Rough paths methods 4: Application to fbm

Transcription

1 Rough paths methods 4: Application to fbm Samy Tindel Purdue University University of Aarhus 2016 Samy T. (Purdue) Rough Paths 4 Aarhus / 67

2 Outline 1 Main result 2 Construction of the Levy area: heuristics 3 Preliminaries on Malliavin calculus 4 Levy area by Malliavin calculus methods 5 Algebraic and analytic properties of the Levy area 6 Levy area by 2d-var methods 7 Some projects Samy T. (Purdue) Rough Paths 4 Aarhus / 67

3 Outline 1 Main result 2 Construction of the Levy area: heuristics 3 Preliminaries on Malliavin calculus 4 Levy area by Malliavin calculus methods 5 Algebraic and analytic properties of the Levy area 6 Levy area by 2d-var methods 7 Some projects Samy T. (Purdue) Rough Paths 4 Aarhus / 67

4 Objective Summary: We have obtained the following picture dx, dxdx Vj (x) dx j Smooth V 0,..., V d Rough paths theory dy = V j (y)dx j Remaining question: How to define dxdx when x is a fbm with H 1/2? Samy T. (Purdue) Rough Paths 4 Aarhus / 67

5 Levy area of fbm Proposition 1. Let B be a d-dimensional fbm, with H > 1/3, and 1/3 < γ < H. Almost surely, the paths of B: 1 Belong to C γ 1 2 Admit a Levy area B 2 C 2γ 2 such that δb 2 = δb δb, i.e. B 2,ij sut = δb i su δb j ut Conclusion: The abstract rough paths theory applies to fbm with H > 1/3 Proof of item 1: Already seen (Kolmogorov criterion) Samy T. (Purdue) Rough Paths 4 Aarhus / 67

6 Geometric and weakly geometric Levy area Remark: The stack B 2 as defined in Proposition 1 is called a weakly geometric second order rough path above X allows a reasonable differential calculus When there exists a family B ε such that B ε is smooth B 2,ε is the iterated Riemann integral of B ε B 2 = lim ε 0 B 2,ε then one has a so-called geometric rough path above B easier physical interpretation Samy T. (Purdue) Rough Paths 4 Aarhus / 67

7 Levy area construction for fbm: history Situation 1: H > 1/4 3 possible geometric rough paths constructions for B. Malliavin calculus tools (Ferreiro-Utzet) Regularization or linearization of the fbm path (Coutin-Qian) Regularization and covariance computations (Friz et al) Situation 2: d = 1 Then one can take B 2 st = (Bt Bs)2 2 Situation 3: H 1/4, d > 1 The constructions by approximation diverge Existence result by dyadic approximation (Lyons-Victoir) Recent advances (Unterberger, Nualart-T) for weakly geometric Levy area construction Samy T. (Purdue) Rough Paths 4 Aarhus / 67

8 Outline 1 Main result 2 Construction of the Levy area: heuristics 3 Preliminaries on Malliavin calculus 4 Levy area by Malliavin calculus methods 5 Algebraic and analytic properties of the Levy area 6 Levy area by 2d-var methods 7 Some projects Samy T. (Purdue) Rough Paths 4 Aarhus / 67

9 fbm kernel Recall: B is a d-dimensional fbm, with Bt i = K t (u) dwu, i t 0, where W is a d-dimensional Wiener process and R K t (u) (t u) H 1 2 1{0<u<t} t K t (u) (t u) H 3 2 1{0<u<t}. Samy T. (Purdue) Rough Paths 4 Aarhus / 67

10 Heuristics: fbm differential Formal differential: we have B j v = v 0 K v(u) dw j u and thus formally for H > 1/2 Ḃ j v = Formal definition of the area: Consider B i. Then formally 1 0 B i v db j v = = v 0 v K v (u) dw j u 1 ( v ) Bv i v K v (u) dwu j dv ( 1 ) v K v (u) Bv i dv dwu j 0 u This works for H > 1/2 since H 3/2 > 1. Samy T. (Purdue) Rough Paths 4 Aarhus / 67

11 Heuristics: fbm differential for H < 1/2 Formal definition of the area for H < 1/2: Use the regularity of B i and write 1 B i v db j v = 0 1 ( 1 = 0 u 1 ( 1 0 u v K v (u) δb i uv dv ) v K v (u) Bv i dv ) dw j u dw j u K 1 (u) B i u dw j u. Control of singularity: v K v (u) δb i uv (v u) H 3/2+H Definition works for 2H 3/2 > 1, i.e. H > 1/4! Hypothesis: 1 0 Bi v db j v well defined as stochastic integral Samy T. (Purdue) Rough Paths 4 Aarhus / 67

12 Outline 1 Main result 2 Construction of the Levy area: heuristics 3 Preliminaries on Malliavin calculus 4 Levy area by Malliavin calculus methods 5 Algebraic and analytic properties of the Levy area 6 Levy area by 2d-var methods 7 Some projects Samy T. (Purdue) Rough Paths 4 Aarhus / 67

13 Space H Notation: Let E be the set of step-functions f : R R B be a 1-d fbm Recall: R H (s, t) = E[B t B s ] = 1 2 ( s 2H + t 2H t s 2H ) Space H: Closure of E with respect to the inner product 1[s1,s 2 ], 1 [t1,t 2 ] H = E [δb s 1 s 2 δb t1 t 2 ] (1) = R H (s 2, t 2 ) R H (s 1, t 2 ) R H (s 2, t 1 ) + R H (s 1, t 1 ) [s1,s 2 ] [t 1,t 2 ]R H Samy T. (Purdue) Rough Paths 4 Aarhus / 67

14 Isonormal process First chaos of B: We set H 1 (B) closure in L 2 (Ω) of linear combinations of δb st Fundamental isometry: The mapping 1 [t,t ] B t B t can be extended to an isometry between H and H 1 (B) We denote this isometry by ϕ B(ϕ). Isonormal process: B can be interpreted as A centered Gaussian family {B(ϕ); ϕ H} Covariance function given by E[B(ϕ 1 )B(ϕ 2 )] = ϕ 1, ϕ 2 H Samy T. (Purdue) Rough Paths 4 Aarhus / 67

15 Underlying Wiener process on compact intervals Volterra type representation for B: with B t = W Wiener process K t (u) defined by R K t (u) dw u, t 0 [ ( ) 1 u 2 K t (u) = c H H (t u) H 1 2 t ( ) 1 t + 2 H u 1 2 H v H 3 2 (v u) H 1 2 dv ]1 {0<u<t} Bounds on K: If H < 1/2 K t (u) (t u) H u H 1 2, and t K t (u) (t u) H 3 2. u Samy T. (Purdue) Rough Paths 4 Aarhus / 67

16 Underlying Wiener process on R Mandelbrot s representation for B: with B t = W two-sided Wiener process R K t (u) dw u, t 0 K t (u) defined by [ ] K t (u) = c H (t u) H 1/2 + ( u) H 1/2 + 1 { <u<t} Bounds on K: If H < 1/2 and 0 < u < t K t (u) (t u) H 1 2, and t K t (u) (t u) H 3 2. Samy T. (Purdue) Rough Paths 4 Aarhus / 67

17 Fractional derivatives Definition: For α (0, 1), u R and f smooth enough, Inversion property: D α f u = α Γ(1 α) I α f u = 1 Γ(α) u 0 f r f u+r dr r 1+α f r dr (r u) 1 α I α ( D α f ) = D α ( I α f ) = f Samy T. (Purdue) Rough Paths 4 Aarhus / 67

18 Fractional derivatives on intervals Notation: For f : [a, b] R, extend f by setting f = f 1 [a,b] Definition: D α f u = D α b f u = I α f u = I α b f u = 1 Γ(α) f u Γ(1 α)(b u) + α α b A related operator: For H < 1/2, u f r dr (r u) 1 α Kf = D 1/2 H f Γ(1 α) b u f u f r dr (r u) 1+α Samy T. (Purdue) Rough Paths 4 Aarhus / 67

19 Wiener space and fractional derivatives Proposition 2. For H < 1/2 we have K isometry between H and a closed subspace of L 2 (R) For φ, ψ H, E [B(φ)B(ψ)] = φ, ψ H = Kφ, Kψ L 2 (R), In particular, for φ H, E [ B(φ) 2] = ϕ H = Kϕ L 2 (R) Notation: B(φ) is called Wiener integral of φ w.r.t B Samy T. (Purdue) Rough Paths 4 Aarhus / 67

20 Cylindrical random variables Definition 3. Let f Cb (R k ; R) ϕ i H, for i {1,..., k} F a random variable defined by F = f (B(ϕ 1 ),..., B(ϕ k )) We say that F is a smooth cylindrical random variable Notation: S Set of smooth cylindrical random variables Samy T. (Purdue) Rough Paths 4 Aarhus / 67

21 Malliavin s derivative operator Definition for cylindrical random variables: If F S, DF H defined by DF = Proposition 4. k i=1 f x i (B(ϕ 1 ),..., B(ϕ k ))ϕ i. D is closable from L p (Ω) into L p (Ω; H). Notation: D 1,2 closure of S with respect to the norm F 2 1,2 = E [ F 2] + E [ DF 2 H]. Samy T. (Purdue) Rough Paths 4 Aarhus / 67

22 Divergence operator Definition 5. Domain of definition: Dom(δ ) = { } φ L 2 (Ω; H); E [ DF, φ H ] c φ F L 2 (Ω) Definition by duality: For φ Dom(I) and F D 1,2 E [F δ (φ)] = E [ DF, φ H ] (2) Samy T. (Purdue) Rough Paths 4 Aarhus / 67

23 Divergence and integrals Case of a simple process: Consider Then n 1 0 t 1 < < t n a i R constants ( n 1 ) n 1 δ a i 1 [ti,t i+1 ) = a i δb ti t i+1 i=0 i=0 Case of a deterministic process: if φ H is deterministic, δ (φ) = B(φ), hence divergence is an extension of Wiener s integral Samy T. (Purdue) Rough Paths 4 Aarhus / 67

24 Divergence and integrals (2) Proposition 6. Let Define B a fbm with Hurst parameter 1/4 < H 1/2 f a C 3 function with exponential growth {Π n st; n 1} set of dyadic partitions of [s, t] S n, = 2 n 1 k=0 f (B tk ) δb tk t k+1. Then S n, converges in L 2 (Ω) to δ (f (B)) Remark: In the Brownian case δ coincides with Itô s integral Samy T. (Purdue) Rough Paths 4 Aarhus / 67

25 Criterion for the definition of divergence Proposition 7. Let a < b, and E [a,b] step functions in [a, b] H 0 ([a, b]) closure of E [a,b] with respect to ϕ 2 H 0 ([a,b]) b Then = a ϕ 2 b u (b u) du + 1 2H a ( b H 0 ([a, b]) is continuously included in H If φ D 1,2 (H 0 ([a, b])), then φ Dom(δ ) u ) 2 ϕ r ϕ u dr du. (r u) 3/2 H Samy T. (Purdue) Rough Paths 4 Aarhus / 67

26 Bound on the divergence Corollary 8. Under the assumptions of Proposition 7, E [ δ (φ) 2] E [ ] φ 2 D 1,2 (H 0 ([a,b])) Samy T. (Purdue) Rough Paths 4 Aarhus / 67

27 Multidimensional extensions Aim: Define a Malliavin calculus for (B 1,..., B d ) First point of view: Rely on Partial derivatives D Bi with respect to each component Divergences δ,bi, defined by duality Related to integrals with respect to each B i Second point of view: Change the underlying Hilbert space and consider Ĥ = H {1,..., d} Samy T. (Purdue) Rough Paths 4 Aarhus / 67

28 Russo-Vallois symmetric integral Definition 9. Let Then φ be a random path b a φ w d Bw i = L 2 1 b ( lim φ w B i ε 0 2ε w+ε Bw ε) i dw, a provided the limit exists. Extension of classical integrals: Russo-Vallois integral coincides with Young s integral if H > 1/2 and φ C 1 H+ε Stratonovich s integral in the Brownian case Samy T. (Purdue) Rough Paths 4 Aarhus / 67

29 Russo-Vallois and Malliavin Proposition 10. Let φ be a stochastic process such that 1 φ1 [a,b] D 1,2 (H 0 ([a, b])), for all < a < b < 2 The following is an almost surely finite random variable: Tr [a,b] Dφ := L 2 1 b lim Dφ u, 1 [u ε,u+ε] H du ε 0 2ε a Then b a φ ud B i u exists, and verifies b a φ u d B i u = δ (φ 1 [a,b] ) + Tr [a,b] Dφ. Samy T. (Purdue) Rough Paths 4 Aarhus / 67

30 Outline 1 Main result 2 Construction of the Levy area: heuristics 3 Preliminaries on Malliavin calculus 4 Levy area by Malliavin calculus methods 5 Algebraic and analytic properties of the Levy area 6 Levy area by 2d-var methods 7 Some projects Samy T. (Purdue) Rough Paths 4 Aarhus / 67

31 Levy area: definition of the divergence Lemma 11. Let H > 1 4 B a d-dimensional fbm(h) 0 s < t < Then for any i, j {1,..., d} (either i = j or i j) we have 1 φ j u δb j su1 [s,t] (u) lies in Dom(δ,Bi ) 2 The following estimate holds true: [ (δ (,B E )) i φ j 2 ] c H t s 4H Samy T. (Purdue) Rough Paths 4 Aarhus / 67

32 Proof Case i = j, strategy: We invoke Corollary 8 We have to prove φ i 1 [s,t] D 1,2,Bi (H 0 ([s, t])) Abbreviation: we write D 1,2,Bi (H 0 ([s, t])) = D 1,2 (H 0 ) Norm of φ i in H 0 : We have E [ ] φ i 2 H 0 = A 1 st + A 2 st A 1 st = t A 2 st = E E [ δb i su 2 ] du s (t u) 1 2H ( t t s u δbur i dr (r u) 3/2 H ) 2 du Samy T. (Purdue) Rough Paths 4 Aarhus / 67

33 Proof (2) Analysis of A 1 st: A 1 st = t s u s 2H u:=s+(t s)v 1 du = (t s) 4H v 2H dv (t u) 1 2H 0 (1 v) 1 2H = c H (t s) 4H Analysis of A 2 st: A 2 st = t s t s du du [ ] [u,t]2 E δb i ur1 δbur i 2 dr 1 dr 2 ( t u dr (r u) 3/2 2H (r 1 u) 3/2 H (r 2 u) 3/2 H ) 2 t c H (t u) 4H 1 du = c H (t s) 4H s Samy T. (Purdue) Rough Paths 4 Aarhus / 67

34 Proof (3) Conclusion for φ i H0 : We have found E [ φ i 2 H 0 ] ch (t s) 4H Derivative term, strategy: setting D = D Bi We have D v φ i u = 1 [s,u] (v) We have to evaluate Dφ i H u 0 H v we have Computation of the H-norm: According to (1), Dφ i 2 H = E [ δb 2 su 2] = u s 2H Samy T. (Purdue) Rough Paths 4 Aarhus / 67

35 Proof (4) Computation for Dφ i : We get E [ ] Dφ i 2 H 0 H = Bst 1 + Bst 2 Moreover: B 1 st = t Bst 2 = E E [ (u s) 2H] du s (t u) 1 2H ( t t s u r s H u s H (r u) 3/2 H dr 0 r s H u s H r u H Hence, as for the terms A 1 st, A 2 st, we get E [ Dφ i 2 H 0 H ] ch (t s) 4H ) 2 du Samy T. (Purdue) Rough Paths 4 Aarhus / 67

36 Proof (5) Summary: We have found ] ch (t s) 4H E [ φ i 2 H 0 ] + E [ Dφ i 2 H 0 H Conclusion for B i : According to Proposition 7 and Corollary 8 δb i s 1 [s,t] Dom(δ,Bi ) We have [ (δ ( )) ],B E i δb i 2 s 1 [s,t] c H t s 4H Samy T. (Purdue) Rough Paths 4 Aarhus / 67

37 Proof (6) Case i j, strategy: Conditioned on F Bj B j and φ j = δbs j are deterministic δ,bi (φ j ) is a Wiener integral Computation: For i j we have [ (δ (,B E )) i φ j 2 ] { [ (δ (,B = E E )) i φ j 2 ]} F B j = E [ ] φ j 2 H c H E [ ] φ j 2 H 0 c H t s 4H, (3) where computations for the last step are the same as for i = j. Samy T. (Purdue) Rough Paths 4 Aarhus / 67

38 Definition of the Levy area Proposition 12. Let H > 1 4 B a d-dimensional fbm(h) 0 s < t < Then for any i, j {1,..., d} (either i = j or i j) we have 1 B 2,ji st t s δbj su d B i u defined in the Russo-Vallois sense 2 The following estimate holds true: E [ B 2,ji st 2] c H t s 4H Samy T. (Purdue) Rough Paths 4 Aarhus / 67

39 Proof Strategy: We apply Proposition 10, and check the assumptions Proposition 10, item 1: proved in Lemma 11 Proposition 10, item 2: need to compute trace term Trace term, case i = j: We have Hence D Bi v φ i u = 1 [s,u] (v) Dφ i u, 1 [u ε,u+ε] H = [s,u] [u ε,u+ε] R H Samy T. (Purdue) Rough Paths 4 Aarhus / 67

40 Proof (2) Computation of the rectangular increment: We have [s,u] [u ε,u+ε] R H = R H (u, u + ε) R H (s, u + ε) R H (u, u ε) + R H (s, u ε) = 1 [ ε 2H + (u s + ε) 2H + ε 2H (u s ε) 2H] 2 = 1 [ (u s + ε) 2H (u s ε) 2H] 2 Computation of the integral: Thanks to an elementary integration, t s = [s,u] [u ε,u+ε] R H du 1 [ (t s + ε) 2H+1 ε 2H+1 (t s ε) 2H+1] 2(2H + 1) Samy T. (Purdue) Rough Paths 4 Aarhus / 67

41 Proof (3) Computation of the trace term: Differentiating we get Tr [s,t] Dφ i = 1 2(2H + 1) lim (t s + ε) 2H+1 ε 2H+1 (t s ε) 2H+1 ε 0 2ε = (t s)2h 2 Expression for the Stratonovich integral: According to Proposition 10 t B 2,ii st = δsud i Bu i = δ,bi (φ i (t s)2h 1 [s,t] ) + s 2 (4) Samy T. (Purdue) Rough Paths 4 Aarhus / 67

42 Proof (4) Moment estimate: Thanks to relation (4) we have E [ B 2,ii st 2] c H t s 4H Case i j: We have Trace term is 0 B 2,ji st = δ,bi (φ j 1 [s,t] ) Moment estimate follows from Lemma 11 Samy T. (Purdue) Rough Paths 4 Aarhus / 67

43 Remark Another expression for B ii : Rules of Stratonovich calculus for B show that Much simpler expression! B ii st = (δbi st) 2 2 Samy T. (Purdue) Rough Paths 4 Aarhus / 67

44 Outline 1 Main result 2 Construction of the Levy area: heuristics 3 Preliminaries on Malliavin calculus 4 Levy area by Malliavin calculus methods 5 Algebraic and analytic properties of the Levy area 6 Levy area by 2d-var methods 7 Some projects Samy T. (Purdue) Rough Paths 4 Aarhus / 67

45 Levy area construction Summary: for 0 s < t τ, we have defined the stochastic integral B 2 st = If i = j: If i j: t u s s B 2 st(i, i) = 1 2 (B t B s ) 2 t u d B v d B u, i. e. B 2,ij st = d Bv i d Bu, j s s B i considered as deterministic path B 2,ij st is a Wiener integral w.r.t B j Samy T. (Purdue) Rough Paths 4 Aarhus / 67

46 Algebraic relation Proposition 13. Let (s, u, t) S 3,τ B 2 as constructed in Proposition 12 Then we have δb 2,ij sut = δb i su δb j ut Samy T. (Purdue) Rough Paths 4 Aarhus / 67

47 Proof Levy area as a limit: from definition of R-V integral we have B 2,ij st t = lim B 2,ε,ij ε 0 st, where B 2,ε,ij st = δbsv i dxv ε,j, s with v Xv ε,j = 0 1 2ε δbj w ε,w+ε dw Increments of B 2,ε,ij : B 2,ε,ij st is a Riemann type integral and δb 2,ε,ij sut = δb i su δx ε,j ut (5) We wish to take limits in (5) Samy T. (Purdue) Rough Paths 4 Aarhus / 67

48 Proof (2) Limit in the lhs of (5): We have seen lim ε 0 δb2,ε,ij L 2 (Ω) sut = δb 2,ij sut Expression for X ε,j : We have Xv ε,j = 1 { v Bw+ε j dw 2ε 0 = 1 2ε = 1 2ε { v+ε ε { v+ε v ε B j w dw B j w dw v 0 v ε ε ε ε } Bw ε j dw } B j w dw } Bw j dw (6) Samy T. (Purdue) Rough Paths 4 Aarhus / 67

49 Proof (3) Limit in the rhs of (5): Invoking Lebesgue s differentiation theorem in (6), we get lim X ε,j ε 0 v = δb j 0v = Bv j = lim δb i ε 0 su δxut ε,j = δbsu i δbut j Conclusion: Taking limits on both sides of (5), we get δb 2,ij sut = δb i su δb j ut Samy T. (Purdue) Rough Paths 4 Aarhus / 67

50 Regularity criterion in C 2 Lemma 14. Let g C 2. Then, for any γ > 0 and p 1 we have g γ c (U γ;p (g) + δg γ ), with ( T T U γ;p (g) = 0 0 g st p ) 1/p ds dt. t s γp+2 Samy T. (Purdue) Rough Paths 4 Aarhus / 67

51 Levy area of fbm: regularity Proposition 15. Let B 2 as constructed in Proposition 12 0 < γ < H Then, up to a modification, we have B 2 C 2γ 2 ([0, τ]; R d,d ) Samy T. (Purdue) Rough Paths 4 Aarhus / 67

52 Proof Strategy: Apply our regularity criterion to g = B 2 Term 2: We have seen: δb 2 = δb δb B C γ 1 δb δb C 2γ 3 Term 1: For p 1 we shall control E [ Uγ;p (B 2 ) p] = T T 0 0 E [ B 2 st p] ds dt t s γp Samy T. (Purdue) Rough Paths 4 Aarhus / 67

53 Proof (2) Aim: Control of E [ B 2 st p] Scaling and stationarity arguments: E [ [ B 2,ij st p] t u = E dbu i s s [ = t s 2pH E dbv j p] 1 u p dbu i dbv j ] 0 0 Stochastic analysis arguments: Since 1 u 0 dbi u 0 dbj v is element of the second chaos of fbm: [ 1 u E dbu i db j p] [ 1 u v c p,1 E dbu i db j 2] v c p, Samy T. (Purdue) Rough Paths 4 Aarhus / 67

54 Proof (3) Recall: B 2 γ c ( U γ;p (B 2 ) + δb 2 γ ) Computations for U γ;p (B 2 ): Let γ < 2H, and p such that γ + 2/p < 2H. Then: E [ Uγ;p (B 2 ) p ] T T E [ B 2 st p] = ds dt 0 0 t s γp+2 T T t s 2pH c p 0 0 t s ds dt c p(γ+2/p) p Conclusion: B 2 C 2γ 2 for any γ < H One can solve RDEs driven by fbm with H > 1/3! Samy T. (Purdue) Rough Paths 4 Aarhus / 67

55 Outline 1 Main result 2 Construction of the Levy area: heuristics 3 Preliminaries on Malliavin calculus 4 Levy area by Malliavin calculus methods 5 Algebraic and analytic properties of the Levy area 6 Levy area by 2d-var methods 7 Some projects Samy T. (Purdue) Rough Paths 4 Aarhus / 67

56 p-variation in R 2 Definition 16. Let We set X centered Gaussian process on [0, T ] R : [0, T ] 2 R covariance function of X 0 s < t T Π st set of partitions of [s, t] R p p var; [s,t] 2 = sup Π 2 st [si,s i+1 ] [t j,t j+1 ]R p i,j and C p var = { f : [0, T ] 2 R; R p var < } Samy T. (Purdue) Rough Paths 4 Aarhus / 67

57 Young s integral in the plane Proposition 17. Let f C p var g C q var p, q such that > 1 p q Then the following integral is defined in Young s sense: [s,u1 ] [s,u 2 ]f dg(u 1, u 2 ) [s,t] 2 Samy T. (Purdue) Rough Paths 4 Aarhus / 67

58 Area and 2d integrals Proposition 18. Let X R d smooth centered Gaussian process on [0, T ] Independent components X j R : [0, T ] 2 R common covariance function of X j s 0 s < t T and i j Define (in the Riemann sense) X 2,ij st E [ X 2,ij st = t s δx i sudx j u. Then 2] = [s,u1 ] [s,u 2 ]R dr(u 1, u 2 ) (7) [s,t] 2 Samy T. (Purdue) Rough Paths 4 Aarhus / 67

59 Proof Expression for the area: We have t t X 2,ij st = δxsudx i u j = δxsu i Ẋ u j du s s Expression for the second moment: E [ X 2,ij st 2] = = = = E [ ] δx i [s,t] 2 su 1 δxsu i 2 Ẋu j 1 Ẋu j du1 2 du 2 E [ ] [Ẋ ] δx i [s,t] 2 su 1 δxsu i 2 E j u1 Ẋu j du1 2 du 2 [s,u1 ] [s,u 2 ]R 2 [s,t] 2 u 1 u 2 R(u 1, u 2 ) du 1 du 2 [s,u1 ] [s,u 2 ]R dr(u 1, u 2 ) [s,t] 2 Samy T. (Purdue) Rough Paths 4 Aarhus / 67

60 Remarks Expression in terms of norms in H: We also have This is similar to (3) Extension: [ X 2,ij E st 2] = E [ ] δb i [s,t] 2 su 1 δbsu i dr(u1 2, u 2 ) = E [ ] δbs, i δbs i Formula (7) makes sense as long as R C p var with p < 2 We will check this assumption for a fbm with H > 1 4 H Samy T. (Purdue) Rough Paths 4 Aarhus / 67

61 p-variation of the fbm covariance Proposition 19. Let Then B a 1-d fbm with H < 1 2 R covariance function of B T > 0 R C 1 2H var Samy T. (Purdue) Rough Paths 4 Aarhus / 67

62 Proof Setting: Let 0 s < t T π = {t i } Π st S π = i,j E [ ] 1 2H δb ti t i+1 δb tj t j+1 For a fixed i, Sπ i = j E [ ] 1 δb ti t i+1 δb 2H tj t j+1 Decomposition: We have with S i,1 π = j i S i π = S i,1 π + S i,2 π, E [ ] 1 2H δb ti t i+1 δb tj t j+1, and Sπ i,2 = [ E (δbti t i+1 ) 2] 1 2H Samy T. (Purdue) Rough Paths 4 Aarhus / 67

63 Proof (2) A deterministic bound: If y j < 0 for all j i then j i y j 1 2H y j j i 1 2H = 1 2H y j j i This applies to y j = E[δB ti t i+1 δb ti t i+1 ] when H < 1 2 Bound for Sπ i,1 : Write Sπ i,1 E [ ] δb ti t i+1 δb tj t j+1 j i E [ ] δb ti t i+1 δb tj t j+1 j 1 2H 1 2H + E [ (δb ti t i+1 ) 2] 1 2H = E [δb ti t i+1 δb st ] 1 2H + E [ (δbti t i+1 ) 2] 1 2H Samy T. (Purdue) Rough Paths 4 Aarhus / 67

64 Proof (3) Bound for S i π: We have found S i π E [δb ti t i+1 δb st ] 1 2H + 2 E [ (δbti t i+1 ) 2] 1 2H = E [δb ti t i+1 δb st ] 1 2H + 2(ti+1 t i ) Bound on increments of R: Let [u, v] [s, t]. Then E [δb uv δb st ] = R(v, t) R(u, t) R(v, s) + R(u, s) = (t v) 2H (t u) 2H (v s) 2H + (u s) 2H (t v) 2H (t u) 2H + (v s) 2H (u s) 2H 2(v u) 2H Samy T. (Purdue) Rough Paths 4 Aarhus / 67

65 Proof (4) Bound for S i π, ctd: Applying the previous estimate, S i π E [δb ti t i+1 δb st ] 1 2H + 2(ti+1 t i ) 4(t i+1 t i ) Bound for S π : We have S π i S i π 4(t s) Conclusion: Since π is arbitrary, we get the finite 1 2H -variation Samy T. (Purdue) Rough Paths 4 Aarhus / 67

66 Construction of the Levy area Strategy: 1 Regularize B as B ε 2 For B ε, the previous estimates hold true 3 Then we take limits This uses the 1 -variation bound, plus rate of convergence 2H Long additional computations Samy T. (Purdue) Rough Paths 4 Aarhus / 67

67 Outline 1 Main result 2 Construction of the Levy area: heuristics 3 Preliminaries on Malliavin calculus 4 Levy area by Malliavin calculus methods 5 Algebraic and analytic properties of the Levy area 6 Levy area by 2d-var methods 7 Some projects Samy T. (Purdue) Rough Paths 4 Aarhus / 67

68 Current research directions Non exhaustive list: Further study of the law of Gaussian SDEs: Gaussian bounds, hypoelliptic cases Ergodicity for rough differential equations Statistical aspects of rough differential equations New formulations for rough PDEs: Weak formulation (example of conservation laws) Krylov s formulation Links between pathwise and probabilistic approaches for SPDEs Example of PAM in R 2 Samy T. (Purdue) Rough Paths 4 Aarhus / 67