Stochastic integration and differential equations for typical paths

The goal of this paper is to define stochastic integrals and to solve stochastic differential equations for typical paths taking values in a possibly infinite dimensional separable Hilbert space without imposing any probabilistic structure. In the spirit of [31, 35] and motivated by the pricing duality result obtained in [4] we introduce an outer measure as a variant of the pathwise minimal superhedging price where agents are allowed to trade not only in $\omega$ but also in $\int\omega\,d\omega:=\omega^2 -\langle \omega \rangle$ and where they are allowed to include beliefs in future paths of the price process expressed by a prediction set. We then call a property to hold true on typical paths if the set of paths where the property fails is null with respect to our outer measure. It turns out that adding the second term $\omega^2 -\langle \omega \rangle$ in the definition of the outer measure enables to directly construct stochastic integrals which are continuous, even for typical paths taking values in an infinite dimensional separable Hilbert space. Moreover, when restricting to continuous paths whose quadratic variation is absolutely continuous with uniformly bounded derivative, a second construction of model-free stochastic integrals for typical paths is presented, which then allows to solve in a model-free way stochastic differential equations for typical paths.


Introduction
Stochastic differential equations (SDEs) are one of the most applied methods to mathematically characterize phenomena in nature. In financial engineering, stochastic differential equations are used to describe risky assets which are tradable in a financial market. This then allows to price financial derivatives and to solve portfolio optimization problems. In biology, SDEs are applied to analyze the dynamics of a population or to investigate the activity of nerve cells. In quantum physics, SDEs have proven to be fruitful to depict kinematics law of quantum fluctuations, to name but a few applications of SDEs. To be able to give sense and to define a solution of a stochastic differential equation, a notion of stochastic integration is necessary. It is well-known that defining a stochastic integral is a highly non-trivial problem and cannot be deduced directly from classical measure-theoretical calculus, as in general, stochastic processes describing the noise of the dynamics do not have finite variation paths. Moreover, in financial engineering, stochastic integrals themselves are an important tool to describe the cumulative gains and losses when trading according to a given investment strategy. The Itô integral and the corresponding Itô calculus has been developed overcoming the obstacle of how to define integrals and differential equations when noise occurs. However, its construction heavily depends on a probabilistic structure and cannot be defined pathwise. More precisely, the construction of the stochastic integral is accomplished by a L 2 (P )-limit procedure, and from the Bichteler-Dellacherie theorem it is known that the only class of good integrators for which the integral is, in a suitable sense, a continuous operator, are semimartingales.
Later, there were several approaches to define stochastic integrals pathwise, without assuming any probabilistic structure. This allows to consider more general paths as integrators, rather than semimartingale paths. Moreover, motivated from mathematical finance, pathwise stochastic calculus can be employed to price financial derivatives without assuming any probabilistic model on the financial market leading to robust prices; see [1,4,5,6,10,15] to name but a few. The first result which provides a construction of a stochastic integral without imposing any probabilistic structure was given in Föllmer [17]. In [11] pathwise stochastic integrals with a directional derivative (whose construction goes back to [16]) of a non-anticipative functional as integrand have been constructed and a change of variable formula for such integrals was obtained. Using this framework, an Itô isometry for such integrals was established in [3], whereas in [33] a pathwise notion for the gain process with respect to corresponding self-financing trading strategies was introduced. Recently in [12] it was shown that one can extend Föllmer's pathwise Itô calculus to paths with arbitrary regularity by employing the concept of p-th variation along a sequence of time partitions. In [22] Föllmer's pathwise stochastic calculus has been extended to obtain prices of American and European options under volatility uncertainty. In [13] Föllmer's pathwise stochastic calculus has been employed to price weighted variance swaps when a finite number of European call and put options for a known price are traded. In [9,20], a pathwise construction of the stochastic integral was proposed for càdlàg integrands which enables to solve the so-called aggregation problem of defining a stochastic integral which coincides with the classical stochastic integral simultaneously for all semimartingale measures. This has been used to price financial derivatives under Knightian uncertainty, see [26,28,34]. A solution for the above aggregation problem, under the continuum hypothesis, was obtained in [27] for general predictable integrands using medial limits. For pathwise construction of stochastic integrals with respect to càdlàg integrators, we refer to [18,21].
Recently, motivated by game-theoretic considerations, Vovk introduced in [35] an outer measure on the space of continuous paths and declared an event to be typical if its complement is null with respect to the defined outer measure. He then showed that typical paths possess a quadratic variation. In other words, it was shown in [35] that paths which do not possess a quadratic variation allow a form of arbitrage. Vovk's approach was employed in [31] to define an outer measure which can be interpreted as the pathwise minimal superhedging price motivated from financial mathematics. Using their outer measure they constructed a model-free stochastic integral which is continuous for typical price paths and connected their typical paths with rough paths by demonstrating that every typical price path possess an Itô rough path. Moreover, [36,37] provide several additional constructions of model-free stochastic integrals for typical paths. In [29,30] Itô calculus with respect to the so-called G-Brownian motion as integrator has been developed, which is motivated from financial mathematics when investigating pricing and portfolio optimization problems under volatility uncertainty. There, by referring to the notion of typical paths, the pathwise integral and the corresponding stochastic calculus is defined for typical paths with respect to the so-called G-expectation.
The goal of this work is to provide a construction of a model-free stochastic integral for typical paths which allows to solve stochastic differential equations pathwise. Motivated by the result of Vovk [35], we restrict our attention to those continuous paths for which a quadratic variation exists. Our setting is similar to the one in [31]. More precisely, we introduce an outer measure which is defined as a variant of the pathwise minimal superhedging price and call a property to hold true on typical paths if the set of paths where the property fails is null with respect to our outer measure. The main difference, compared to the outer measure in [31], is that in our definition hedging is not only allowed in ω representing the price path of the risky security, but also in the second security ω 2 − ω . This roughly means that superhedging strategies both in ω and ω dω are permitted. Moreover, the pathwise superhedging property only needs to hold with respect to a predefined prediction set of paths. Such a superhedging price, which can be seen as a second-order Vovk approach, was introduced in [4] and enabled to provide a pricing duality result when the financial agent is allowed to include beliefs in future paths of the price process expressed by a prediction set Ξ, while eliminating all those which are seen as impossible. This reduces the (robust) superhedging price, which typically leads to too high prices, see [14,25]. We refer to [5,19,24] for related works regarding prediction sets and its relation to pricing of financial derivatives. It turns out that adding the second term in the definition of the outer measure enables to directly define stochastic integrals which are continuous, even for paths taking values in an infinite dimensional separable Hilbert space, see Theorem 2.2. Its proof is based on an elementary, but crucial observation provided in Lemma 3.4 using heavily the second order term in the definition of the outer measure, which is then employed to derive a Burkholder-Davis-Gundy (BDG) type of inequality, see Proposition 3.6. We point out that no condition on the prediction set is imposed so far. To be able to solve stochastic differential equations pathwise, a second construction of model-free stochastic integrals is provided under the condition that the prediction set consists of all paths possessing an absolutely continuous quadratic variation whose derivative is uniformly bounded, see Theorem 2.6. This notion of a model-free stochastic integral allows us to solve stochastic differential equations for typical paths taking values in a possibly infinite dimensional Hilbert space, see Theorem 2.8.
The remainder of this paper is organized as follows. In Section 2, we introduce the setup and state our main results of this paper, whose proofs are then provided in Section 3.

Setup and main results
Let H be a separable Hilbert space with scalar product ·, · H and respective norm h H = h, h Let Ω be the Borel set of all ω ∈ C([0, T ], H) for which the pathwise quadratic variation ω given by exists as a limit in the supremum norm in C([0, T ], R) along the dyadic partition where 0 = τ 0 ≤ · · · ≤ τ n ≤ τ n+1 ≤ · · · ≤ T are F + -stopping times such that for each ω there is n(ω) such that τ n(ω) (ω) = T and the functions f n : Ω → L(H, K) are F τn+measurable. For such a simple integrand F the stochastic integral (F · N ) against any process N : where N t s := N s∧t . Notice that processes in H s (H, R) can be viewed as simple integrands with values in H by identifying L(H, R) with H.
Our results strongly rely on the following modified version of Vovk's outer measure. If not explicitly stated otherwise, all (in-)equalities between functions X : Ω → [−∞, +∞] are understood pointwise on Ω. there are (F n ) in H s (H, R) and (G n ) in H s (R, R) such that λ + (F n · S) t + (G n · S) t ≥ 0 on Ξ for all n and t ∈ [0, T ], and λ + lim inf n (F n · S) T + (G n · S) T ≥ X on Ξ    . Moreover, we say that a property holds for typical paths (on Ξ) if E(1 N ) = 0 for the set N where the property fails.
From now on we fix a prediction set Ξ ⊆ Ω and consider the outer measure E(·) with respect to Ξ. Further, we denote by M(Ξ) the set of martingale measures supported on Ξ, i.e. all Borel probability measures P on Ω such that (S t ) is a P -F-martingale and P (Ξ) = 1.
The function t → ω t is continuous and nondecreasing for all ω ∈ Ω, thus induces a finite measure on [0, T ]. Therefore, we denote the Lebesgue-Stieltjes integral with respect to a function F : [0, T ] × Ω → R such that F (ω) is measurable and T 0 |F u (ω)| d S(ω) u < +∞ for all ω ∈ Ω and set (F · S ) t (ω) := +∞ otherwise. Now, we start with our first result stating that for any prediction set Ξ ⊆ Ω we can define for typical paths stochastic integrals which are continuous. To that end, for any and define the space of integrands : : Ω → C([0, T ], K) exists and satisfies the following weak Burkholder-Davis-Gundy (BDG) type of inequality Moreover, the space H ∞ (H, K) and the stochastic integral are linear (for typical paths) and the latter coincides with the classical stochastic integral under every martingale measure P ∈ M(Ξ).

Remark 2.3.
If K is a general (not finite dimensional) Hilbert space, then the stochastic integral (F · S) exists for every F ∈ H ∞ (H, K). However, it remains open whether it has a continuous modification. We refer to Remark 3.7 for further details.
Remark 2.4. Throughout this paper we work with the real-valued quadratic variation S of the H-valued processes S. However, for K = R one can instead consider the tensor-valued process S defined by for all k, m ∈ N, and where ⊗ denotes the tensor product. Then the processes S and S := S ⊗ S − S take values in the tensor space H ⊗ H. In this setting, E(·) can be defined as before with the difference that the integrands G n are elements of H s (H ⊗H, R). In the weak BDG inequality of Theorem 2.2, the term ( F 2 L(H,R) · S ) has to be replaced by "(F ⊗F · S )", see e.g. [23,Chapter 20] for more details on tensor quadratic variation. Note that in case H = R d it holds H ⊗ H = R d×d , the process S is the symmetric matrix containing the pairwise covariation of all components of S, Replacing S by S might be of interest for the following reason: The prediction set Ξ may include different predictions for the quadratic variation and covariation of different components of S. While this is ignored in , and the integral (F · S) can potentially be defined for a larger space of integrands F .
To be able to not only define stochastic integrals for typical paths, but also solve stochastic differential equations, we need to control the quadratic variation of typical paths.
Assumption 2.5. There exists a constant c ∈ [0, +∞) such that ω is Hölder continuous and ω is If the prediction set Ξ satisfies Assumption 2.5, then it turns out that stochastic integrals can be defined for integrands lying in the set such that the stopping times (τ n ) are deterministic and f n : Ω → L(H, K) is continuous for each n.
Note that H 2 (H, K) is a linear space. More precisely, the following result holds true.
Theorem 2.6. Assume that Ξ satisfies (1) and let F ∈ H 2 (H, K). Then the stochastic integral (F · S) : Ω → C([0, T ], K) exists and satisfies the following weak BDG-type inequality Moreover, it coincides with the classical stochastic integral under every martingale measure P ∈ M(Ξ). In addition, if f : For the rest of this Section, let Assumption 2.5 hold true with respect to the fixed constant c > 0. To be able to define a notion of a solution of a stochastic differential equation for typical paths, let A : [0, T ] × Ω → R be a process such that ω → A t (ω) is continuous for all t and t → |A(ω) t | is absolutely continuous with d|A|(ω)/dt ≤ c for all ω ∈ Ω. Moreover, let µ : [0, T ] × K → L(R, K) and σ : [0, T ] × K → L(H, K) be two functions which satisfy the following.
Then we can state our third main result stating the existence of solutions of stochastic differential equations for typical paths.
Theorem 2.8. Assume that Ξ satisfies (1) and Assumption 2.7 holds. Moreover, let x 0 ∈ K. Then there exists a unique (up to typical paths) X : Ω → C([0, T ], K) such that X ∈ H 2 (K, R) and X solves the SDE For the precise definition of (µ(·, X) · A) and (σ(·, X) · S) see Lemma 3.15 & Remark 3.16 and Lemma 3.13 & Remark 3.14, respectively. Remark 2.9. We point out that with our methods we cannot solve SDEs for typical paths on the space H ∞ (H, K) instead of H 2 (H, K), even when Ξ satisfies (1). The reason is that the corresponding norm · H ∞ (H,K) is too strong to obtain a (similar) But such a relation is the key property necessary to solve SDEs. We refer to Lemma 3.13 for further details.

Properties of E(·).
In this subsection, we analyze properties of the outer measure E which will be crucial to define stochastic integrals and solutions of stochastic differential equation for typical paths. Throughout this subsection, we work with the conventions 0 · (+∞) = (+∞) · 0 = 0 and +∞ − ∞ = −∞ + ∞ = +∞. First, observe that directly from its definition, the outer measure E is sublinear, positive homogenous, and satisfies E(λ) ≤ λ for all λ ∈ [0, +∞). In addition, E satisfies the following properties.
and satisfies for every sequence X n : Ω → [0, +∞], n ∈ N. Furthermore, it fulfills Hölder's inequality Proof. The proof of countable subadditivity is the same as in [35,Lemma 4.1] and [31,Lemma 2.3]. However, due to the different setting and in order to be self contained, we provide a proof. Without loss of generality assume that n E(X n ) < +∞. Fix ε > 0, a sequence (c n ) in (0, +∞) such that n c n = ε, and let λ n := E(X n ) + c n , as well as λ := n λ n . Then, by definition of E(X n ), for every n there are two sequences of simple integrands (F n,m ) m and (G n,m ) m such that λ n + (F n,m · S) t + (G n,m · S) t ≥ 0 for all t ∈ Passing to the limit in k yields E( n X n ) ≤ λ = n E(X n ) + ε. Since ε > 0 was arbitrary, we obtain the first inequality.
Proposition 3.2. Given an arbitrary Banach space (B, · B ), we define X := E( X 2 B ) 1 2 for X : Ω → B. Then the following hold: (i) The functional · is a semi-norm, i.e. it only takes non-negative values, is absolutely homogeneous, and satisfies the triangle inequality. (ii) Every Cauchy sequence X n : Ω → B, n ∈ N, w.r.t. · has a limit X : Ω → B, i.e. X n − X → 0, and there is a subsequence (n k ) such that X n k (ω) → X(ω) for typical paths.
Proof. It is clear that · only takes values in [0, +∞] and is absolutely homogeneous.
To show the triangle inequality, let X, Y : Ω → B. It follows from Proposition 3.1 that To see that (ii) holds true, let (X n ) be a Cauchy sequence and choose a subsequence (X n k ) such that X n k+1 − X n k ≤ 2 −k . By Proposition 3.1 it holds This implies that the set N := { k X n k+1 − X n k B = +∞} satisfies E(1 N ) = 0. As B is complete, for every ω ∈ N c , the sequence (X n k (ω)) k has a limit. Therefore, X := lim k X n k 1 N c is a mapping from Ω to B and Proposition 3.1 yields as k tends to infinity. Since (X n ) is a Cauchy sequence, the triangle inequality shows that X − X n ≤ X − X n k + X n k − X n → 0 as k, n → ∞. such that sup n≤N f n H < ∞ and sup n≤N |g n | < ∞ for some N ∈ N, then the process (F · S) + (G · S) is a continuous martingale (the martingale property follows e.g. by approximating f n P -a.s. by functions with finite range and dominated convergence). Second, let λ ≥ 0, F = n f n 1 (τn,τn+1] in H s (H, R), and G = n g n 1 (τn,τn+1] in H s (R, R) be such that λ + (F · S) t + (G · S) t ≥ 0 on Ξ for all t. Define the stopping times Finally, let λ ≥ 0, (F n ) a sequence in H s (H, R), and (G n ) a sequence in H s (R, R) such that λ + (F n · S) t + (G n · S) t ≥ 0 on Ξ for all t and λ + lim inf n (F n · S) T + (G n · S) T ≥ X on Ξ. Since (F n · S) + (G n · S) is a supermartingale by the previous arguments, it follows from Fatou's lemma that As λ was arbitrary, this shows E P [X] ≤ E(X).

3.2.
Proof of Theorem 2.2. The goal of this subsection is to prove Theorem 2.2. Lemma 3.4, though of elementary nature, is the key observation in what follows. More precisely, it is exactly in this Lemma that we see that adding the second order, i.e. integrals with respect to S in the definition of E, leads to a simple Itô isometry and, with the help of a pathwise inequality, to a BDG-inequality. This, in turn, allows us to directly define stochastic integrals for typical paths which are continuous.
In particular, the following weak Itô isometry holds true. For every F ∈ H s (H, K), it holds that Proof. Fix a simple integrand F = n f n 1 (τn,τn+1] ∈ H s (H, K). By definition, when denoting f * n the adjoint operator of f n , we see that where all sums are finite by definition of simple integrands. Using the inequality For the second claim let λ := sup ω∈Ξ ( F 2 L(H,K) · S ) T (ω). SinceF ∈ H s (H, R), . We first recall an inequality from [2] which turns out to be crucial for the proof of the weak BDG inequality. The connection between pathwise inequalities as in (3) and martingale inequalities are studied in [2,7,8].
If K is finite dimensional (say with orthonormal basis {k 1 , . . . , k d }), one has (F · S) t t with F i = F, k i K . Therefore, one can apply the previous step to every F i to obtain the desired result also when K is finite dimensional.
Proof of Theorem 2.2. Fix F ∈ H ∞ (H, K) and a sequence ( Therefore Proposition 3.2 (applied to the Banach space B := C([0, T ], K)) implies the existence of a limit (F · S) : Ω → B such that The proof that (for typical paths) (F ·S) does not depend on the choice of the sequence (F n ) which converges to F and that the BDG inequality extends to F ∈ H ∞ (H, K) follows from the triangle inequality by standard arguments. Moreover, we derive from the well-known L 2 (P )-limit procedure for the construction of the classical stochastic integral (see, e.g., [32]) and Lemma 3.3 that indeed, the constructed stochastic integral for typical paths coincides with the classical stochastic integral under every martingale measure P ∈ M(Ξ).
Remark 3.7. By arguing like in the proof of Theorem 2.2, but using the weak Itô isometry introduced in Lemma 3.4 instead of the weak BDG-inequality defined in Proposition 3.6, we can define stochastic integrals with respect to integrands F ∈ H ∞ (H, K) without imposing that K is finite dimensional. Moreover, using standard arguments involving the triangle inequality, we see that the weak Itô isometry introduced in Lemma 3.4 for simple integrands in H s (H, K) also holds true for integrands in H ∞ (H, K). However, it remains open if the stochastic integral with respect to integrands in H ∞ (H, K) possess a continuous modification, since the weak Itô isometry, compared to the weak BDG inequality whose proof depends on the fact that K is finite dimensional, is too weak to guarantee that the sequence of simple integrals converge uniformly.

3.3.
Duality Result for Second-oder Vovk's outer measure. The goal of this subsection is to provide a duality result for the outer measure E.   for (t,ω) ∈ [0, T ] ×Ω, where 0 =τ 0 ≤ · · · ≤τ n ≤τ n+1 ≤ · · · ≤ T areF∆ + -stopping times such that for eachω only finitely many stopping times are strictly smaller than T , and f n areF∆ τn+ -measurable functions onΩ with values in L(H, R) and L(R, R), respectively. The functionF is called finite simple, ifτ n = T for all n ≥ N for some N ∈ N. The reason to consider the enlarged spaceΩ is that the duality arguments in the proof of Theorem 3.8 build on topological arguments and the set Ξ in contrast toΞ is not regular enough. The following transfer principle is the reason why duality on Ω can be recovered from duality on the enlarged spaceΩ. Proof. The proof is similar to [4,Lemma 4.6].
We are now ready for the proof of the duality theorem.
This showsĒ(X) ≥ sup nĒn (X n ). To prove the reverse inequality, which then implies that all inequalities in (6) are actually equalities, one may assume without loss of generality that m := sup nĒn (X n ) < +∞. Given some fixed ε > 0, for every n there exist by definition finite simple integrandsḠ n andH n such that m + ε/2 + (Ḡ n ·S) T + (H n ·S) T ≥X n on∆ ∩Ξ n .
The proof of the theorem is now readily completed using the transfer principle derived in Lemma 3.9.
3.4. Proof of Theorem 2.6. Throughout this subsection we assume that the prediction set Ξ satisfies (1), i.e. every ω ∈ Ξ is Hölder continuous with d ω /dt ≤ c. (F · S) t (ω) 2 K defines its lower semicontinuous extension on C([0, T ], H). Moreover, since by assumption Ξ satisfies (1), the conditions of Theorem 3.8 are satisfied. Therefore, we can apply Theorem 3.8 to obtain that E sup Now, under every P ∈ M(Ξ), the process (F · S) K is a (real-valued) submartingale. Therefore, Doob's maximal inequality implies where the last inequality is the weak Itô-Isometry (apply e.g. the same arguments as in the proof of Lemma 3.4 and integrate w.r.t. P ). Finally, by assumption and Lemma 3.3 one has which proves the claim.
Proof of Theorem 2.6. The part of Theorem 2.2 which states that the stochastic integral exists for integrands in H 2 (H, K) follows from Proposition 3.12 using the exact same arguments as in the proof of Theorem 2.2. The second part is shown in Lemma 3.13 below.
In the next Lemma, we extend the inequality obtained in Proposition 3.12 to integrands lying in H 2 (H, K). Moreover, we prove for Lipschitz continuous functions f : [0, T ] × K → L(H, K), that f (·, (F · S)) ∈ H 2 (H, K) whenever F ∈ H 2 (H, K). This is the crucial property allowing to solve stochastic differential equations for typical paths.
In addition, if f : Proof. The first part follows from Proposition 3.12 and the triangle inequality.
It remains to prove that f (·, (F · S)) ∈ H 2 (H, K). Assume first that F ∈ H s,c (H, K) and define for every n. Since ω → (F · S) iT /n (ω) is continuous for ever i, one has H n ∈ H s,c (H, K). Moreover, with π n (t) := max{iT /n : i ∈ N such that iT /n ≤ t} and L f being the Lipschitz constant of f , one has Now Theorem 3.8, the weak Itô-Isometry (argue as in Proposition 3.12), and weak duality (Lemma 3.3) imply for every t that   K is dominated by c T 0 E F s 2 L(H,K) ds < +∞ and converges pointwise to 0 when n goes to infinity since then π n (t) → t. Therefore, dominated convergence implies f (·, (F ·S))−H n H 2 (H,K) → 0, which shows that f (·, (F · S)) ∈ H 2 (H, K).
The general case follows by approximating F ∈ H 2 (H, K) by F n ∈ H s,c (H, K), and using the inequality f (·, (F n · S)) − f (·, (F · S)) 2 , where the last inequality is ensured by the first part.
Remark 3.14. Let F ∈ H 2 (H, K). Similar to the proof of Lemma 3.13 one can verify (1) f (·, F ) ∈ H 2 (H, K) for every function f : [0, T ]×K → L(H, K) which is Lipschitz continuous, and (2) (F · S) can be identified with an element in H 2 (K, R), by considering i(F · S) for the isometric isomorphism i : K → L(K, R) given by the Riesz representation theorem. In Subsection 3.5 we will frequently use this identification.
Proof. For F, G ∈ H s,c (R, K), Theorem 3.8 and Hölder's inequality implies The rest follows by the same arguments as in the proof of Theorem 2.2. The second part can be proved as in Lemma 3.13.
In line with Remark 3.14 the following holds.
As for the uniqueness, suppose there exist two solutions X, Y : Ω → C([0, T ], K) in H 2 (K, R) of the SDE. Similar to the proof of Lemma 3. with C := (2c 2 T + 8c)L 2 . Iterating this estimate yields g(t) ≤ g(T ) (Ct) k k! for all k ∈ N and t ∈ [0, T ], so that X − Y = g(T ) 1 2 = 0. This shows that X and Y coincide for typical paths.