Quasi-shuffles, Hoffman’s exponential and applications to SDEs Kurusch Ebrahimi–Fard, Simon J.A. Malham , Frederic Patras and Anke Wiese ACPMS Seminar 29th May 2020 14:30 KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential
Outline 1 Motivation. 2 Quasi-shuffle algebra. 3 Hoffman’s exponential map. 4 Application to exponential Lie series. KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential
Motivation: SDEs o stochastic differential system for Y t P R N : Itˆ ż t d ÿ V i p Y τ q d X i Y t “ Y 0 ` τ , 0 i “ 1 X i t driving scalar continuous semimartingales. Quadratic covariations: r X i , X j s “ 0 for all i ‰ j . V i governing non-commuting vector fields. Goal: compute the logarithm of the flowmap; Lie series? KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential
Itˆ o chain rule and flowmap The Itˆ o chain rule ù ñ ż t d ÿ f p Y τ q d X i ` ˘ f p Y t q “ f p Y 0 q ` V i ¨ B τ 0 i “ 1 ż t d ÿ ` 1 V i b V i : B 2 ˘ f p Y τ q d r X i , X i s τ , ` 2 0 i “ 1 Use p V i ¨ Bq f p Y q or V i ˝ f ˝ Y . Flowmap: ϕ t : f ˝ Y 0 ÞÑ f ˝ Y t . Solution: Y t “ ϕ t ˝ id ˝ Y 0 . KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential
Driving processes: extend For i “ 1 , . . . , d set D i : “ V i ¨ B and X r i , i s : “ r X i , X i s , 2 V i b V i : B 2 . D r i , i s : “ 1 ż t ÿ D a ˝ f ˝ Y τ d X a ñ f ˝ Y t “ f ˝ Y 0 ` τ , 0 a P A A : “ t 1 , . . . , d , r 1 , 1 s , . . . , r d , d su . KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential
Flowmap Iterating chain rule ù ñ ÿ ÿ ϕ t “ I w p t q D w Ñ w b w . w P A ˚ For word w “ a 1 ¨ ¨ ¨ a n P A ˚ : D w : “ D a 1 ˝ ¨ ¨ ¨ ˝ D a n ż d X a 1 τ 1 ¨ ¨ ¨ d X a n I w : “ τ n . 0 ď τ 1 﨨¨ď τ n ď t KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential
Algebra: structures Assume: D commutative & associative prod. r ¨ , ¨ s on KA . Definition (Bilinear form) x ¨ , ¨ y : K x A y b K x A y Ñ K for any u , v P A ˚ : # 1 , if u “ v , x u , v y : “ 0 , if u ‰ v . For this scalar product, A ˚ forms an orthonormal basis. KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential
Quasi-shuffle product K x A y Ñ concatenation algebra: uv P K x A y . Definition (Quasi-Shuffle product) ˚ generated recursively, u ˚ 1 “ 1 ˚ u “ u , where ‘1’ “ empty word: ua ˚ vb “ p u ˚ vb q a ` p ua ˚ v q b ` p u ˚ v q r a , b s . K x A y ˚ commutative and associative algebra (Hoffman). r ¨ , ¨ s ” 0 Ñ shuffle algebra K x A y . D KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential
Examples Example The quasi-shuffle of 12 and 34 is: 12 ˚ 34 “ 1234 ` 3412 ` 1342 ` 3142 ` 1324 ` 3124 ` 1 r 2 , 3 s 4 ` r 1 , 3 s 42 ` 3 r 1 , 4 s 2 ` r 1 , 3 s 24 ` 13 r 2 , 4 s ` 31 r 2 , 4 s ` r 1 , 3 sr 2 , 4 s . KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential
Co-products Definition (Deconcatenation and de-quasi-shuffle coproducts) Deconcatenation coproduct ∆: K x A y Ñ K x A y b K x A y : ÿ ∆ p w q : “ x uv , w y u b v . u , v Also D de-quasi-shuffle coproduct ∆ 1 (finiteness cond.). ù ñ K x A y ¨ , ∆ 1 and K x A y ˚ , ∆ are Hopf algebras. KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential
Convolution products End p K x A y ˚ q the K -module of linear endomorphisms of K x A y ˚ . Definition (Convolution products) X , Y P End p K x A y ˚ q , quasi-shuffle convolution prod.: X ˚ Y : “ quas ˝ p X b Y q ˝ ∆ . ÿ ` ˘ X ˚ Y p w q “ X p u q ˚ Y p v q . uv “ w Same notation: quasi-shuffle conv. prod. Ø underlying prod. Concatenation conv. prod.: conc ˝ p X b Y q ˝ ∆ 1 . KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential
Tensor Hopf algebra Associative Hopf algebra (Reutenauer). K x A y ˚ b K x A y . p u b v qp u 1 b v 1 q “ p u ˚ u 1 q b p vv 1 q . Natural abstract setting for the flowmap. KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential
Endomorphism characterisation X P End p K x A y ˚ q comp. described by image in K x A y ˚ b K x A y : ÿ X ÞÑ X p w q b w , w P A ˚ ÿ eg . id ÞÑ w b w . w P A ˚ KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential
Logarithm endomorphism ¸ k ˜ ÿ ¸ ˜ ÿ ÿ log w b w “ c k w b w ´ 1 b 1 w P A ˚ w P A ˚ k ě 1 ¸ k ˜ ÿ ÿ “ c k w b w w P A ˚ zt 1 u k ě 1 ÿ ÿ “ c k p u 1 ˚ ¨ ¨ ¨ ˚ u k q b p u 1 ¨ ¨ ¨ u k q k ě 1 u 1 ,..., u k P A ˚ zt 1 u ˜ | w | ¸ ÿ ÿ ÿ “ u 1 ˚ ¨ ¨ ¨ ˚ u k b w c k w P A ˚ u 1 ¨¨¨ u k “ w k “ 1 ˜ ÿ ¸ ÿ c k J ˚ k “ ˝ w b w . w P A ˚ k ě 1 J ˚ k p w q is zero if | w | ă k . KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential
Logarithm convolution power series Lemma (Logarithm convolution power series) The logarithm of ř w w b w is given by ˜ ÿ ¸ ÿ log ˚ p id q ˝ w b w , log w b w “ w P A ˚ w P A ˚ where p´ 1 q k ´ 1 log ˚ p id q : “ ÿ J ˚ k . k k ě 1 Often abbreviate log ˚ p id q ˝ w to log ˚ p w q . KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential
Adjoint endomorphisms Embedding End p K x A yq Ñ K x A y ˚ b K x A y given by ÿ Y ÞÑ w b Y p w q , w P A ˚ an algebra homomorphism for concatenation conv. prod. Definition (Adjoint endomorphisms) X and Y adjoints if images match: ÿ ÿ X ÞÑ w b X p w q and Y ÞÑ Y p w q b w . w w X : p u q , v @ D @ D Reutenaeur ù ñ equiv to “ u , X p v q . KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential
Composition action on words � ( For ℓ ď n : C p n q : “ λ “ p λ 1 , . . . , λ ℓ q : λ 1 ` ¨ ¨ ¨ ` λ ℓ “ n . Set | λ | : “ ℓ and Σ p λ q : “ λ 1 ` ¨ ¨ ¨ ` λ ℓ , Π p λ q : “ λ 1 ¨ ¨ ¨ λ ℓ and Γ p λ q : “ λ 1 ! ¨ ¨ ¨ λ ℓ ! . Definition (Composition action) Given word w “ a 1 ¨ ¨ ¨ a n and λ “ p λ 1 , . . . , λ ℓ q P C p n q : action: λ ˝ w : “ r a 1 ¨ ¨ ¨ a λ 1 sr a λ 1 ` 1 ¨ ¨ ¨ a λ 1 ` λ 2 s ¨ ¨ ¨ r a λ 1 `¨¨¨` λ ℓ ´ 1 ` 1 ¨ ¨ ¨ a n s , Brackets are concatenated; r w s is the nested bracket above. KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential
Hoffman exponential Definition (Hoffman exponential) exp H : K x A y Ñ K x A y ˚ defined by (‘1’ unchanged): D 1 ÿ exp H p w q : “ Γ p λ q λ ˝ w . λ P C p| w |q Inverse log H : K x A y ˚ Ñ K x A y given by D p´ 1 q Σ p λ q´| λ | ÿ log H p w q : “ λ ˝ w . Π p λ q λ P C p| w |q Hoffman ù ñ exp H : K x A y , ∆ Ñ K x A y ˚ , ∆ isomorphism. D Example exp H p a 1 a 2 a 3 q “ a 1 a 2 a 3 ` 1 2 r a 1 , a 2 s a 3 ` 1 2 a 1 r a 2 , a 3 s ` 1 6 r a 1 , a 2 , a 3 s . KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential
Hoffman adjoints exp : H : K x A y conc , ∆ 1 Ñ K x A y conc ,δ isomorphism, for a P A : 1 exp : ÿ ÿ H p a q : “ n ! a 1 . . . a n , n ě 1 r a 1 ,..., a n s“ a ( δ “ deshuffle coprod.). log : H : K x A y conc ,δ Ñ K x A y conc , ∆ 1 given by: p´ 1 q n ´ 1 log : ÿ ÿ H p a q : “ a 1 . . . a n . n n ě 1 r a 1 ,..., a n s“ a K x A y , ∆ and K x A y conc ,δ duals; K x A y ˚ , ∆ and K x A y conc , ∆ 1 also. D KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential
Recap Diagram KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential
Word to integral/partial differential operator maps Orthog. conts semimartingales: r ¨ , ¨ s nilpotent of deg. 3. ÿ ÿ I w D w P I b D Ý Ñ w b w P K x A y ˚ b K x A y . w w P A Definition (Itˆ o word-to-integral and PDO maps) µ : K x A y ˚ Ñ I : µ : w ÞÑ I w , µ p u ˚ v q “ µ p u q µ p v q ¯ µ : K x A y Ñ D : µ : i ÞÑ D i , ¯ µ p uv q “ ¯ ¯ µ p u q ¯ µ p v q . KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential
Abstract flowmap µ b ¯ µ : K x A y ˚ b K x A y Ñ I b D also a homomorphism, and: ˆÿ ˙ ÿ I w b D w “ p µ b ¯ µ q ˝ w b w . w w Compute a logarithm of ř w w b w . In terms of Lie polynomials/brackets of vector fields? KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential
Fisk–Stratonovich integral Definition (Fisk–Stratonovich integral) Continuous semimartingales H and Z , FS integral defined as ż t ż t H τ o d Z τ : “ H τ d Z τ ` 1 2 r H , Z s t . 0 0 Lemma (Itˆ o to Fisk–Stratonovich conversion) ż t ż t f p Y τ q o d X i ` ˘ f p Y τ q d X i ` ˘ τ ´ 1 “` ˘ f p Y q , X i ‰ V i ¨ B τ “ V i ¨ B V i ¨ B t , 2 0 0 ż t “` ˘ f p Y q , X i ‰ ` ˘` ˘ f p Y τ q d r X i , X i s τ . V i ¨ B t “ V i ¨ B V i ¨ B 0 Proof. Definition ‘ Itˆ o chain rule. KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential
FS chain rule Corollary (Fisk–Stratonovich chain rule) “ p V i b V i q : B 2 ` p V i ¨ B V i q ¨ B ` ˘` ˘ V i ¨ B V i ¨ B ż t d ÿ ` ˘ f p Y τ q o d X i ù ñ f p Y q “ f p Y 0 q ` V i ¨ B τ 0 i “ 1 ż t d ÿ ´ 1 f p Y τ q d r X i , X i s τ . ` ˘ p V i ¨ B V i q ¨ B 2 0 i “ 1 Set V r i , i s : “ ´ 1 ` V i ¨ B V i q ¨ B . Chain rule ù ñ FS flowmap: 2 ÿ J w V w . w P A V w are compositions of vector fields. KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential
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