Variational estimates for martingale transforms Pavel Zorin-Kranich - PowerPoint PPT Presentation

Variational estimates for martingale transforms Pavel Zorin-Kranich University of Bonn 2020-06-09 Joint work with P. Friz. Pavel Zorin-Kranich (U Bonn) – Variational estimates for martingale transforms 1/ 9

Rough paths Defjnition Pavel Zorin-Kranich (U Bonn) – Variational estimates for martingale transforms . 1 / p |𝕐 u l − 1 , u l | p ) l = 1 ∑ l max ( l max , u 0 <⋯< u lmax sup V p 𝕐 = , 1 / p l = 1 ∑ l max A p -rough path, 2 < p < 3, is a pair X ∶ [ 0 , ∞) → H , 𝕐 ∶ Δ = {( s , t ) | 0 ≤ s < t < ∞} → H ⊗ H such that X ∈ V p loc , 𝕐 ∈ V p / 2 2/ 9 (Chen’s relation) p-variation: V p X = sup l max , u 0 <⋯< u lmax ( loc , and for s < t < u 𝕐 s , u = 𝕐 s , t + 𝕐 t , u + ( X u − X t ) ⊗ ( X t − X s ). | X u l − X u l − 1 | p ) How to check the conditions X ∈ V p and 𝕐 ∈ V p / 2 ?

Rough path lifts of martingales Let M = ( M t ) be a (Hilbert space valued) càdlàg martingale. Let Pavel Zorin-Kranich (U Bonn) – Variational estimates for martingale transforms How to incorporate e.g. fractional Brownian motion? Q: what is the appropriate generality for these lifting results? There exist rough path lifts of over processes, e.g. Lévy processes. 3/ 9 ( s , t ] Then, a.s., the pair ( M , 𝕅) is a p -rough path for any p > 2. 𝕅 s , t ∶= ∫ ( M u − − M s ) ⊗ dM u . ▶ Chen’s relation – from Itô integration ▶ Bound for V p M : Lépingle 1976. ▶ Bounds for V p / 2 𝕅 : ▶ M Brownian motion: Lyons 1998 ▶ M has continuous paths: Friz+Victoir 2006 ▶ M dyadic: Do+Muscalu+Thiele 2010, ▶ M has càdlàg paths: Chevyrev+Friz 2017, Kovač+ZK 2018.

Joint rough path lifts All martingales and processes are adapted, càdlàg, Hilbert space valued. Pavel Zorin-Kranich (U Bonn) – Variational estimates for martingale transforms The proof also recovers existence of Itô integrals and estimates for New in this result: Motivation: dY = a ( Y ) d X + b ( Y ) dM is a p-rough path. ) 4/ 9 𝕐 ( X p-rough path ( 2 < p < 3 ). Then, a.s., the pair of processes Let M = ( M t ) be a càdlàg martingale and ( X , 𝕐) a deterministic càdlàg Theorem (Friz+ZK 2020+) ∫ X u − ⊗ dM u M ) , ( ∫ M u − ⊗ dX u ∫ M u − ⊗ dM u ▶ Variational estimates for Itô integrals ∫ X dM , ▶ existence of and estimates for integrals ∫ M dX . 𝕅 = ∫ M dM from previous slide.

Martingale transforms 𝜐 Pavel Zorin-Kranich (U Bonn) – Variational estimates for martingale transforms RHS is ≲ ‖ Sf ‖ L q 0 ‖ Sg ‖ L q 0 . In this case, any r > 1 works. The supremum is taken over increasing sequences of stopping times 𝜐 = (𝜐 k ) . ‖ L q 1 ‖ Sg ‖ L q 0 . 1 / p ‖ 𝜐 k − 1 < j ≤𝜐 k sup ( k ‖(∑ ‖ 5/ 9 ‖ Theorem (Main estimate) Martingale in t variable, discrete version of area integral. ‖ V r Π( f , g )‖ s < j ≤ t Let ( f n ) n ∈ℕ be a discrete time adapted process and ( g n ) n ∈ℕ a discrete time martingale. Defjne paraproduct Π s , t ( f , g ) ∶= ∑ ( f j − 1 − f s ) dg j , dg j = g j − g j − 1 . Let 1 ≤ p ≤ ∞ , 0 < q 1 ≤ ∞ , 1 ≤ q 0 < ∞ . Defjne q by 1 / q = 1 / q 0 + 1 / q 1 and suppose 1 / r < 1 / 2 + 1 / p. Then, with ‖ g ‖ L q = (𝔽| g | q ) 1 / q , Sg = [ g ] 1 / 2 , ‖ L q ≲ sup | f j − 1 − f 𝜐 k − 1 |) p ) ▶ If f is a martingale, p = 2, 1 ≤ q 1 < ∞ , then by BDG inequality the ▶ For general f , RHS is ≤ ‖ V p f ‖ L q 0 ‖ Sg ‖ L q 0 and r = p / 2 works.

Discrete approximation of adapted processes lim Pavel Zorin-Kranich (U Bonn) – Variational estimates for martingale transforms Given 𝜗 > 0, consider the adapted partition Proof. p ∈ ( p , ∞) ∪ {∞} . for any ̃ p ) in Defjnition 6/ 9 If f ∈ L q ( V p ) for some q > 0 and p > 1 , then Lemma ∶= f ⌊ t ,𝜌⌋ . t f (𝜌) ⌊ t , 𝜌⌋ ∶= max { s ∈ 𝜌 | s ≤ t }, For an adapted partition 𝜌 , let An adapted partition 𝜌 = (𝜌 j ) j is an increasing sequence of stopping times. Adapted partitions are ordered by a.s. inclusion of the sets {𝜌 j | j ∈ ℕ} . The set of adapted partitions is directed, so lim 𝜌 makes sense. 𝜌 f (𝜌) = f L q ( V ̃ 𝜌 j + 1 (𝜕) ∶= inf { t > 𝜌 j (𝜕) | 𝜌 0 ∶= 0 , | | f t − f 𝜌 j (𝜕) |(𝜕) > 𝜗}.

Stopping time reduction ∞ Pavel Zorin-Kranich (U Bonn) – Variational estimates for martingale transforms where the supremum is taken over all adapted partitions 𝜐 . (1) ‖ L q , 1 /𝜍 ‖ 𝜍 ) |Π t , t ′ |) 𝜐 j − 1 ≤ t < t ′ ≤𝜐 j sup ( j = 1 ∑ ‖( f adapted process, g martingale ‖ 𝜐 Then, for every 0 < 𝜍 < r < ∞ and q ∈ ( 0 , ∞] , we have Lemma Suppose 1 / r < 1 / p + 1 / 2 . Then Theorem (Main estimate) Square function: Sg = [ g ] 1 / 2 , 7/ 9 Martingale transform: Π s , t ( f , g ) = ∑ s < j ≤ t ( f j − 1 − f s ) dg j Hölder exponents: 1 / q = 1 / q 0 + 1 / q 1 . ‖ V r Π‖ L q (Ω) ≲ ‖ V p f ‖ L q 1 (Ω) ‖ Sg ‖ L q 0 (Ω) . The V r norm is estimated as follows. Let (Π s , t ) s ≤ t be a càdlàg adapted sequence with Π t , t = 0 for all t. ‖ V r Π‖ L q ≲ sup

Lépingle’s inequality Above stopping time argument fjrst used in the following result. Pavel Zorin-Kranich (U Bonn) – Variational estimates for martingale transforms inequalities that follow from weighted inequalities by Osȩkowski. For dealing with martingale transforms, we use vector-valued BDG Weighted inequalities imply vector-valued inequalities. Classical Lépingle’s inequality is the case w ≡ 1, A p ( w ) = 1. p − 1 ‖ L ∞ ( w ) 𝜐 ) ‖𝔽( w | ℱ 𝜐 stopping time sup A p ( w ) ∶= For 1 < p < ∞ and 2 < r, we have Let M be a martingale and w a positive random variable. Theorem (ZK 2019) 8/ 9 ‖ V r M ‖ L p ( w ) ≤ C p , r A p ( w ) max ( 1 , 1 /( p − 1 )) ‖ M ‖ L p ( w ) , where the A p charactersitic is given by 𝜐 )𝔽( w − 1 /( p − 1 ) | ℱ

Integration by parts ‖ L q ‖( ∞ ∑ j = 1 ( ∑ 𝜐 j − 1 < u ≤𝜐 j |Δ X u Δ u M | 2 ) 𝜍/ 2 ) 1 /𝜍 ‖ Hölder ≲ ≤ ‖( ∞ ∑ j = 1 ∑ 𝜐 j − 1 < u ≤𝜐 j |Δ u M | 2 ) 1 / 2 ‖ ‖ L q Pavel Zorin-Kranich (U Bonn) – Variational estimates for martingale transforms ‖ vector BDG ( X , 𝕐) rough path, M martingale ‖ We do this by partial integration: Π( M , X ) ∶= 𝜀 M 𝜀 X − Π( X , M ) − 𝜀[ X , M ]. The bracket is given by u ≤ T Δ X u Δ M u , Variation norm estimate for the bracket: ‖ V r [ X , M ]‖ L q ‖ L q ≲ stopping ‖( ∞ ∑ j = 1 ( sup 𝜐 j − 1 < t < t ′ ≤𝜐 j |𝜀[ X , M ] t , t ′ |) 𝜍 ) 1 /𝜍 ‖ 9/ 9 So far we have estimated ∫ X dM and ∫ M dM . Next, we want to construct and estimate Π( M , X ) = ∫ M dX . [ X , M ] T = ∑ Δ M u = M u − M u − . V p X ⋅ ‖