σ m k and ρ m k,i are defined analogously as in Section 4.1.

Lemma 4 . 7 .
Typical price paths are price paths in the sense of Definition 3.1, with any p ∈ (2, 3).

t 0 Y
s (ω) dX s (ω) = t 0 Y s− dX s (ω), for all t ∈ [0, ∞), Allan gratefully acknowledges financial support by the Swiss National Science Foundation via Project 200021 184647.C. Liu gratefully acknowledges support from the Early Postdoc.Mobility Fellowship (No. P2EZP2 188068) of the Swiss National Science Foundation, and from the G. H. Hardy Junior Research Fellowship in Mathematics awarded by New College, Oxford.
Definition 2.8.A path S ∈ D([0, T ]; R d ) is said to satisfy (RIE) with respect to p and (P n ) n∈N , if p, (P n ) n∈N and S together satisfy Property (RIE).
• and vertically differentiable, i.e. for all (t, S) ∈ Λ d T , ∇ x F (t, S) = (∂ i F (t, S)) i=1,...,d , with h↓0 F (t + h, S •∧t ) − F (t, S •∧t )h exists, and DF is continuous at fixed times, t be the second level component of the Itô lift of X.By [34, Proposition 3.4], we know that X possesses finite p 2 -variation; that is, there exists a control function w X, p 2. For every n ∈ N we set ⊗ dX u , for (s, t) ∈ ∆ [0,T ] .We seek a control function c such that sup [26,llows us to conclude the existence of a control function c X such that Next we use the local estimates of Young integration to show that there exists a control c XY such that the above bound holds for XY n and c XY .Indeed, by[26, Proposition 2.4], for all s < t in [0, T ], we have ≤ C p,q Y p/2 q,[s,t]| X n p/2 p,[s,t] ≤ C p,q Y Theorem II.21], we have that(4.12)uniformly in probability, for t ∈ [0, T ].By taking a subsequence if necessary, we can then assume that the (uniform) convergence in (4.12) holds almost surely.On the other hand, by Theorem 2.14, we know that, for almost every ω ∈ Ω,(4.13) uniformly for t ∈ [0, T ].Combining (4.12) and (4.13), we deduce that, almost surely, t 0 Y s dX s = t 0 Y s− dX s for all t ∈ [0, T ].Since T > 0 was arbitrary, the result follows.