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

Variational estimates for martingale transforms Pavel Zorin-Kranich University of Bonn 2020-06-04 Joint work with P. Friz. https://www.math.uni-bonn.de/~pzorin/slides/2020-06-04-var-mart-transforms.pdf Pavel Zorin-Kranich (U Bonn) – Variational estimates for martingale transforms 1/ 12

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/ 12 (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/ 12 ( 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: is a p-rough path. ) 4/ 12 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/ 12 ‖ 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/ 12 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}.

Discrete approximation of Itô integrals T ∈ 𝜌. Pavel Zorin-Kranich (U Bonn) – Variational estimates for martingale transforms integral. Cauchy net in the space L q ( V r ) , so we also reprove the existence of the Itô In fact, the discrete estimate gives more: the discrete approximations are a ( s , t ] where 1 / r > 1 / 2 + 1 / p and Since it converges to the Itô integral, we get the same estimate for it: The RHS is a martingale transform, to which our main estimate applies. j ∶𝜌 j ≤ T ∑ f (𝜌) 0 T ∫ 7/ 12 The Itô integral of the discretized process f (𝜌) is given by u − dM u = f 𝜌 j − 1 ( M 𝜌 j − M 𝜌 j − 1 ), ‖ V r Π( f , g )‖ L q ≲ ‖ V p f ‖ L q 1 ‖ Sg ‖ L q 0 , Π( f , g ) s , t = ∫ ( f u − − f s ) dg u .

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 , 8/ 12 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

Stopping time construction ( V r X ) ∑ m = 0 ( 2 − m V ∞ ∞ ) r −𝜍 ∞ ∑ j = 1 | X 𝜐 ( m ) j − X 𝜐 ( m ) ∞ − X 𝜐 ( m ) ∑ m = 0 ( 2 − m ) r −𝜍 ∞ ∑ j = 1 | X 𝜐 ( m ) j − X 𝜐 ( m ) Pavel Zorin-Kranich (U Bonn) – Variational estimates for martingale transforms ∞ ≤ C j | > 2 − m V ∞ Let V ∞ Construct stopping times with m ∈ ℕ : 𝜐 ( m ) 0 ∶= 0 , 𝜐 ( m ) j | j | X 𝜐 ( m ) 9/ 12 ∑ ( V r X ) j = 1 ∞ ∑ Then m = 0 ∞ For simplicity, we consider processes Π s , t = X t − X s . n ∶= sup n ″ ≤ n ′ ≤ n | X n ″ − X n ′ | . j + 1 ∶= inf { t > 𝜐 ( m ) | | X t − X 𝜐 ( m ) t / 10 }. r ≤ C j − 1 | r j − 1 | 𝜍 Since V ∞ ≤ V r , and assuming V r < ∞ , this implies 𝜍 ≤ C j − 1 | 𝜍 .

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) 10/ 12 ‖ 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

Sketch of proof of the main estimate k j = 1 f adapted process, g martingale k | f ( j ) k − 1 | 2 | g ( j ) k − 1 | 2 ) 1 / 2 ( here f ( j ) t ≤ 𝔽 ∞ ∑ j = 1 ( f ( j ) | g ( j ) ∞ j = 1 Pavel Zorin-Kranich (U Bonn) – Variational estimates for martingale transforms 1 / 2 k − 1 | 2 ) | g ( j ) k ∑ ∑ k − 1 | 2 ) ∞ 1 / 2 (𝔽 ( f ( j ) j = 1 ∑ ∞ ∑ (∑ = 𝔽 sup Square function: Sg = [ g ] 1 / 2 , For an adapted partition 𝜐 , want to show ‖ ‖(∑ l sup 𝜐 l − 1 ≤ t ≤ t ′ ≤𝜐 l |Π t , t ′ | 𝜍 ) 1 /𝜍 ‖ ‖ ∞ ‖ S Π 𝜐 j − 1 ,𝜐 j ‖ 1 ∑ j = 1 11/ 12 [𝜐 j − 1 ,𝜐 j ] ‖ sup j = 1 ∑ ∞ ≲ BDG |Π|‖ 1 [𝜐 j − 1 ,𝜐 j ] ∑ j = 1 ∞ Martingale transform: Π s , t ( f , g ) = ∑ s < j ≤ t ( f j − 1 − f s ) dg j exponents: 1 / q = 1 / q 0 + 1 / q 1 , 1 /𝜍 = 1 / p + 1 / 2. ‖ L q (Ω) ≲ ‖ V p f ‖ L q 1 (Ω) ‖ Sg ‖ L q 0 (Ω) . Simple case: q 1 = q 0 = p = 2, q = 𝜍 = 1. |Π|‖ 1 = k − g ( j ) = f t ∧𝜐 j − f t ∧𝜐 j − 1 ) 1 / 2 ≤ (𝔽 ∗ )(∑ k − g ( j ) ∗ ) 2 ) k − g ( j ) If one of the conditions q 1 = p , q = 𝜍 , q 0 = 2 fails, things get more tricky.