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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

Itˆ

• stochastic differential system for Yt P RN:

Yt “ Y0 `

d

ÿ

i“1

ż t VipYτq dX i

τ,

X i

t driving scalar continuous semimartingales.

Quadratic covariations: rX i, X js “ 0 for all i ‰ j. Vi governing non-commuting vector fields. Goal: compute the logarithm of the flowmap; Lie series?

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Itˆ

• chain rule and flowmap

The Itˆ

• chain rule ù

ñ f pYtq “ f pY0q `

d

ÿ

i“1

ż t ` Vi ¨ B ˘ f pYτq dX i

τ

` 1

2 d

ÿ

i“1

ż t ` Vi b Vi : B2˘ f pYτq drX i, X isτ, Use pVi ¨ Bqf pY q or Vi ˝ f ˝ Y . Flowmap: ϕt : f ˝ Y0 ÞÑ f ˝ Yt. Solution: Yt “ ϕt ˝ id ˝ Y0.

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Driving processes: extend

For i “ 1, . . . , d set Di :“ Vi ¨ B and X ri,is :“ rX i, X is, Dri,is :“ 1

2Vi b Vi : B2.

ñ f ˝ Yt “ f ˝ Y0 ` ÿ

aPA

ż t Da ˝ f ˝ Yτ dX a

τ ,

A :“ t1, . . . , d, r1, 1s, . . . , rd, dsu.

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Flowmap

Iterating chain rule ù ñ ϕt “ ÿ

wPA˚

IwptqDw Ñ ÿ w b w. For word w “ a1 ¨ ¨ ¨ an P A˚: Dw :“ Da1 ˝ ¨ ¨ ¨ ˝ Dan Iw :“ ż

0ďτ1﨨¨ďτnďt

dX a1

τ1 ¨ ¨ ¨ dX an τn .

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: KxAy b KxAy Ñ K for any u, v P A˚: xu, vy :“ # 1, if u “ v, 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

KxAy Ñ concatenation algebra: uv P KxAy. Definition (Quasi-Shuffle product) ˚ generated recursively, u ˚ 1 “ 1 ˚ u “ u, where ‘1’ “ empty word: ua ˚ vb “ pu ˚ vbq a ` pua ˚ vq b ` pu ˚ vq ra, bs. KxAy˚ commutative and associative algebra (Hoffman). r ¨ , ¨ s ” 0 Ñ shuffle algebra KxAy 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 ` 1r2, 3s4 ` r1, 3s42 ` 3r1, 4s2 ` r1, 3s24 ` 13r2, 4s ` 31r2, 4s ` r1, 3sr2, 4s.

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Co-products

Definition (Deconcatenation and de-quasi-shuffle coproducts) Deconcatenation coproduct ∆: KxAy Ñ KxAy b KxAy: ∆pwq :“ ÿ

u,v

xuv, wy u b v. Also D de-quasi-shuffle coproduct ∆1 (finiteness cond.). ù ñ KxAy ¨ ,∆1 and KxAy˚,∆ are Hopf algebras.

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Convolution products

EndpKxAy˚q the K-module of linear endomorphisms of KxAy˚. Definition (Convolution products) X, Y P EndpKxAy˚q, quasi-shuffle convolution prod.: X ˚ Y :“ quas ˝ pX b Y q ˝ ∆. ` X ˚ Y ˘ pwq “ ÿ

uv“w

Xpuq ˚ Y pvq. Same notation: quasi-shuffle conv. prod. Ø underlying prod. Concatenation conv. prod.: conc ˝ pX b Y q ˝ ∆1.

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Tensor Hopf algebra

Associative Hopf algebra (Reutenauer). KxAy˚b KxAy. pu b vqpu1 b v1q “ pu ˚ u1q b pvv1q. Natural abstract setting for the flowmap.

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Endomorphism characterisation

X P EndpKxAy˚q comp. described by image in KxAy˚b KxAy: X ÞÑ ÿ

wPA˚

Xpwq b w, eg. id ÞÑ ÿ

wPA˚

w b w.

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Logarithm endomorphism

log ˜ ÿ

wPA˚

w b w ¸ “ ÿ

kě1

ck ˜ ÿ

wPA˚

w b w ´ 1 b 1 ¸k “ ÿ

kě1

ck ˜ ÿ

wPA˚zt1u

w b w ¸k “ ÿ

kě1

ck ÿ

u1,...,ukPA˚zt1u

pu1 ˚ ¨ ¨ ¨ ˚ ukq b pu1 ¨ ¨ ¨ ukq “ ÿ

wPA˚

˜ |w| ÿ

k“1

ck ÿ

u1¨¨¨uk“w

u1 ˚ ¨ ¨ ¨ ˚ uk ¸ b w “ ÿ

wPA˚

˜ ÿ

kě1

ckJ˚k ¸ ˝ w b w. J˚kpwq 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

log ˜ ÿ

wPA˚

w b w ¸ “ ÿ

wPA˚

log˚pidq ˝ w b w, where log˚pidq :“ ÿ

kě1

p´1qk´1 k J˚k. Often abbreviate log˚pidq ˝ w to log˚pwq.

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Adjoint endomorphisms

Embedding EndpKxAyq Ñ KxAy˚b KxAy given by Y ÞÑ ÿ

wPA˚

w b Y pwq, an algebra homomorphism for concatenation conv. prod. Definition (Adjoint endomorphisms) X and Y adjoints if images match: X ÞÑ ÿ

w

w b Xpwq and Y ÞÑ ÿ

w

Y pwq b w. Reutenaeur ù ñ equiv to @ X :puq, v D “ @ u, Xpvq D .

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Composition action on words

For ℓ ď n: Cpnq :“

• λ “ pλ1, . . . , λℓq: λ1 ` ¨ ¨ ¨ ` λℓ “ n

( . Set |λ| :“ ℓ and Σpλq :“ λ1 ` ¨ ¨ ¨ ` λℓ, Πpλq :“ λ1 ¨ ¨ ¨ λℓ and Γpλq :“ λ1! ¨ ¨ ¨ λℓ!. Definition (Composition action) Given word w “ a1 ¨ ¨ ¨ an and λ “ pλ1, . . . , λℓq P Cpnq: action: λ ˝ w :“ ra1 ¨ ¨ ¨ aλ1sraλ1`1 ¨ ¨ ¨ aλ1`λ2s ¨ ¨ ¨ raλ1`¨¨¨`λℓ´1`1 ¨ ¨ ¨ ans, Brackets are concatenated; rws is the nested bracket above.

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Hoffman exponential

Definition (Hoffman exponential) expH : KxAy D Ñ KxAy˚ defined by (‘1’ unchanged): expHpwq :“ ÿ

λPCp|w|q

1 Γpλqλ ˝ w. Inverse logH : KxAy˚ Ñ KxAy D given by logHpwq :“ ÿ

λPCp|w|q

p´1qΣpλq´|λ| Πpλq λ ˝ w. Hoffman ù ñ expH : KxAy D

,∆ Ñ KxAy˚,∆ isomorphism.

Example expHpa1a2a3q “ a1a2a3 ` 1

2ra1, a2sa3 ` 1 2a1ra2, a3s ` 1 6ra1, a2, a3s.

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Hoffman adjoints

exp:

H : KxAyconc,∆1 Ñ KxAyconc,δ isomorphism, for a P A:

exp:

Hpaq :“

ÿ

ně1

ÿ

ra1,...,ans“a

1 n!a1 . . . an, (δ “ deshuffle coprod.). log:

H : KxAyconc,δ Ñ KxAyconc,∆1 given by:

log:

Hpaq :“

ÿ

ně1

ÿ

ra1,...,ans“a

p´1qn´1 n a1 . . . an. KxAy D

,∆ and KxAyconc,δ duals; KxAy˚,∆ and KxAyconc,∆1 also.

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

wPA

IwDw P I b D Ý Ñ ÿ

w

w b w P KxAy˚ b KxAy. Definition (Itˆ

• word-to-integral and PDO maps)

µ: KxAy˚ Ñ I : µ: w ÞÑ Iw, µpu ˚ vq “ µpuqµpvq ¯ µ: KxAy Ñ D : ¯ µ: i ÞÑ Di, ¯ µpuvq “ ¯ µpuq¯ µpvq.

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Abstract flowmap

µ b ¯ µ: KxAy˚ b KxAy Ñ I b D also a homomorphism, and: ÿ

w

Iw b Dw “ pµ b ¯ µq ˝ ˆÿ

w

w b 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 Hτ o dZτ :“ ż t Hτ dZτ ` 1

2rH, Zst.

Lemma (Itˆ

• to Fisk–Stratonovich conversion)

ż t ` Vi ¨ B ˘ f pYτq dX i

τ “

ż t ` Vi ¨ B ˘ f pYτq o dX i

τ ´ 1 2

“` Vi ¨ B ˘ f pY q, X i‰

t,

“` Vi ¨ B ˘ f pY q, X i‰

t “

ż t ` Vi ¨ B ˘` Vi ¨ B ˘ f pYτq d rX i, X isτ. Proof. Definition ‘ Itˆ

• chain rule.

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

FS chain rule

Corollary (Fisk–Stratonovich chain rule) ` Vi ¨ B ˘` Vi ¨ B ˘ “ pVi b Viq: B2 ` pVi ¨ BViq ¨ B ù ñ f pY q “ f pY0q `

d

ÿ

i“1

ż t ` Vi ¨ B ˘ f pYτq o dX i

τ

´ 1

2 d

ÿ

i“1

ż t ` pVi ¨ BViq ¨ B ˘ f pYτqdrX i, X isτ. Set Vri,is :“ ´1

2

` Vi ¨ BViq ¨ B. Chain rule ù ñ FS flowmap: ÿ

wPA

JwVw. Vw are compositions of vector fields.

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

FS word to integral/VF maps

Definition (FS word-to-vector field/VF maps) ν : KxAy D Ñ J : ν : w ÞÑ Jw, νpu D vq “ νpuqνpvq ¯ ν : KxAy Ñ V : ¯ ν : a ÞÑ Va, ¯ νpuvq “ ¯ νpuq¯ νpvq. FS flowmap setting KxAy D b KxAy. ν b ¯ ν : KxAy D b KxAy Ñ J b V is a homomorphism.

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Chen–Strichartz Lie series

ÿ

w

IwDw ” ÿ

w

JwVw and log ˆÿ

w

IwDw ˙ ” log ˆÿ

w

JwVw ˙ . Theorem (Chen–Strichartz Lie series) log ˆÿ

w

JwVw ˙ “ ÿ

w

1 |w|Jlog

D pwqVrwsL

where Jlog

D pwq “ ν ˝ log D

pwq and log

D

pwq “ ÿ

σPS|w|

cσ σ´1pwq with cσ :“ p´1qdpσq |σ| ˆ|σ| ´ 1 dpσq ˙´1 . VrwsL :“ rVa1, rVa2, . . . , rVan´1, VansL ¨ ¨ ¨ sLsL are Lie polynomials.

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Proof (part 1): DFSW Theorem

Proof. log ˆÿ JwVw ˙ “ log ˝pν b ¯ νq ˝ ˆÿ w b w ˙ “ pν b ¯ νq ˝ log ˝ ˆÿ w b w ˙ “ pν b ¯ νq ˝ ˆÿ log

D

pidq ˝ w b w ˙ , ` log

D

pidqq: “ logpidq Lie idempotent ô Eulerian/Solomon idem. θ :“ 1

pr1 ¨ ¨ ¨ psL Dynkin idem. on KrSps.

Theorem (Dynkin–Friedrichs–Specht–Wever Theorem) Any polynomial P P KxAy: P P LFpKxAyq ô θP “ P.

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Proof (part 2): DFSW Theorem

Proof. Reutenauer ñ logpwq P LFpKxAyq. DFSW ñθ ` logpwq ˘ ” logpwq. log ˆÿ w b w ˙ “ ÿ log

D

pwq b w “ ÿ w b logpwq “ ÿ w b θ ` logpwq ˘ “ ÿ log

D

pwq b θpwq “ ÿ 1 |w| log

D

pwq b rwsL,

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Itˆ

• to Fisk–Stratonovich: Hoffman exponential

Proposition (Itˆ

• to Fisk–Stratonovich: Hoffman exponential)

For continuous semimartingales: Jw “ IexpHpwq where IexpHpwq “ ÿ

λPCp|w|q

1 2Σpλq´|λ| Iλ˝w “ Iw ` ÿ

uPrrwss

1 2|w|´|u| Iu. Nilpotency of r ¨ , ¨ s ù ñ λ P Cp|w|q only contain 1 and 2 ù ñ Γpλq “ 2Σpλq´|λ|. rrwss “ words constructed from w by successively replacing neighbouring pairs ii by ri, is.

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Proof

Proof. Def of Fisk–Stratonovich integral ù ñ Ja1¨¨¨an “ ż Ja1¨¨¨an´1 dIan ` 1 2 ż Ja1¨¨¨an´2 drIan´1, Ians. Recursively apply for Ja1¨¨¨an´1 . . .

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Itˆ

• Lie series

Corollary (Itˆ

• Lie series)

ñ log ˆÿ

w

JwVw ˙ “ ÿ

w

ÿ

σPS|w|

cσ |σ|IexpHpσ´1pwqqVrwsL “ ÿ

w

Iw ˜ ÿ

σPS|w|

cσ |σ|VrσpwqsL ` ÿ

uPsswrr

1 2|u|´|w| ÿ

σPS|u|

cσ |σ|VrσpuqsL ¸ . sswrr“ words constructed from w by successively replacing letters ri, is by ii.

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Recall: Itˆ

• and Fisk–Stratonovich map relations

¯ µ: # i ÞÑ Vi ¨ B, ri, is ÞÑ 1

2Vi b Vi : B2,

¯ ν : # i ÞÑ Vi ¨ B, ri, is ÞÑ ´1

2pVi ¨ BViq ¨ B.

µ, ν : # i ÞÑ Xi, ri, is ÞÑ rX i, X is.

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Itˆ

• and Fisk–Stratonovich map relations II

Theorem (Itˆ

• and Fisk–Stratonovich map relations)

ν ” µ ˝ expH and ¯ ν ” ¯ µ ˝ log:

H .

Proof. Jw “ IexpHpwq ô ν ˝ w “ µ ˝ expH ˝w. ` Vi ¨ B ˘` Vi ¨ B ˘ “ pVi b Viq: B2 ` pVi ¨ BViq ¨ B

1 2 ¯

µ ˝ ii “ ¯ µ ˝ ri, is ´ ¯ ν ˝ ri, is ô ¯ ν ˝ ri, is “ ¯ µ ˝ ` ri, is ´ 1

2ii

˘ ô ¯ ν ˝ ri, is “ ¯ µ ˝ log:

H ˝ri, is.

Note: ¯ µpiiq “ ¯ νpiiq; log:

H ˝i “ i and log: H is concat homom.

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Itˆ

• and Fisk–Stratanovich representations equivalent

ÿ IwDw “ ÿ µ ˝ w b ¯ µ ˝ w “ ÿ µ ˝ w b ¯ ν ˝ exp:

H ˝w

“ pµ b ¯ νq ˝ ˆÿ w b exp:

H ˝w

˙ “ pµ b ¯ νq ˝ ˆÿ expH ˝w b w ˙ “ ÿ µ ˝ expH ˝w b ¯ ν ˝ w “ ÿ ν ˝ w b ¯ ν ˝ w “ ÿ JwVw.

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Exponential Lie series, refrain

log ˆÿ IwDw ˙ “ log ˝pµ b ¯ µq ˝ ˆÿ w b w ˙ “ log ˝pν b ¯ νq ˝ ˆÿ w b w ˙ “ pν b ¯ νq ˝ log ˝ ˆÿ w b w ˙ “ pν b ¯ νq ˝ ˆÿ log

D

˝w b w ˙ “ pν b ¯ νq ˝ ˆÿ 1 |w| log

D

˝w b rwsL ˙ “ pµ b ¯ νq ˝ ˆÿ 1 |w| expH ˝ log

D

˝w b rwsL ˙ “ ÿ 1 |w|IexpHplog

D pwqqVrwsL,

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Resummation, refrain

log ˆÿ JwVw ˙ “ pν b ¯ νq ˝ ˆÿ log

D

˝w b θ ˝ w ˙ “ pµ b ¯ νq ˝ ˆÿ expH ˝ log

D

˝w b θ ˝ w ˙ “ pµ b ¯ νq ˝ ˆÿ w b θ ˝ log ˝ exp:

H ˝w

˙ , where log ˝ exp:

H ˝w “ log ˝

˜ w ` ÿ

uPsswrr

1 2|u|´|w| u ¸ “ ÿ

σPS|w|

cσ σpwq ` ÿ

uPsswrr

1 2|u|´|w| ÿ

σPS|u|

cσ σpuq.

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Thank you for listening!

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Bibliography I

Curry C, Ebrahimi–Fard K, Malham SJA, Wiese A. 2018. Algebraic structures and stochastic differential equations driven by L´ evy processes. PRSA 475. Ebrahimi–Fard K, Lundervold A, Malham SJA, Munthe–Kaas H, Wiese A. 2012. Algebraic structure of stochastic expansions and efficient simulation. Proc. R. Soc. A doi:10.1098/rspa.2012.0024. Ebrahimi–Fard K, Malham SJA, Patras F, Wiese A. 2015. Flows and stochastic Taylor series in Itˆ

• calculus, to appear in
• J. Phys. A: Math. Theor.

Hoffman ME. 2000. Quasi-shuffle products. Journal of Algebraic Combinatorics 11, pp. 49–68. Hoffman ME, Ihara K. 2012. Quasi-shuffle products revisited. Max-Planck-Institut f¨ ur Mathematik Preprint Series 2012 (16).

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential

Bibliography II

Malham SJA, Wiese A. 2008. Stochastic Lie group integrators, SIAM J. Sci. Comput. 30(2), pp. 597–617. Malham SJA, Wiese A. 2009. Stochastic expansions and Hopf

• algebras. Proc. R. Soc. A 465, pp. 3729–3749.

Reutenauer C. 1993. Free Lie algebras. London Mathematical Society Monographs New Series 7, Oxford Science Publications. Strichartz RS. 1987. The Campbell–Baker–Hausdorff–Dynkin formula and solutions of differential equations, J. Funct. Anal. 72, pp. 320–345.

KEF, Malham, Patras Wiese Quasi-shuffles: Hoffman’s exponential