Stochastic calculus without probability: Pathwise integration and functional calculus for functionals of paths with arbitrary Hölder regularity

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1 Stochastic calculus without probability: Pathwise integration and functional calculus for functionals of paths with arbitrary Hölder regularity Rama Cont Joint work with: Anna ANANOVA (Imperial) Nicolas PERKOWSKI (Humboldt Univ, Berlin), Yi LU (Paris VI). Oxford, 218.

2 Model-free approaches in mathematical finance The classical approach to continuous-time finance theory starts by specifying a probability measure P on a space of scenarios (price trajectories). Gain of a strategy is defined as an Ito integral, constructed as a limit in probability of Riemann sums under P. Stochastic calculus plays a key role in derivation of key results in continuous-time finance. A lot of results rely on probabilistic assumptions such as the Markov property: pricing PDEs, sensitivity analysis, HJB equations for optimal investment... However, P is not known in pratice Knightian uncertainty. Also: many results are almost-sure statements about gains, losses or strategies. What is the role of P? How far can one go without knowing it?

3 Pathwise approaches to stochastic calculus, N Perkowski (218) Pathwise integration and change of variable formulas for paths with arbitrary regularity, ARXIV. A Ananova, (217) Pathwise integration with respect to paths of finite quadratic variation, Journal de Mathématiques Pures et appliquées. Functional Ito Calculus and Functional Kolmogorov Equations, (Lectures Notes of the Barcelona Summer School on Stochastic Analysis, July 212), Springer. and D Fournié (21) Change of variable formulas for non-anticipative functional on path space, Journal of Functional Analysis, 259, H Föllmer (1981) Calcul d Ito sans probabilités, Séminaire de Probabilités.

4 Notations D([, T ], R d ) space of cadlag functions (right continuous with left limits). C α ([, T ], R d ) α Holder functions For a path ω D([, T ], R d ), denote by ω(t) R d the value of ω at t ω t = ω(t.): path stopped at t ω t = ω 1 [,t[ + ω(t ) 1 [t,t ] For a process X we denote X (t) its value and X t = X (t.) its path stopped at t.

5 Young integration W p ([, T ], R d ): space of path with finite p-variation Let p, q > with 1 p + 1 q > 1. Then for any H W p([, T ], R d ), S W q ([, T ], R d ) and any sequence π n = (t n = <.. < tn j <... < t n m(n) = T ). [, T ] of partitions of [, T ] with π n, the Riemann sums t n i π n H(t n i ).(S(t n i+1) S(t i n)) converge as n to a limit T H.dS which does not depend on the choice of the partition (π, α) and satisfies T H.dS C p,q ( H + H p var ) S q var

6 Young integration Young integration allows to define HdS when H W p ([, T ], R d ), S W q ([, T ], R d ) with 1 p + 1 q > 1. This fails for typical Brownian or semimartingale paths for which p = q = 2. Rough path theory has provided an integration concepts which extends to such situations by adding an extra structure to the path S: a (non-unique) rough path associated with S. This construction, however, is neither canonical nor pathwise: one needs to first define multiple integrals of S wrt itself, which is usually done probabilistically. Moreover, its physical meaning is not always obvious. In this work we exploit a different, a canonical and strictly pathwise approach to integration against such irregular paths, purely based on the path S itself.

7 Calcul d Itô Sans Probabilités (Föllmer 1981) Let X C ([, T ], R d ) and f C 2 (R d, R). The main idea in the proof of the Ito formula is to consider a sequence of partitions π n = ( = t n < tn 1.. < tn N(π n) = T ) of [, T ] with step size decreasing to zero and expand increments of f (X (t)) along the partition using a 2nd order Taylor expansion: f (X (t)) f (X ()) = π n f (X (t n i+1)) f (X (t n i )) = π n f (X (t n i )).(X (t n i+1) X (t n i )) t (X (t n i+1) X (t n i )) 2 f (X (t n i )).(X (t n i+1) X (t n i )) + r(x (t n i+1), X (t n i ))

8 Summing over π n we get f (X (t)) f (X ()) = S 1 (π n, f ) + S 2 (π n, f ) + R(π n, f ) By uniform continuity of r(x, y) = f (y) f (x) f (x).(y x).5 t (y x) 2 f (x)(y x), r(x, y) ϕ( x y ) x y 2 with ϕ(u) u R(π n, f ) = ɛ n π n X (t n i+1 ) X (tn i ) 2. So both this term and the quadratic Riemann sum S 2 (π n, f ) = 1 t (X (t n 2 i+1) X (ti n )) 2 f (X (ti n )).(X (ti+1) n X (ti n )) π n are controlled by the convergence of (weighted) sums of squared increments of X along π n.

9 Quadratic Riemann sums For d=1: given a path of X, pointwise convergence of quadratic Riemann sums S 2 (π n, f ) = 1 2 π n 2 f (X (t n i )).(X (t n i+1) X (t n i )) 2 along the path for every f C 2 (R d, R) is exactly equivalent to the weak convergence of the sequence of discrete measures µ n = t j π n (X (t n j+1) X (t n j )) 2 δ tj where δ t denotes a point mass at t. This is a joint property of the path X and (π n ). This motivated Föllmer s (1981) definition of pathwise quadratic variation along a sequence of partitions.

10 Quadratic variation along a partition sequence Definition (Follmer 1981) Let π n = ( = t n < tn 1.. < tn N(π n) = T ) be a sequence of partitions of [, T ] with step π n decreasing to zero. A càdlàg function x D([, T ], R) is said to have finite quadratic variation along the sequence of partitions π = (π n ) n 1 if the weak limit µ := lim (x(tj+1) n x(tj n )) 2 δ tj n t j π n exists and [x] c π defined by [x] c π(t) = µ([, t]) <s t x(s) 2 is a continuous and increasing function. We denote [x] π = µ([, t]). We denote Q π ([, T ], R) the set of functions with this property

11 Characterization in continuous case Proposition (R.C. & P. Das (217)) Let x C ([, T ], R) and define [x] πn (t) = The following properties are equivalent: t j π n (x(t n j+1 t) x(t n j t)) 2 1 x has finite quadratic variation along the sequence of partitions (π n ) n 1. 2 The sequence [x] πn converges uniformly on [, T ] to a continuous function [x] π. 3 The sequence [x] πn converges pointwise on [, T ] to a continuous function [x] π.

12 Multidimensional paths Definition (Paths of finite quadratic variation) x Q π ([, T ], d ) if, for all 1 i, j d, x i, x i + x j in Q π ([, T ], R). Then [x] π : [, T ] S + d defined by ([x] π) i,j (t) = 1 ( ) [x i + x j ] π(t) [x i ] π(t) [x j ] π(t) 2 = [x i, x j ] c π(t) + x i (s) x j (s), i, j = 1,..., d <s t is an increasing function with values positive symmetric d d matrices.

13 Quadratic Riemann sums Denote < A, B >= i,j A ijb ij = tr( t A.B) for d d matrices A, B Follmer s definition guarantees the convergence of quadratic Riemann sums, which is the condition for deriving the Ito formula: Let ω Q π ([, T ], R d ) be a path with finite quadratic variation along (π n, n 1). Then f C b ([, T ], Rd d ), π n < f (t n i ), t (ω(t n i+1) ω(t n i ))(ω(t n i+1) ω(t n i )) > n T uniformly on [, T ]. < f, d[ω] >.

14 Föllmer s pathwise Ito formula Proposition (Föllmer, 1981) f C 2 (R d, R), ω Q π ([, T ], R d ), the non-anticipative Riemann sums along π π n converge pointwise and f (ω(t)) f (ω()) = f (ω(t n i )).(ω(t n i+1) ω(t n i )) n T + s t f (ω).d π ω f (ω(t)).d π ω < 2 f (ω), d[ω] c π > f (ω(s)) f (ω(s )) f (ω(s )). ω(s)

15 Pathwise construction of stochastic integrals Föllmer s result allows to define integrals of the type f (ω(t))dω f C 2 (R d, R) as a (pathwise) limit of Riemann sums, for any path ω with finite quadratic variation along (π n ). Admissible integrands are gradients of C 2 functions.

16 Dependence on the partition Consider now two sequences of partitions π, τ and a continuous path ω Q π ([, T ], R d ) Q τ ([, T ], R d ). Since f C 2 (R d ), f (ω(t)) f (ω()) = = f (ω).d π ω f (ω).d τ ω < 2 f (ω), d[ω] π > < 2 f (ω), d[ω] τ > (1) the pathwise integrals are equal if and only if [ω] π = [ω] τ. But the pathwise quadratic variation does depend on the sequence of partition...

17 Dependence on the sequence of partitions Proposition (Friedman) Let ω C ([, T ], R d ). There exists a sequence of partitions (π n ) such that [ω] π =. Proof: We construct recursively partitions π n such that π n 1 n and π n ω(t n k+1) ω(t n k ) 2 1 n. Assume we have constructed π n with this property. Adding to π n the points k/(n + 1), k = 1..n we obtain a partition σ n = (s n i, i =..M n) with σ n 1/(n + 1).

18 For i =..(M n 1), we further refine [si n, sn i+1 ] as follows. Let J(i) be an integer with J(i) (n + 1)M n ω(si+1) n ω(si n ) 2, τi,k+1 n = inf{t τi,k, n ω(t) = ω(si n ) + k ( ω(si+1 n ) ω(sn i )) }. J(i) Then points (τ n i,k, k = 1..J(i)) defines a partition of [sn i, sn i+1 ] with τ n i,k+1 τ n i,k 1 n + 1 and ω(τ n i,k+1) ω(τ n i,k) = ω(sn i+1 ) ω(sn i ) J(i) so J(i) k=1 ω(τi,k+1) n ω(τi,k) n 2 J(i) ω(sn i+1 ) ω(sn i ) 2 1 J(i) 2 =. (n + 1)M n Sorting (τi,k n, i =..M n, k = 1..J(i)) gives π n+1 = (t n+1 ) such that π n+1 1 n + 1, π n+1 ω(t n i+1) ω(t n i ) 2 1 n + 1. j

19 Definition (Well-balanced sequence of partitions) Let π n = inf i=..n(πn) 1 t n i+1 tn i. The sequence of partitions (π n ) n 1 well-balanced if c >, n 1, π n π n c. (2) Theorem (R.C. & P. Das, 217) Let α >, f C α ([, T ], R d ) and τ = (τ n ) n 1 and σ = (σ n ) n 1 two well-balanced partition sequences such that f Q τ ([, T ], R d ) Q σ ([, T ], R d ) and [f ] σ >, [f ] τ >. Then: t [, T ], [f ] σ (t) = [f ] τ (t)

20 p-th variation along a sequence of partitions Define the oscillation of S C([, T ], R) along π n as osc(s, π n ) := max max S(s) S(r). [t j,t j+1 ] π n r,s [t j,t j+1 ] Definition (p-th variation along a sequence of partitions) Let p >. A continuous path S C([, T ], R) is said to have a p-th variation along a sequence of partitions π = (π n ) n 1 if osc(s, π n ) and the sequence of measures µ n = [t j,t j+1 ] π n δ( t j ) S(t j+1 ) S(t j ) p converges weakly to a measure µ without atoms. We write S V p (π) and call [S] p (t) := µ([, t]) for t [, T ]the p-th variation of S.

21 p-th variation along a sequence of partitions Functions in V p (π) do not necessarily have finite p-variation. Example: If B is a fractional Brownian motion with Hurst index H (, 1) and π n = {kt /n : k N } [, T ], then B V 1/H (π) and [B] 1/H (t) = te[ B 1 1/H ], while B p var = almost surely for p = 1/H. Lemma Let S C([, T ], R). S V p (π) if and only if there exists a continuous increasing function [S] p such that t [, T ], [t j,t j+1 ] π n: t j t If this property holds then the convergence is uniform. S(t j+1 ) S(t j ) p n [S] p (t).

22 Rough Change of variable formula Theorem ( R.C- Perkowski (218)) Let p 2N be even, S V p (π). Then for every f C p (R, R) f (S(t)) f (S()) = < T p 1 f (S(s)), d p 1 S > + 1 p! f (p) (S(s))d[S] p (s), where the integral is defined as a (pointwise) limit of compensated Riemann sums: lim n p 1 π n k=1 < T p 1 f (S(s)), d p 1 S(s) >:= f (k) (S(t j )) (S(t j+1 t) S(t j t)) k k!

23 Pathwise integral The pathwise integral < T p 1 f (S(s)), d p 1 S(s) >:= lim n R p 1 (f, S, π n ) is a pointwise limit of compensated Riemann sums R p 1 (f, S, π n ) = π n p 1 k=1 f (k) (S(t j )) (S(t j+1 t) S(t j t)) k k! It should be really seen as an integral of the (p 1) jet of f T p 1 f (x) = (f (k) (x), k =, 1,..., p 1) with respect to a differential structure of order p 1 constructed along S V p (π) using the powers of increments up to order p 1.

24 Pathwise local time of order p Definition (Local time of order p) Let p N be an even integer and let q [1, ]. A continuous path S C([, T ], R) has an L q -local time of order p 1 along a sequence of partitions π = (π n ) n 1 if osc(s, π n ) and L πn,p 1 t ( ) = 1 S(tj t),s tj+1 t ( ) S(t j+1 t) p 1 t j π converges weakly in L q (R) to a weakly continuous map L: [, T ] L q (R) which we call the order p local time of S. We denote L q p(π) the set of continuous paths S with this property. Intuitively, the limit L t (x) then measures the rate at which the path S accumulates p-th order variation near x.

25 Theorem (Higher order Tanaka-Wuermli formula) Let p 2N be an even integer, q [1, ] with conjugate exponent q = q/(q 1). Let f C p 1 (R, R) and assume that f (p 1) is weakly differentiable with derivative in L q (R). Then for any S L q p(π) f (S(s))dS(s) := lim p 1 n [t j,t j+1 ] π n k=1 f (k) (S(t j )) (S(t j+1 t) S(t j t)) k k! exists and the following change of variable formula holds: f (S(t)) f (S()) = < T p 1f (S(s)), d p 1 1 S(s) > + f (p) (x)l t(x)dx. (p 1)! R

26 Multidimensional paths: Symmetric tensors A symmetric p-tensor T on R d is a p-tensor invariant under any permutation σ S p of its arguments: for (v 1, v 2,..., v p ) (R d ) p σt (v 1,..., v p ) := T (v σ1, v σ2,..., v σp ) = T (v 1, v 2,..., v p ) The space Sym p (R d ) of symmetric tensors of order p on R d is naturally isomorphic to the dual of the space H p [X 1,..., X d ] of homogeneous polynomials of degree p on R d. p S p (R d ) = Sym k (R d ). k= For any tensor T T p (R d ) we define the symmetric part Sym(T ) := 1 σt Sym p! p (R d ) σ S k where S p of {1,..., k} is the group of permutations of {1, 2,..., p}

27 Extension to multidimensional functions Consider now a continuous R d -valued path S C([, T ], R d ) and a sequence of partitions π n = {t n,..., tn N(π n) } with tn = <... < tn k <... < tn N(π n) = T. Then µ n = (S(t j+1 ) S(t j ))... (S(t j+1 ) S(t j )) δ( t j ) }{{} π n p times defines a tensor-valued measure on [, T ] with values in Sym p (R d ). This space of measures is in duality with the space C([, T ], H p [X 1,..., X d ]) of continuous functions taking values in homogeneous polynomials of degree p i.e. homogeneous polynomials of degree p with continuous time-dependent coefficients. This motivates the following definition:

28 Definition (p-th variation of a multidimensional function) Let p 2N be an (even) integer, and S C([, T ], R d ) a continuous path and π = (π n ) n 1 a sequence of partitions of [, T ]. S C([, T ], R d ) is said to have a p-th variation along π = (π n ) n 1 if osc(s, π n ) and the sequence of tensor-valued measures µ n S = π n (S(t j+1 ) S(t j )) p δ( t j ) converges to a Sym p (R d )-valued measure µ S without atoms in the following sense: f C([, T ], S p (R d )), < f, µ n >= π n < f (t j ), (S(t j+1 ) S(t j )) p > n < f, µ S >. (3) We write S V p (π) and call [S] p (t) := µ([, t]) the p-th variation of S.

29 Theorem (Rough change of variable formula: multi-dim case) Let p 2N be an even integer, let (π n ) be a sequence of partitions of [, T ] and S V p (π) C([, T ], R d ). Then for every f C p (R, R) the limit of compensated Riemann sums < T p 1f (S), d p 1 S >= lim n π n p 1 k=1 exists for every t [, T ] and satisfies f (S(t)) f (S()) = + 1 p! 1 k! < k f (S(t j )), (S(t j+1 t) S(t j t)) k > < T p 1 f (S(u)), d p 1 S(u) > < p f (S(t))), d[s] p (u) >.

30 Extension to non-anticipative Functionals Denote ω t = ω(t.) the past i.e. the path stopped at t. Definition (Non-anticipative Functionals) A causal, or non-anticipative functional is a functional F : [, T ] D([, T ], R d ) R whose value only depends on the past: ω Ω, t [, T ], F (t, ω) = F (t, ω t ). (4) Causal functional= map on the space Λ d T of stopped paths, defined as the quotient space: Λ d T := ( [, T ] D([, T ], R d ) ) / where (t, x) (t, x ) t = t, x t = x t. Λ d T is equipped with a metric d ((t, x), (t, x )) = C, (Λ d T ) = continuous maps (Λd T, d ) R. sup x(u t) x (u t ) + t t. u [,T ]

31 Definition (Horizontal and vertical derivatives) A non-anticipative functional F is said to be: horizontally differentiable at (t, ω) Λ d T DF (t, ω) := lim h + if the finite limit exists F (t + h, ω t ) F (t, ω t ). h vertically differentiable at (t, ω) Λ d T if the map R d R, e F (t, ω(t.) + e1 [t,t ] ) is differentiable at ; its gradient at is denoted by ω F (t, ω). Note that DF (t, ω) is not the partial derivative in t: F (t + h, ω) F (t, ω) DF (t, ω) t F (t, ω) = lim. h h

32 Smooth functionals Definition (C 1,p b (Λd T ) functionals) We denote by C 1,p b (Λd T ) the set of non-anticipative functionals F C, l (Λ d T ), such that F is horizontally differentiable with DF continuous at fixed times, F is p times vertically differentiable with j ωf C, l (Λ d T ) for j = 1..p DF, j ωf B(Λ d T ).

33 Examples of smooth functionals Example (Cylindrical functionals) For g C (R d n ), h C k (R d ) with h() =. Then is in C 1,k b F (t, ω) = h (ω(t) ω(t n )) 1 t tn g(ω(t 1 ), ω(t 2 )..., ω(t n )) and D t F (ω) =, and j = 1..k, j ωf (t, ω) = h (j) (ω(t) ω(t n )) 1 t tn g (ω(t 1 ), ω(t 2 )..., ω(t n )) S(Λ T, π n ) := space of simple predictable cylindrical functionals piecewise constant along π n, S(Λ T, π) := n>1 S(Λ T, π n )x

34 Examples of smooth functionals Example (Integral functionals) For g C (R d ), Y (t) = g(x (u))ρ(u)du = F (t, X t) where F C 1, b, with: F (t, ω) = g(ω(u))ρ(u)du (5) D t F (ω) = g(ω(t))ρ(t) j ωf (t, ω) = (6)

35 Conditional expectations as smooth functionals Let σ C, (W T ) be such that ( X (t) = X () exp σ(u)dw (u) 1 ) σ 2 (u).du 2 ( ) is a martingale, i.e. E(X (T )) = 1 and denote by Q σ the law of (*). Proposition (Cont & Riga 215) Let h : (D([, T ], R),. ) R be Q σ -integrable and Lipschitz. Assume that for (t, ω) W T, the map g h (.; t, ω) : e R d g h (e) = h ( ω + e1 [t,t ] ) ), (7) is twice differentiable at, with derivatives bounded uniformly in (t, ω) W T in a neighborhood of. Then, there exists F C,2 b (W T ) such that F (t, X t ) = E Qσ [H Ft X ] Q σ a.s.

36 Weak Euler schemes as smooth functionals Let σ : (Λ T, d ) R d d be a Lipschitz map. Then nx (t j+1, ω) = n X (t j, ω) + σ(t j, n X tj (ω)) (ω(t j+1 ) ω(t j )). (8) defines a non-anticipative functional n X which approximates X (t) = X () + σ(u, X u )dw (u) (9) For a Lipschitz functional g : (D([, T ], R d ),. ) R, consider the weak Euler approximation of E [ ] g(x T ) Ft W : F n (t, ω) = E [ g ( n X T (W T )) F W t ] (ω). (1) (- Yi Lu, SPA 216): F n C 1, b (W T ).

37 Functional change of variable formula: p = 2 Theorem (R.C.- Fournié,21) Let ω V 2 (π) and ω n := m(n) 1 i= ω(ti+1 n )1 [ti n,ti+1 n ) + ω(t )1 {T }. Then for any F C 1,2 b (Λd T ), the pointwise limit of Riemann sums T ω F (t, ω)d π ω := lim n m(n) 1 i= ω F (t n i, ωn )(ω(t n i+1 ) ω(tn i )) T exists and F (T, ω) = F (, ω) + ω F (t, ω) d π ω + T T DF (t, ω)dt tr ( 2 ωf (t, ω)d[ω] c π(t) ).

38 These result allows to construct. ωf as a pointwise limit of non-anticipative Riemann sums : T m(n) 1 ω F (t, ω t ) d π ω = lim ω F (ti n, ω n )(ω(ti+1) n ω(ti n )) n i= Remark F R(Λ d T ), ω C 1 2 ([, T ], R d ) ω F (t, ω) C 1 2 ([, T ], R d ). The pathwise integral is a strict extension of the Young integral.

39 Denote C 1 2 ([, T ], R d ) = ν<1/2 C ν ([, T ], R d ) Theorem (Pathwise Isometry formula, A. Ananova, R. C. 216) If F C 1,2 (Λ T ) is Lipschitz-continuous and ω F C 1,1 b (Λd T ) then for any ω Q π ([, T ], R) C 1/2 ([, T ], R d ) and any sequence of partitions π = (π n ) n 1 satisfying osc(f (., ω), π n ) n + we have [. π ω F (s, ω).d ω] π (t) = [F (t, ω)] π (t) = t ω F (s, ω). ω F (s, ω), d[ω] π (s). (Isometry)

40 Theorem Let p be an even integer, let F C 1,p b (Λ T ), and let S V p (π) for a sequence of partitions (π n ) with π n. Then the limit < T p 1F (s, S s ), d p 1 S(s) >= lim p 1 n [t j,t j+1 ] π n k=1 1 k! k ωf (t j, S n t j )(S(t j+1 t) S(t j t)) k, and F (t, S t ) = F (, S ) + DF (s, S s )ds + < T p 1F (s, S s ), d p S > + 1 t p! p ωf (s, S s )d[s] p (s)

41 A higher order isometry formula The following result extends the pathwise isometry formula (Ananova-C. 217) for the pathwise integral to paths with lower regularity: Theorem ( Isometry formula) Let p N be an even integer, α > ((1 + 4 p )1/2 1)/2, (π n ) a sequence of partitions with mesh size going to zero, and S V p (π) C α ([, T ], R). Let F C 1,p b (Λ T ) Lip(Λ T, d ) be such that ω F C 1,p 1 b (Λ T ). Then F (, S) V p (π) and [F (, S)] p (t) = ω F (s, S s ) p d[s] p (s).

42 Higher-order Dirichlet decomposition The following result gives a Doob-Meyer/ Dirichlet decomposition for functionals of processes with arbitrary regularity: Theorem Let p N be an even integer, let α > ((1 + 4 p )1/2 1)/2, and let S V p (π) C α ([, T ], R) be a path with strictly increasing p-th variation [S] p along (π n ). Then any X C 1,p b (S) admits a unique decomposition X = X () + A + < φ, d p 1 S > where φ is a (p 1)-form and [A] p =.

43 Relation with rough path integration Define a control function as a continuous map c : T R + such that c(t, t) = and c(s, u) + c(u, t) c(s, t). Definition Reduced rough path of order p Let p 1. A reduced rough path of finite p-variation is a map X = (1, X 1,..., X p ): T S p (R d ), such that p X k s,t p/k c(s, t), (s, t) T ; k=1 for some control function c and the reduced Chen relation holds X s,t = Sym(X s,u X u,t ), (s, u), (u, t) T.

44 The canonical reduced rough path for S V p (π) Lemma Let S C([, T ], R d ) and let (π n ) be the dyadic Lebesgue partition generated by S. Let p 1 and assume that S V p (π). Then for any q > p with q = p we obtain a reduced rough path of finite q-variation by setting X s,t := 1, X k s,t := 1 k! (S(t) S(s)) k, k = 1,..., p 1, X p s,t := 1 p! (S(t) S(s)) p 1 p! ([S]p (t) [S] p (s)).

45 Proposition Let p 1, let X be a reduced rough path of finite p-variation and let Y D p /p X ([, T ]). Then the rough path integral I X (Y )(t) = Y (s), dx(s) = lim π Π([,t]) π p [t j,t j+1 ] π k=1 Y k (t j ), X k t j,t j+1, defines a function in C([, T ], R), and it is the unique function with I X (Y )() = for which there exists a control function c with s p Y (r), dx(r) Y k (s), X k s,t c(s, t) p +1 p, (s, t) T. k=1

46 Pathwise integral as canonical rough integral Proposition (R.C- N. Perkowski 218) Let p 2N be an even integer, S V p (π) and X the canonical reduced rough path of order p associated to S, defined above. Then f (S(s)), dx(s) = } {{ } Rough integral T p 1 f (S), d p 1 S, } {{ } Pathwise integral where the right hand side is the pathwise integral defined as a limit of compensated Riemann sums.