Unified Signature Cumulants and Generalized Magnus Expansions
UUNIFIED SIGNATURE CUMULANTS AND GENERALIZED MAGNUSEXPANSIONS
PETER K. FRIZ : , ; , PAUL HAGER : , AND NIKOLAS TAPIA : , ; Abstract.
The signature of a path can be described as its full non-commutative exponential.Following T. Lyons we regard its expectation, the expected signature , as path space analogue of theclassical moment generating function. The logarithm thereof, taken in the tensor algebra, defines the signature cumulant . We establish a universal functional relation in a general semimartingale context.Our work exhibits the importance of Magnus expansions in the algorithmic problem of computingexpected signature cumulants, and further offers a far-reaching generalization of recent results oncharacteristic exponents dubbed diamond and cumulant expansions; with motivation ranging fromfinancial mathematics to statistical physics. From an affine process perspective, the functional relationmay be interpreted as infinite-dimensional, non-commutative (“Hausdorff”) variation of Riccati’sequation. Many examples are given.
Contents
1. Introduction and main results
2. Preliminaries
3. Expected signatures and signature cumulants
4. Main Results
5. Two special cases
6. Applications
7. Proofs References : Institut für Mathematik, TU Berlin, Str. des 17. Juni 136, 10586 Berlin, Germany. ; Weierstrass Institute, Mohrenstr. 39, 10117 Berlin, Germany.
E-mail addresses : {friz,phager,tapia}@math.tu-berlin.de .2020 Mathematics Subject Classification.
Key words and phrases.
Signatures, Lévy processes, Markov processes, stochastic Volterra processes, universalsignature relations for semimartingales, moment-cumulant relations, characteristic functions, diamond product, Magnusexpansion. a r X i v : . [ m a t h . P R ] F e b P. FRIZ, P. HAGER, AND N. TAPIA Introduction and main results
Write T – T pp R d qq “ Π k ě p R d q b k for the tensor series over R d , equipped with concatenationproduct, elements of which are written indifferently as x “ p x p q , x p q , x p q , . . . q ” x p q ` x p q ` x p q ` ¨ ¨ ¨ . The affine subspace T (resp. T ) with scalar component x p q “ “
1) has a natural Lie algebra(resp. formal Lie group) structure.Let further S “ S p R d q , resp. S c “ S c p R d q , denote the class of càdlàg, resp. continuous, d -dimensional semimartingales on some filtered probability space p Ω , p F t q t ě , P q . The formal sum ofiterated Stratonovich-integrals, the signature of X P S c Sig p X q s,t “ ` X s,t ` ż ts X s,u ˝ d X u ` ż ts ˆż u s X s,u ˝ d X u ˙ ˝ d X u ` ¨ ¨ ¨ for 0 ď s ď t defines a random element in T and, as a process, a formal T -valued semimartingale. Byregarding the d -dimensional semimartingale X as T -valued semimartingale ( X Ø X “ p , X, , . . . )),we see that the signature of X satisfies the Stratonovich stochastic differential equationd S “ S ˝ d X . (1.1)The solution is a.k.a. the Lie group valued stochastic exponential (or development) of X P S p T q ,with classical references [ McK69 , HDL86 ]; the càdlàg case [
Est92 ] is consistent with the geometric orMarcus [
Mar78 , Mar81 , KPP95 , App09 , FS17 ] interpretation of ( ) with jump behavior S t “ e ∆ X t S t ´ .From a stochastic differential geometry point of view, one aims for an intrinsic understanding of ( )valid for arbitrary Lie groups. For instance, if X takes values in any sub Lie algebra L Ă T , then S takes values in the group G “ exp L . In case of a d -dimensional semimartingale X , the minimal choiceis Lie pp R d qq , see e.g. [ Reu03 ], the resulting log-Lie structure of iterated integrals (both in the smoothand Stratonovich semimartingale case) is well-known. The extrinsic linear ambient space T Ą exp L will be important to us. Indeed, writing S t “ Sig p X q ,t for the (unique, global) T -valued solutionof ( ) driven by T -valued X , started at S “
1, we define, whenever Sig p X q ,T is (componentwise)integrable, the expected signature and signature cumulants (SigCum) µµµ p T q – E p Sig p X q ,T q P T , κκκ p T q – log µµµ p T q P T . Already when X is deterministic, and sufficiently regular to make ( ) meaningful, this leads toan interesting (ordinary differential) equation for κκκ with accompanying (Magnus) expansion, wellunderstood as effective computational tool [ IMKNZ05 , BCOR09 ]. The importance of the stochastic case X “ X p ω q , with expectation and logarithm thereof, was developed by Lyons and coworkers; see [ Lyo14 ]and references therein, with a variety of applications, ranging from machine learning to numericalalgorithms on Wiener space known as cubature, see e.g. [
LV04 ]. In case of d “ X “ p , X, , . . . q with a single scalar semimartingale X , this is nothing but the sequence of moments and cumulants of thereal valued random variable X T ´ X . When d ą
1, expected signature / cumulants provides an effectiveway to describe the process X on r , T s , see [ LQ11 , Lyo14 , CL16 ]. The question arises how to compute.If one takes X as d -dimensional Brownian motion, the signature cumulant κκκ p T q equals p T { q I d , where I d is the identity 2-tensor over R d . This is known as Fawcett’s formula , [
LV04 , FH20 ]. Loosely speaking,and postponing precise definitions, our main result is a vast generalization of Fawcett’s formula.
Theorem 1.1 (FunctEqu S -SigCum) . For sufficiently integrable X P S p T q , the (time- t ) conditionalsignature cumulants κκκ t p T q ” κκκ t – log E t p Sig p X q t,T q , is the unique solution of the functional equation κκκ t p T q “ E t " ż p t,T s H p ad κκκ u ´ qp d X u q ` ż Tt H p ad κκκ u ´ qp d x X c y u q` ż Tt H p ad κκκ u ´ q ˝ Q p ad κκκ u ´ qp d (cid:74) κκκ, κκκ (cid:75) cu q ` ż Tt H p ad κκκ u ´ q ˝ p Id d G p ad κκκ u ´ qqp d (cid:74) X , κκκ (cid:75) cu q` ÿ t ă u ď T ˆ H p ad κκκ u ´ q ´ exp p ∆ X u q exp p κκκ u q exp p´ κκκ u ´ q ´ ´ ∆ X u ¯ ´ ∆ κκκ u ˙* , (1.2) Diamond notation for Marcus SDEs, d S “ S ˛ d X , cf. [ App09 ], will not be used here to avoid notational clashwith [
AGR20 , FGR20 ]. NIFIED SIGNATURE CUMULANTS AND GENERALIZED MAGNUS EXPANSIONS 3 where all integrals are understood in Itô- and Riemann–Stieltjes sense respectively. The functions
H, G, Q are defined in ( ) below, cf. also Section for further notation. As displayed in Figures 1 and 2, this theorem has an avalanche of consequences on which we nowcomment. ‚ Equation ( ) allows to compute κκκ p n q “ π p n q p κκκ q P p R d q b n as function of κκκ p q , . . . , κκκ p n ´ q . (Thisremark applies mutatis mutandis to all special cases seen as vertices in Figure 1.) The resultingexpansions, displayed in Figure , are of computational interest. ‚ The most classical consequence of ( ) appears when X is a deterministic continuous semimartin-gale, i.e. X P F V c p T q , which also covers the absolutely continuous case with, X P L p T q .In this case all bracket terms and the final jump-sum disappear. What remains is a classicaldifferential equation due to [ Hau06 ], here in backward form ´ d κκκ t p T q “ H p ad κκκ t q d X t , ´ κκκ t p T q “ H p ad κκκ t q X t , (1.3)the accompanying expansions is then precisely Magnus expansion [ Mag54 , IN99 , IMKNZ05 , BCOR09 ]. By taking X continuous and piecewise linear on two adjacent intervals, say r , q Yr , q , one obtains the Baker–Campbell–Hausdorff formula (see e.g. [ Mil72 , Theorem 5.5]) κκκ p q “ log ` exp p x q exp p x q ˘ — BCH p x , x q“ x ` ż Ψ p exp p ad t x q ˝ exp p ad x qqp x q d t, (1.4)with Ψ p z q – ln p z q z ´ “ ÿ n ě p´ q n n ` p z ´ q n It is also instructive to let X piecewise constant on these intervals, with ∆ X “ x , ∆ X “ x ,in which case ( ) reduces to the first equality in ( ). Such jump variations of the Magnusexpansion are discussed in Section . ‚ Writing x ÞÑ ˆ x for the projection from T to the symmetric algebra S as the linear spaceidentified with symmetric tensor series, equation ( ), in its projected and commutative formbecomesFunctEqu S -Cum: ˆ κκκ t p T q “ E t " ˆ X t,T ` A p ˆ X ` ˆ κκκ q c E t,T ` ÿ t ă u ď T ˆ exp ´ ∆ ˆ X u ` ∆ˆ κκκ u ¯ ´ ´ p ∆ ˆ X u ` ∆ˆ κκκ u q ˙* (1.5)where ˆ X is a S -valued semimartingale, and exp : S ÞÑ S defined by the usual power series.This includes of course semimartingales with values in R d , canonically embedded in S . Moreinterestingly, the case ˆ X “ p , aX, b x X y , , . . . q , for a d -dimensional continuous martingale X can be seen to underlie the expansions of [ FGR20 ], which improves and unifies previousresults [
LRV19 , AGR20 ], treating p a, b q “ p , q and p a, b q “ p , ´ { q , with motivation fromQFT and mathematical finance, respectively. Following Gatheral and coworkers, ( ) andsubsequent expansions involve “diamond” products of semimartingales, given, whenever well-defined, by p A ˛ B q t p T q – E t ` x A c , B c y t,T ˘ . All this is discussed in Section . With regard to the existing (commutative) literature, ouralgebraic setup is ideally suited to work under finite moment assumptions, we are able to dealwith jumps, not treated in [
LRV19 , AGR20 ]. Equation ( ) has a remarkable interpretationin that it can be viewed as (with jumps: generalized) infinite-dimensional Riccati differentialequation and indeed reduces to the finite-dimensional equation when specialized to (sufficientlyintegrable) “affine” continuous (resp. general) semimartingales [
DFS ` , CFMT11 , KRST11 ].Of recent interest, explicit diamond expansions have been obtained for “rough affine” processes,non-Markov by nature, with cumulant generating function characterized by Riccati Volterraequations, see [
AJLP19 , GKR19 , FGR20 ]. It is remarkable that analytic tractability remainsintact when one passes to path space and considers signature cumulants, Section . P. FRIZ, P. HAGER, AND N. TAPIA S c -SigCum F V c HausdorffFunctEqu S -SigCum F V
ODEHausdorffFunctEqu S c -Cum trivial FunctEqu S -Cum trivial continuouscommutative deterministic Figure 1.
FunctEqu S -SigCum (Theorem ) and implications S c -SigCum F V c MagnusRecursion S -SigCum F V
MagnusExpansionDiamondExpansion trivial
Recursion S -Cum trivial continuouscommutative deterministic Figure 2.
Computational consequence: accompanying recursions
Acknowledgment.
PKF has received funding from the European Research Council (ERC) underthe European Union’s Horizon 2020 research and innovation program (grant agreement No. 683164)and the DFG Research Unit FOR 2402. PH and NT are supported by the DFG MATH ` ExcellenceCluster. 2.
Preliminaries
The tensor algebra and tensor series.
Denote by T p R d q the tensor algebra over R d , i.e. T p R d q – à k “ p R d q b k , elements of which are finite sums (a.k.a. tensor polynomials) of the form x “ ÿ k ě x p k q “ ÿ w P W d x w e w (2.1)with x p k q P p R d q b k , x w P R and linear basis vectors e w – e i ¨ ¨ ¨ e i k P p R d q b k where w ranges over allwords w “ i ¨ ¨ ¨ i k P W d over the alphabet t , . . . , d u . Note x p k q “ ř | w |“ k x w e w where | w | denotes thelength a word w . The element e H “ P p R d q b – R is neutral element of the concatenation (a.k.a.tensor) product, is obtained by linear extension of e w e w “ e ww where ww P W d denotes concatenationof two words. We thus have, for x , y P T p R d q , xy “ ÿ k ě k ÿ ‘ “ x p ‘ q y p k ´ ‘ q “ ÿ w P W d ˜ ÿ w w “ w x w y w ¸ e w P T p R d q . Here ˝ denotes composition, not to be confused with Stratonovich integration ˝ d X . NIFIED SIGNATURE CUMULANTS AND GENERALIZED MAGNUS EXPANSIONS 5
This extends naturally to infinite sums, a.k.a tensor series, elements of the “completed” tensor algebra T – T pp R d qq – ź k “ p R d q b k , which are written as in ( ), but now as formal infinite sums with identical notation and multiplicationrules; the resulting algebra T obviously extends T p R d q . For any n P N ě define the projection to tensorlevels by π n : T Ñ p R d q b n , x ÞÑ x p n q . Denote by T and T the subspaces of tensor series starting with 0 and 1 respectively; that is, x P T (resp. T ) if and only if x H “ x H “ T and T respectively, the exponentialand logarithm in T , defined by the usual series,exp : T Ñ T , x ÞÑ exp p x q – ` ÿ k “ k ! p x q k , log : T Ñ T , ` x ÞÑ log p ` x q – ÿ k “ p´ q k ` k p x q k , are globally defined and inverse to each other. The vector space T becomes a Lie algebra with r x , y s – xy ´ yx , ad y : T Ñ T , x ÞÑ r y , x s . Its exponential image T “ exp p T q is a Lie group, at least formally so. We refrain from equippingthe infinite-dimensional T with a differentiable structure, not necessary in view of the “locally finite”nature of the group law p x , y q ÞÑ xy .Let p a k q k ě be a sequence of real numbers then we can always define a linear operator on T by « ÿ k ě a k p ad x q k ff : T Ñ T , y ÞÑ ÿ k ě a k p ad x q k p y q , where p ad x q “ Id is the identity operator and p ad x q n “ ad x ˝ p ad x q n ´ for any n P N ě . Indeed,there is no convergence issue due to the graded structure as can be seen by projecting to some tensorlevel n P N ě π n ˜ ÿ n ě a k p ad x q k p y q ¸ “ n ´ ÿ k “ a k π n ` p ad x q k p y q ˘ “ a y p n q ` n ´ ÿ k “ a k ÿ } ‘ }“ n, | ‘ |“ k ` p ad x p l q ˝ ¨ ¨ ¨ ˝ ad x p l k ` q qp y p l q q , (2.2)where the inner summation in the right-hand side is over a finite set of multi-indices ‘ “ p l , . . . , l k ` q Pp N ě q k ` where | ‘ | – k ` } ‘ } – l ` ¨ ¨ ¨ ` l k ` . In the following we will simply write p ad x ad y q ”p ad x ˝ ad y q for the composition of adjoint operators. Further, when ‘ “ p l q is a multi-index oflength one, we will use the notation p ad x p l q ¨ ¨ ¨ ad x p l k ` q q ” Id. Note also that the iteration of adjointoperations can be explicitly expanded in terms of left- and right-multiplication as followsad y p l q ¨ ¨ ¨ ad y p l k ` q p x p l q u q “ ÿ I Y J “t ,...,k u p´ q | J | ˜ź i P I y p l i ` q ¸ x p l q u ˜ź j P J y p l j ` q ¸ . (2.3)For a word w P W d with | w | ą f : T Ñ R by pB w f qp a q – B t p f p a ` te w qq ˇˇ t “ , for any a P T such that the right-hand derivative exists.Write m : T b T Ñ T for multiplication (concatenation) map, i.e. m p a b b q “ ab , in general differentfrom ba , extended by linearity. For linear maps g, f : T Ñ T we define g d f “ m ˝ p g b f q , i.e. p g d f qp a b b q “ g p a q f p b q , a, b P T , extended by linearity. P. FRIZ, P. HAGER, AND N. TAPIA
Some quotients of the tensor algebra.
The symmetric algebra over R d , denoted by S p R d q is the quotient of T p R d q by the two-sided ideal I generated by t xy ´ yx : x, y P R d u . The canonicalprojection T p R d q (cid:16) S p R d q , x ÞÑ ˆ x , is an algebra epimorphism. A linear basis of S p R d q is then givenby t ˆ e w u over non-decreasing words, w “ p i , . . . , i n q P x W d , with 1 ď i ď ¨ ¨ ¨ ď i n ď d, n ě
0. Every˜ x P S p R d q can be written as finite sum, ˜ x “ ÿ w P x W d ˜ x w ˆ e w , and we have an immediate identification with polynomials in d commuting indeterminates. Thecanonical projection map extends to an epimorphism T (cid:16) S where T “ T pp R d qq and S “ S pp R d qq are the respective completions, identifiable as formal series in d non-commuting (resp. commuting)indeterminates. As a vector space, S can be identified with symmetric formal tensor series. Denote by S and S the affine space determined by ˜ x H “ x H “ S define y exp : S Ñ S with inverse x log : S Ñ S and we have { exp p x ` y q “ y exp p ˆ x q y exp p ˆ y q , x , y P T { log p xy q “ x log p ˆ x q ` x log p ˆ y q , x , y P T . We shall abuse notation in what follows and write exp (resp. log), instead of y exp (resp. x log).2.2.1. The (step- n ) truncated tensor algebra. For n P N , the subspace I n – ź k “ n ` p R d q b k is a two sided ideal of T . Therefore, the quotient space T { I n has a natural algebra structure. Wedenote the projection map by π p ,n q . We can identify T { I n with T n – n à k “ p R d q b k , equipped with truncated tensor product, xy “ n ÿ k “ ÿ ‘ ` ‘ “ k x p ‘ q y p ‘ q “ ÿ w P W d , | w |ď n ˜ ÿ w w “ w x w y w ¸ e w P T n . The sequence of algebras p T n : n ě q forms an inverse system with limit T . There are also canonicalinclusions T k ã Ñ T n for k ď n ; in fact, this forms a direct system with limit T p R d q . The usual powerseries in T n define exp n : T n Ñ T n with inverse log n : T n Ñ T n , we may again abuse notation andwrite exp and log when no confusion arises. As before, T n has a natural Lie algebra structure, and T n (now finite dimensional) is a bona fide Lie group.We equip T p R d q with the norm | a | T p R d q – max k P N | a p n q | p R d q b k , where | ¨ | p R d q b k is the euclidean norm on p R d q b k – R d k , which makes it a Banach space. The samenorm makes sense in T n , and since the definition is consistent in the sense that | a | T k “ | a | T n for any a P T n and k ě n and | a | T n “ | a | p R d q b n for any a P p R d q b n . We will drop the index whenever it ispossible and write simply | a | .2.3. Semimartingales.
Let D be the space of adapted càdlàg process X : Ω ˆ r , T q Ñ R with T P p , defined on some filtered probability space p Ω , p F t q ď t ď T , P q . The space of semimartingales S is given by the processes X P D that can be decomposed as X t “ X ` M t ` A t , where M P M loc is a càdlàg local martingale, and A P V is a càdlàg adapted process of locally boundedvariation, both started at zero. Recall that every X P S has a well-defined continuous local martingalepart denoted by X c P M c loc . The quadratic variation process of X is then given by r X s t “ x X c y t ` ÿ ă u ď t p ∆ X u q , ď t ď T, NIFIED SIGNATURE CUMULANTS AND GENERALIZED MAGNUS EXPANSIONS 7 where x¨y denotes the (predictable) quadratic variation of a continuous semimartingale. Covariationsquare resp. angle brackets r X, Y s and x X c , Y c y , for another real-valued semimartingale Y , are definedby polarization. For q P r , , write L q “ L q p Ω , F , P q , then a Banach space H q Ă S is given bythose X P S with X “ } X } H q – inf X “ M ` A ›››› r M s { T ` ż T | d A s | ›››› L q ă 8 . Note that for local martingale M P M loc it holds (see [ Pro05 , Ch. V, p. 245]) } M } H q “ ››› r M s { T ››› L q . For a process X P D we define } X } S q – ››› sup ď t ď T | X t | ››› L q and define the space S q Ă S of semimartingales X P S such that } X } S q ă 8 . Note that there exitsa constant c q ą q such that (see [ Pro05 , Ch. V, Theorem 2]) } X } S q ď c q } X } H q . (2.4)We view d -dimensional semimartingales, X “ ř di “ X i e i P S p R d q , as special cases of tensor seriesvalued semimartingales S p T q of the form X “ ÿ w P W d X w e w with each component X w a real-valued semimartingale. (This extends mutatis mutandis to the spaces D , M , V . Note also that we typically deal with T -valued semimartingales which amounts to have onlywords with length | w | ě X c and jump process∆ X t “ X t ´ X t ´ are defined componentwise. Brackets : Now let X and Y be T -valued semimartingales. We define the (non-commutative) outerquadratic covariation bracket of X and Y by (cid:74) X , Y (cid:75) t – ÿ w ,w P W d r X w , Y w s t e w b e w P T b T . Similarly, define the (non-commutative) inner quadratic covariation bracket by r X , Y s t – m p (cid:74) X , Y (cid:75) q “ ÿ w P W d ˜ ÿ w w “ w r X w , Y w s t ¸ e w P T ;for continuous T -valued semimartingales X , Y , this coincides with the predictable quadratic covariation x X , Y y t – ÿ w P W d ˜ ÿ w w “ w x X w , Y w y t ¸ e w P T . As usual, we may write (cid:74) X (cid:75) ” (cid:74) X , X (cid:75) and x X y ” x X , X y . H -spaces : The definition of H q -norm naturally extends to tensor valued martingales. Moreprecisely, for X p n q P S pp R d q b n q with n P N ě and q P r , we define } X p n q } H q – } X p n q } H q pp R d q b n q – inf X p n q “ M ` A ››› |r M s| { T ` | A | ´ var; r T s ››› L q , where the infimum is taken over all possible decompositions X p n q “ M ` A with M P M loc pp R d q b n q and A P V pp R d q b n q , where | A | ´ var; r T s – sup ď t 﨨¨ď t k ď T ÿ t i ˇˇ A t i ` ´ A t i ˇˇ ď ÿ w P W d , | w |“ n ż T | d A ws | , with the supremum taken over all partitions of the interval r , T s . One may readily check that } X p n q } H q ď ÿ w P W d , | w |“ n } X w } H q ; and for X p n q P M loc : } X p n q } H q “ }|r X p n q s| T } L q . Further define the following subspace H q,N Ă S p T N q of homogeneously integrable semimartingales H q,N – ! X P S p T N q ˇˇˇ X “ , ||| X ||| H q,N ă 8 ) , P. FRIZ, P. HAGER, AND N. TAPIA where for any X P S p T N q we define ||| X ||| H q,N – N ÿ n “ ` } X p n q } H qN { n ˘ { n . Note that |||¨||| H q,N is sub-additive and positive definite on H q,N and it is homogeneous under dilationin the sense that ||| δ λ X ||| H q,N “ | λ | ||| X ||| H q,N , δ λ X – p X p q , λ X p q , . . . , λ N X p N q q , λ P R . We also introduce the following subspace of S p T q H p T q – t X P S p T q : X w P H q , @ ď q ă 8 , w P W d u . Note that if X P S p T q such that ||| X p ,N q ||| H ,N ă 8 for all N P N ě then it also holds X P H p T q . Stochastic integrals : We are now going to introduce a notation for the stochastic integration withrespect to tensor valued semimartingales. Let F : Ω ˆ r , T s Ñ L p T ; T q with p t, ω q ÞÑ F t p ω ; ¨q suchthat it holds p F t p x qq ď t ď T P D p T q , for all x P T (2.5)and F t p ω ; I n q Ă I n , for all n P N , p ω, t q P Ω ˆ r , T s , (2.6)where I n Ă T was introduced in Section , consisting of series with tensors of level n and higher.In this case, we can define the stochastic Itô-integral (and then analogously the Stratonovich/Marcusintegral) of F with respect to X P S p T q by ż p , ¨s F t ´ p d X t q : “ ÿ w P W d ÿ v P W d , | v |ď| w | ż p , ¨s F t ´ p e v q w d X vt e w P S p T q . (2.7)For example, let Y , Z P D p T q and define F : “ Y Id Z , i.e. F t p x q “ Y t x Z t for all x P T . Then we seethat F indeed satisfies the conditions ( ) and ( ) and we have ż p , ¨s p Y t ´ Id Z t ´ qp d X t q “ ż p , ¨s Y t ´ d X t Z t ´ “ ÿ w P W d ˜ ÿ w w w “ w ż p , ¨s Z w t ´ Y w t ´ d X w t ¸ e w . (2.8)Another important example is given by F “ p ad Y q k for any Y P D p T q and k P N . Indeed, weimmediately see F satisfies the condition ( ) and recalling from ( ) that the iteration of adjointoperations can be expanded in terms of left- and right-multiplication, we also see that F satisfies ( ).More generally, let p a k q k “ Ă R and let X P S p T q , then the following integral ż p , ¨s « ÿ k “ a k p ad Y t ´ q k ff p d X t q “ ÿ n “ n ´ ÿ k “ ÿ } ‘ }“ n, | ‘ |“ k ` ż p , ¨s ad Y p l q t ´ ¨ ¨ ¨ ad Y p l k ` q t ´ p d X p l q t q (2.9)is well define in the sense ( ). The definition of the integral with integrands of the form F : Ω ˆr , T s Ñ L p T b T ; T q with respect to processes X P S p T b T q is completely analogous. Quotient algebras:
All of this extends in a straight forward way to the case of semimartingales inthe quotient algebra of Section , i.e. symmetric and truncated algebra. In particular, given X and Y in S p S q have well-defined continuous local martingale parts denoted by X c , Y c respectively, with inner (predictable) quadratic covariation given by x X c , Y c y “ ÿ w ,w P x W d x X w ,c , Y w ,c y ˆ e w ˆ e w . Write S N for the truncated symmetric algebra, linearly spanned by t ˆ e w : w P x W d , | w | ď N u and S N for those elements with zero scalar entry. In complete analogy with non-commutative settingdiscussed above, we then write x H q,N Ă S p S N q for the corresponding space homogeneously q -integrablesemimartingales. NIFIED SIGNATURE CUMULANTS AND GENERALIZED MAGNUS EXPANSIONS 9
Diamond Products.
We extend the notion of the diamond product introduced in [
AGR20 ] forcontinuous scalar semimartingales to our setting.
Definition 2.1.
For X and Y in S p T q define p X ˛ Y q t p T q – E t ` x X c , Y c y t,T ˘ “ ÿ w P W d ˜ ÿ w w “ w p X w ˛ Y w q t p T q ¸ e w P T whenever the T -valued quadratic covariation which appears on the right-hand side is integrable. Similarto the previous section, we also define an outer diamond , for X , Y P T , by p X ˛ Y q t p T q – E t p (cid:74) X c , Y c (cid:75) t,T q “ ÿ w ,w P W d p X w ˛ Y w q t p T q e w b e w P T b T . This definition extends immediately to semimartingales with values in the quotient algebras ofSection . In particular, given ˜ X and ˜ Y in S p S q , we have p ˜ X ˛ ˜ Y q t p T q – E t ` x ˜ X c , ˜ Y c y t,T ˘ “ ÿ w ,w P W d p ˜ X w ˛ ˜ Y w q t p T q ˆ e w ˆ e w P S , where the last expression is given in terms of diamond products of scalar semimartingales. Lemma 2.2.
Let p, q, r
P r , such that { p ` { q ` { r ă and let X P M c loc pp R d q b l q , Y P M c loc pp R d q b m q , and Z P D pp R d q b n q with l, m, n P N , such that } X } H p , } Y } H q , } Z } S r ă 8 then it holds for all ď t ď T E t ˜ż Tt Z u ´ d p X ˛ Y q u p T q ¸ “ ´ E t ˜ż Tt Z u ´ d x X, Y y u ¸ . Proof.
Using the Kunita-Watanabe inequality (Lemma ) we see that the expectation on the righthand side is well defined. Further note that it follows from Emery’s inequality (Lemma ) and Doob’smaximal inequality that the local martingale ż ¨ Z u ´ d p E u x X, Y y T q is a true martingale. Recall the definition of the diamond product and observe that the difference ofleft- and right-hand side of the above equation is a conditional expectation of a martingale intermentand is hence zero. (cid:3) Generalized signatures.
We now give the precise meaning of ( ), that is d S “ S ˝ d X , orcomponent-wise, for every word w P W d ,d S w “ ÿ w w “ w S w ˝ d X w , where the driving noise X is a T -valued semimartingale, so that X H ”
0. Following [
Mar78 , Mar81 , KPP95 , FS17 , BCEF20 ] the integral meaning of this equation, started at time s from ξ P T , for times t ě s , is given by S t “ ξ ` ż p s,t s S u ´ d X u ` ż ts S u ´ d x X c y u ` ÿ s ă u ď t S u ´ ` exp p ∆ X u q ´ ´ ∆ X u ˘ , (2.10)leaving the component-wise version to the reader. We have Proposition 2.3.
Let ξ P T and suppose X takes values in T . For every s ě and ξ P T , equation ( ) has a unique global solution on T starting from S s “ ξ .Proof. Note that S solves ( ) iff ξ ´ S solves the same equation started from 1 P T . We may thustake ξ “ X “ p , X, X , . . . q in ( ) has no scalar component, shows that the (necessarily) unique solutionis given explicitly by iterated integration, as may be seen explicitly when writing out S p q ” S p q t “ ş ts d X “ X s,t P R d , S p q t “ ż p s,t s S p q u ´ d X u ` X t ´ X s ` x X c y s,t ` ÿ s ă u ď t p ∆ X u q P p R d q b . and so on. (In particular, we do not need to rely on abstract existence, uniqueness results for MarcusSDEs [ KPP95 ] or Lie group stochastic exponentials [
HDL86 ].) (cid:3)
Definition 2.4.
Let X be a T -valued semimartingale defined on some interval r s, t s . ThenSig p X | r s,t s q ” Sig p X q s,t is defined to be the unique solution to ( ) on r s, t s , such that Sig p X q s,s “ Lemma 2.5.
Let X be a T -valued semimartingales on r , T s and ď s ď t ď u ď T . Then thefollowing identity holds with probability one, for all such s, t, u , Sig p X q s,t Sig p X q t,u “ Sig p X q s,u . (2.11) Proof.
Call Φ t Ð s ξ – S t the solution to ( ) at time t ě s , started from S s “ ξ . By uniqueness of thesolution flow, we have Φ u Ð t ˝ Φ t Ð s “ Φ u Ð s . It now suffices to remark that, thanks to the multiplicativestructure of ( ) we have Φ t Ð s ξ “ ξ Sig p X q s,t . (cid:3) Expected signatures and signature cumulants
Definitions and existence.
Throughout this section let X P S p T q be defined on a filteredprobability space p Ω , F , p F t q ď t ď T , P q . When E p| Sig p X q w ,t |q ă 8 for all 0 ď t ď T and all words w P W d , then the (conditional) expected signature µµµ t p T q – E t p Sig p X q t,T q “ ÿ w P W d E t p Sig p X q wt,T q e w P T , ď t ď T, is well defined with E t denoting the conditional expectation with respect to the sigma algebra F t . Inthis case, we can also define the (conditional) signature cumulant of X by κκκ t p T q – log p E t p µµµ t p T qqq P T , ď t ď T. An important observation is the following
Lemma 3.1.
Given E p| Sig p X q w ,t |q ă 8 for all ď t ď T and words w P W d , then µµµ p T q P S p T q and κκκ p T q P S p T q .Proof. It follows from the relation ( ) that µµµ t p T q “ E t p Sig p X q t,T q “ E t ` Sig p X q ´ ,t Sig p X q ,T ˘ “ Sig p X q ´ ,t E t p Sig p X q ,T q . Therefore projecting to the tensor components we have µµµ t p T q w “ ÿ w w “ w p´ q | w | S p X q w ,t E t ´ S p X q w ,T ¯ , ď t ď T, w P W d . Since p Sig p X q w ,t q ď t ď T and p E t p Sig p X q w ,T q ď t ď T are semimartingales (the latter in fact a martingale),it follows from Itô’s product rule that µµµ w p T q is also a semimartingale for all words w P W d , hence µµµ p T q P S p T q . Further recall that κκκ p T q “ log p µµµ p T qq and therefore it follows from the definition ofthe logarithm on T that each component κκκ p T q w with w P W d is a polynomial of p µµµ p T q v q v P W d , | v |ď| w | .Hence it follows again by Itô’s product rule that κκκ p T q P S p T q . (cid:3) It is of strong interest to have a more explicit necessary condition for the existence of the expectedsignature. The following theorem below, the proof of which can be found in Section , yields such acriterion.
Theorem 3.2.
Let q P r , and N P N ě , then there exist two constants c, C ą depending only on d , N and q , such that for all X P H q,N c ||| X ||| H q,N ď ||| Sig p X q , ¨ ||| H q,N ď C ||| X ||| H q,N . In particular, if X P H p T q then Sig p X q , ¨ P H p T q and the expected signature exists. Remark 3.3.
Let X “ p , M, , . . . , q where M P M p R d q is a martingale, then ||| X ||| H q,N “ } M } H qN “ }|r M s T | { } L qN , and we see that the above estimate implies that max n “ ,...,N } Sig p X q p n q , ¨ } { n S qN { n ď C } M } H qN . This estimate is already known and follows from the Burkholder-Davis-Gundy inequality for enhancedmartingales, which was first proved in the continuous case in [
FV06 ] and for the general case in [
CF19 ]. NIFIED SIGNATURE CUMULANTS AND GENERALIZED MAGNUS EXPANSIONS 11
Remark 3.4.
When q ą , the above estimate also holds true when the signature Sig p X q , ¨ is replacedby the conditional expected signature µµµ p T q or the conditional signature cumulant κκκ p T q . This will beseen in the proof of Theorem below (more precisely in Claim ). Moments and cumulants.
We quickly discuss the development of a symmetric algebra valuedsemimartingale, more precisely ˜ X P S p S q , in the group S . That is, we considerd ˜ S “ ˜ S ˝ d ˜ X . (3.1)It is immediate (validity of chain rule) that the unique solution to this equation, at time t ě s ,started at ˜ S s “ ˜ ξ P S is given by ˜ S t – exp ` ˜ X t ´ ˜ X s ˘ ˜ ξ P S and we also write ˜ S s,t “ exp ` ˜ X t ´ ˜ X s ˘ for this solution started at time s from 1 P S . The relation tosignatures is as follows. Recall that the hat denotes the canonical projection from T to S . Proposition 3.5. (i) Let X , Y P S p T q and Z “ ş X d Y in Itô sense. Then ˆ X , ˆ Y P S p S q and, in thesense of indistinguishable processes, p Z “ ż ˆ X d ˆ Y . (3.2) (ii) Let X P S p T q . Then { Sig p X q s, ¨ solves ( ) started at time s from P S and driven by ˆ X P S p S q .In particular { Sig p X q s,t “ exp p ˆ X t ´ ˆ X s q .Proof. (i) That the projections ˆ X , ˆ Y define S -valued semimartingales follows from the componentwisedefinition and the fact that the canonical projection is linear. In particular, the right-hand side ofeq. ( ) is well defined. Now, eq. ( ) is true whenever X is piece-wise constant. By a limitingprocedure, we immediately see that it is also true for general semimartingales. Part (ii) is thenimmediate. (cid:3) Assuming componentwise integrability, we then define symmetric moments and cumulants by˜ µµµ t p T q – E t exp ` ˜ X T ´ ˜ X t ˘ “ ÿ w E t ´ exp ` ˜ X T ´ ˜ X t ˘ wt,T ¯ ˆ e w P S , ˜ κκκ t p T q – log ˜ µµµ t p T q P S , ď t ď T. If ˜ X “ ˆ X , for X P S p T q , with expected signature and signature cumulants µµµ and κκκ , it is then clearthat the symmetric moments and cumulants of ˆ X are obtained by projection, µµµ ÞÑ ˆ µµµ, κκκ ÞÑ ˆ κκκ. Example 3.6.
Let X be an R d -valued martingale in H , and ˜ X t – ř di “ X it ˆ e i . Then˜ µµµ t p T q “ ÿ n “ n ! E t p X T ´ X t q n , consists of the (time- t conditional) multivariate moments of X T ´ X t P R d . And it readily follows, alsonoted in [ BO20 , Example 3.3], that ˜ κκκ t p T q “ log µµµ t p T q consists precisely of the multivariate cumulantsof X T ´ X t . Note that the symmetric moments and cumulants of the scaled process aX , a P R , isprecisely given by δ a µµµ and δ a κκκ where the linear dilation map is defined by δ a : ˆ e w ÞÑ a | w | ˆ e w . Thesituation is similar for a ¨ X , a P R d , but now with δ a : ˆ e w ÞÑ a w ˆ e | w | with a w “ a n ¨ ¨ ¨ a n d d where n i denotes the multiplicity of the letter i P t , . . . , d u in the word w . ♦ We next consider linear combinations, ˜ X “ aX ` b x X y , for general pairs a, b P R , having already dealtwith b “
0. The special case b “ ´ a {
2, by scaling there is no loss in generality to take p a, b q “ p , ´ { q ,yields a (at least formally) familiar exponential martingale identity. Example 3.7.
Let X be an R d -valued martingale in H , and define˜ X t – d ÿ i “ X it ˆ e i ´ ÿ ď i ď j ď d x X i , X j y t ˆ e ij . In this case we have trivial symmetric cumulants, ˜ κκκ t p T q “ ď t ď T . Indeed, Itô’s formulashows that t ÞÑ exp ` ˜ X t ˘ is an S -valued martingale, so that˜ µµµ t p T q “ E t exp p ˜ X T ´ ˜ X t q “ exp p´ ˜ X t q E t exp p ˜ X T q “ . ♦ While the symmetric cumulants of the last example carries no information, it suffices to work with˜ X “ d ÿ i “ a i X ` d ÿ j,k “ b jk x X j , X k y in which case µµµ “ µµµ p a, b q , κκκ “ κκκ p a, b q contains full information of the joint moments of X and itsquadratic variation process. A recursion of these was constructed as diamond expansion in [ FGR20 ].4.
Main Results
Functional equation for signature cumulants.
Let X P S p T q defined one a filtered proba-bility space p Ω , F , p F t q ď t ď T ă8 , P q satisfying the usual conditions. For all x P T (or T N ) define thefollowing operators, with Bernoulli numbers p B k q k ě “ p , ´ , . . . q , G p ad x q “ ÿ k “ p ad x q k p k ` q ! , Q p ad x q “ ÿ m,n “ p ad x q n d p ad x q m p n ` q ! p m q ! p n ` m ` q ,H p ad x q – ÿ k “ B k k ! p ad x q k , (4.1)noting G p z q “ p exp p z q ´ q{ z , H p z q “ G ´ p z q “ z {p exp p z q ´ q . Our main result is the following Theorem 4.1.
Let X P H p T q , then the signature cumulant κκκ “ κκκ p T q “ p log E t p Sig p X q t,T qq ď t ď T is the unique solution (up to indistinguishably) of the following functional equation: for all ď t ď T “ E t " X t,T ` x X c y t,T ` ż p t,T s G p ad κκκ u ´ qp d κκκ u q ` ż Tt Q p ad κκκ u ´ qp d (cid:74) κκκ c , κκκ c (cid:75) u q` ż Tt p Id d G p ad κκκ u ´ qqp d (cid:74) X c , κκκ c (cid:75) u q` ÿ t ă u ď T ´ exp p ∆ X u q exp p κκκ u q exp p´ κκκ u ´ q ´ ´ ∆ X u ´ G p ad κκκ u ´ qp ∆ κκκ u q ¯* . (4.2) Equivalently, κκκ “ κκκ p T q is the unique solution to κκκ t “ E t " ż p t,T s H p ad κκκ u ´ qp d X u q ` ż Tt H p ad κκκ u ´ qp d x X c y u q` ż Tt H p ad κκκ u ´ q ˝ Q p ad κκκ u ´ qp d (cid:74) κκκ c , κκκ c (cid:75) u q` ż Tt H p ad κκκ u ´ q ˝ p Id d G p ad κκκ u ´ qqp d (cid:74) X c , κκκ c (cid:75) u q` ÿ t ă u ď T ˆ H p ad κκκ u ´ q ´ exp p ∆ X u q exp p κκκ u q exp p´ κκκ u ´ q ´ ´ ∆ X u ¯ ´ ∆ κκκ u ˙* . (4.3) Furthermore, if X P H ,N for some N P N ě , then the identities ( ) and ( ) still hold true forthe truncated signature cumulant κκκ – p log E t p Sig p X p ,N q q t,T qq ď t ď T .Proof. We postpone the proof for the fact that κκκ satisfies the equations ( ) and ( ) to sectionSection . The uniqueness part of the statement can be easily seen as follows: Regarding equation( ) we first note that it holds E t p t,T s G p ad κκκ u ´ qp d κκκ u q + “ E t p t,T s p G p ad κκκ u ´ q ´ Id qp d κκκ u q + ´ κκκ t , ď t ď T, where we have used that κκκ T ” G , we can bring κκκ t to the left-hand side in ( ). This identity is an equality of tensor series in T and can be projectedto yield an equality for each tensor level of the series. As presented in more detail in the followingsubsection, we see that projecting the latter equation to tensor level say n P N ě , the right-hand sideonly depends on κκκ p k q for k ă n , hence giving an explicit representation κκκ p n q in terms of X and strictlylower tensor levels of κκκ . Therefore the equation ( ) characterizes κκκ up to a modification and then due NIFIED SIGNATURE CUMULANTS AND GENERALIZED MAGNUS EXPANSIONS 13 to right-continuity up to indistinguishably. The same argument applies to the equation ( ), referringto the following subsections for details on the recursion. (cid:3)
Diamond formulation:
The functional equations given in Theorem above, can be phrased interms of the diamond product between T -valued semimartingales. Writing J t p T q “ ř t ă u ď T p . . . q forthe last (jump) sum in ( ), this equation can be written, thanks to Lemma , which applies justthe same with outer diamonds,12 p X ˛ X q t p T q ` E t " X t,T ` ż p t,T s G p ad κκκ u ´ qp d κκκ u q ` J t p T q * “ E t " ż Tt Q p ad κκκ u ´ q d p κκκ ˛ κκκ q u p T q ` ż Tt p Id d G p ad κκκ u ´ qq d p X ˛ κκκ q u p T q * and a similar form may be given for ( ). While one may, or may not, prefer this equation to ( ),diamonds become very natural in d “ ).In this case G “ Id, Q “ Id d Id and with identities of the form ż Tt p Id d Id q d p X ˛ Y q u p T q “ p X ˛ Y q u p T q| Tu “ t “ ´p X ˛ Y q t p T q some simple rearrangement, using bilinearity of the diamond product, gives κκκ t p T q “ E t t X t,T u ` pp X ` κκκ q ˛ p X ` κκκ qq t p T q ` E t t J t p T qu . (4.4)If we further impose martingality and continuity, we arrive at κκκ t p T q “ pp X ` κκκ q ˛ p X ` κκκ qq t p T q . Recursive formulas for signature cumulants.
Theorem allows for an iterative computationof signature cumulants, trivially started from κκκ p q t “ µµµ p q t “ E t ´ X p q t,T ¯ . The second signature cumulant, obtained from Theorem , or from first principles, reads κκκ p q t “ E t " X p q t,T ` A X p q c E t,T ` ż p t,T s ” κκκ p q u ´ , d κκκ p q u ı ` A κκκ p q c E t,T ` A X p q c , κκκ p q c E t,T ` ÿ t ă u ď T ˆ ´ ∆ X p q u ¯ ` ∆ X p q u ∆ κκκ p q u ` ´ ∆ κκκ p q u ¯ ˙* For instance, consider the special case with vanishing higher order components, X p i q ”
0, for i ‰
1, and X “ X p q ” M , a d -dimensional continuous square-integrable martingale. In this case, κκκ p q “ µµµ p q ” κκκ and µµµ , we have κκκ p q “ µµµ p q ´ µµµ p q µµµ p q “ µµµ p q .It then follows from Stratonovich-Ito correction that κκκ p q t “ E t ż Tt p M u ´ M s q ˝ d M u “ E t x M y t,T “ E t A X p q E t,T which is indeed a (very) special case of the general expression for κκκ p q . We now treat general higherorder signature cumulants. Corollary 4.2.
Let X P H ,N for some N P N ě , then we have κκκ p q t “ E t ´ X p q t,T ¯ , for all ď t ď T and for n P t , . . . , N u we have recursively (the r.h.s. only depends on κκκ p j q , j ă n ) κκκ p n q t “ E t ´ X p n q t,T ¯ ` n ´ ÿ k “ E t ˆA X p k q c , X p n ´ k q c E t,T ˙ ` ÿ | ‘ |ě , } ‘ }“ n E t ´ Mag p κκκ ; ‘ q t,T ` Qua p κκκ ; ‘ q t,T ` Cov p X , κκκ ; ‘ q t,T ` Jmp p X , κκκ ; ‘ q t,T ¯ (4.5) with ‘ “ p l , . . . , l k q , l i P N ě , | ‘ | – k P N ě , } ‘ } – l ` ¨ ¨ ¨ ` l k and Mag p κκκ ; l , . . . , l k q t,T “ k ! ż p t,T s ad κκκ p l q u ´ ¨ ¨ ¨ ad κκκ p l k q u ´ p d κκκ p l q u q Qua p κκκ ; l , . . . , l k q t,T “ k ! k ÿ m “ ˆ k ´ m ´ ˙ ˆ ż Tt ´ ad κκκ p l q u ´ ¨ ¨ ¨ ad κκκ p l m q u ´ d ad κκκ p l m ` q u ´ ¨ ¨ ¨ ad κκκ p l k q u ´ ¯´ d (cid:114) κκκ p l q c , κκκ p l q c (cid:122) u ¯ Cov p X , κκκ ; l , . . . , l k q t,T “ p k ´ q ! ż Tt ´ Id d ad κκκ p l q u ´ ¨ ¨ ¨ ad κκκ p l k q u ´ ¯´ d (cid:114) X p l q c , κκκ p l q c (cid:122) u ¯ Jmp p X , κκκ ; l , . . . , l k q t,T “ ÿ t ă u ď T ÿ ď m ď j ď k ˜ p´ q k ´ j ∆ X p l q u ¨ ¨ ¨ ∆ X p l m q u κκκ p l m ` q u ¨ ¨ ¨ κκκ p l j q u κκκ p l j ` q u ´ ¨ ¨ ¨ κκκ p l k q u ´ m ! p m ´ j q ! p k ´ j q ! ¸ ´ k ! ad κκκ p l q u ´ ¨ ¨ ¨ ad κκκ p l k q u ´ ´ ∆ κκκ p l q u ¯ . Proof.
Recall from Section , more specifically ( ), the definition of the stochastic Itô integral ofa power series of adjoint operations with respect to a tensor valued semimartingale. As in the proofof Theorem above, in ( ), we can separate the identity from G and bring the resulting κκκ t tothe left-hand side. The recursion then follows from projecting the resulting form of the equation totensors of level n P t , . . . , N u . We demonstrate this projection for the first appearing term, which isthe stochastic integral with respect to κκκ . It holds π n E t p t,T s p G p ad κκκ u ´ q ´ Id qp d κκκ u q + “ E t $&% n ÿ k “ k ! ÿ } ‘ }“ n, | ‘ |“ k ż p t,T s ad κκκ p l q u ´ ¨ ¨ ¨ ad κκκ p l k q u ´ p d κκκ p l q u q ,.- “ E t $&% ÿ } ‘ }“ n Mag p κκκ ; ‘ q t,T ,.- , for all 0 ď t ď T , where in the first equality we have used the linearity to interchange π n with theexpectation and the explicit form of the projection of a power series of adjoint operations given in ( ).The projection of the remaining terms in equation ( ) follows analogously except for the jump part.Regarding the latter, we note again that due to the linearity we can interchange the projection π n withthe expectation and the sum over the interval p t, T s . The remaining steps in order to arrive at theabove form of the Jmp p X , κκκ q term are a simple combinatorial exercise. (cid:3) We obtain another recursion for the signature cumulants from projecting the functional equation( ). Note that, apart from the first two levels, it is far from trivial to see that the following recursionis equivalent to the recursion in Corollary . Corollary 4.3.
Let X P H ,N for some N P N ě , then we have κκκ p n q t “ E t ´ X p n q t,T ¯ ` ÿ | ‘ |ě , || ‘ ||“ n E t ˆ HMag p X , κκκ ; ‘ q t,T `
12 HMag p X , κκκ ; ‘ q t,T ` HQua p κκκ ; ‘ q t,T ` HCov p X , κκκ ; ‘ q t,T ` HJmp p X , κκκ ; ‘ q t,T ˙ (4.6) NIFIED SIGNATURE CUMULANTS AND GENERALIZED MAGNUS EXPANSIONS 15 with ‘ “ p l , . . . , l k q , l i ě , | ‘ | “ k , || ‘ || “ l ` ¨ ¨ ¨ ` l k and HMag p X , κκκ ; l , . . . , l k q t,T “ B k ´ p k ´ q ! ż p t,T s ad κκκ p l q u ´ ¨ ¨ ¨ ad κκκ p l k q u ´ ´ d X p l q u ¯ HMag p X , κκκ ; l , . . . , l k q t,T “ B k ´ p k ´ q ! ż Tt ad κκκ p l q u ´ ¨ ¨ ¨ ad κκκ p l k q u ´ ´ d A X p l q c , X p l q c E u ¯ HQua p κκκ ; l , . . . , l k q t,T “ ż Tt k ÿ j “ B k ´ j p k ´ j q ! ad κκκ p l j ` q u ´ ¨ ¨ ¨ ad κκκ p l k q u ´ p dQua p κκκ ; l , . . . , l j q u q HCov p X , κκκ ; l , . . . , l k ` q t,T “ ż Tt k ÿ j “ B k ´ j p k ´ j q ! ad κκκ p l j ` q u ´ ¨ ¨ ¨ ad κκκ p l k q u ´ ` dCov p X , κκκ ; l , . . . , l j q u ˘ HJmp p X , κκκ ; l , . . . , l k q t,T “ ÿ t ă u ď T ÿ ď m ď j ď i ď k p´ q k ´ j ˜ B k ´ i p k ´ i q ! ˆ ad κκκ p l i ` q u ´ ¨ ¨ ¨ ad κκκ p l k q u ´ ˜ ∆ X p l q u ¨ ¨ ¨ ∆ X p l m q u κκκ p l m ` q u ¨ ¨ ¨ κκκ p l j q u κκκ p l j ` q u ´ ¨ ¨ ¨ κκκ p l i q u ´ m ! p m ´ j q ! p k ´ j q ! ¸¸ . Proof.
The recursion follows from projecting the equation ( ) to each tensor level, analogously to theway that the recursion of Corollary follows from ( ) (see the proof of Corollary ). (cid:3) Diamonds.
All recursions here can be rewritten in terms of diamonds. In a first step, by definitionthe second term in Corollary can be rewritten as12 n ÿ k “ p X p k q ˛ X p n ´ k q q t p T q . Thanks to Lemma we may also write E t Qua p κκκ ; ‘ q t,T “ ´ E t k ! k ÿ m “ ˆ k ´ m ´ ˙ ż Tt ´ ad κκκ p ‘ q u ´ ¨ ¨ ¨ ad κκκ p ‘ m q u ´ d ad κκκ p ‘ m ` q u ´ ¨ ¨ ¨ ad κκκ p ‘ k q u ´ ¯´ d p κκκ p ‘ q ˛ κκκ p ‘ q q u p T q ¯+ . Similarly, E t Cov p X , κκκ ; ‘ q t,T “ ´ E t p k ´ q ! ż Tt ´ Id d ad κκκ p ‘ q u ´ ¨ ¨ ¨ ad κκκ p ‘ k q u ´ ¯´ d p X p ‘ q ˛ κκκ p ‘ q q u p T q ¯+ . Inserting these expressions into Equation ( ) we may obtain a “diamond” form of the recursions in Hform.When d “ ) the recursions take aparticularly simple form, since ad x ” x P T , for d “ )then becomes κκκ p n q t p T q “ E t ´ X p n q t,T ¯ ` n ´ ÿ k “ pp X p k q ` κκκ p k q q ˛ p X p n ´ k q ` κκκ p n ´ k q qq t p T q ` E t ´ J p n q t p T q ¯ where J p n q t p T q “ ř | ‘ |ě , } ‘ }“ n Jmp p X, κκκ ; ‘ q t,T contains the n -th tensor component of the jump contri-bution. The above diamond recursion can also be obtained by projecting the functional relation ( )to the n -th tensor level. We shall revisit this in a multivariate setting and comment on related works inSection . 5. Two special cases and application of the Lie bracket, coming from the ad operator.
Variations on Hausdorff, Magnus and Baker–Campbell–Hausdorff.
We now consider adeterministic driver X of finite variation. This includes the case when X is absolutely continuous, inwhich case we recover, up to a harmless time reversal, t Ø T ´ t , Hausdorff’s ODE and the classicalMagnus expansion for the solution to a linear ODE in a Lie group [ Hau06 , Mag54 , Che54 , IN99 ]. Ourextension with regard to discontinuities seems to be new and somewhat unifies Hausdorff’s equationwith multivariate Baker–Campbell–Hausdorff integral formulas.
Theorem 5.1.
Let X P V p T q , and more specifically X : r , T s Ñ T deterministic, càdlàg of boundedvariation. The log-signature Ω t “ Ω t p T q – log p Sig p X q t,T q satisfies the integral equation Ω t p T q “ ż Tt H p ad Ω u ´ qp d X cu q ` ÿ t ă u ď T ż Ψ p exp p ad θ ∆ X u q ˝ exp p ad Ω u qqp ∆ X u q d θ, (5.1) with Ψ p z q – H p log z q “ log z {p z ´ q as in the introduction. The sum in ( ) is absolutely convergent,over (at most countably many) jump times of X , vanishes when X ” X c , in which case eq. ( ) reducesto Hausdorff’s ODE.(i) The accompanying Jump Magnus expansion becomes Ω p q t p T q “ X p q t,T followed by Ω p n q t p T q “ X p n q t,T ` ÿ | ‘ |ě , } ‘ }“ n ` HMag p X , Ω; ‘ q t,T ` HJmp p X , Ω; ‘ q t,T ˘ where the right-hand side only depends on Ω p k q , k ă n .(ii) If X P V p V q for some linear subspace V Ă T “ T pp R d qq , it follows that, for all t P r , T s , Ω t p T q P L – Lie pp V qq Ă T , Sig p X q t,T P exp p L q Ă T , we say that Ω t p T q is Lie in V . In case V “ R d one speaks of (free) Lie series, cf. [ Lyo14 , Def. 6.2].Proof.
Since we are in a purely deterministic setting the signature cumulant coincides with the log-signature κκκ t p T q “ Ω t p T q and Theorem applies without any expectation and angle brackets.Using ∆Ω u “ Ω u ´ Ω u ´ “ Ω u ´ log p e ∆ X u e Ω u q we see thatΩ t p T q “ ż Tt H p ad Ω u ´ qp d X cu q ´ ÿ t ă u ď T ∆Ω u “ ż Tt H p ad Ω u ´ qp d X cu q ´ ÿ t ă u ď T p Ω u ´ BCH p ∆ X u , Ω u qq“ ż Tt H p ad Ω u ´ qp d X cu q ` ÿ t ă u ď T ż Ψ p exp p θ ad ∆ X u q ˝ exp p ad Ω u qqp ∆ X u q d θ, where we used the identityBCH p x , x q ´ x “ log ` exp p x q exp p x q ˘ ´ x “ ż Ψ p exp p θ ad x q ˝ exp p ad x qqp x q d θ. (5.2) (cid:3) Remark 5.2 (Baker–Campbell–Hausdorff) . The identity ( ) is well-known, but also easy to obtain en passant , thereby rendering the above proof self-contained. We treat directly the n -fold case. Given x , . . . , x n P T one defines a continuous piecewise affine linear path p X t : 0 ď t ď n q with X i ´ X i ´ “ x i . Then Sig p X | r i ´ ,i s q “ Sig p X q i ´ ,i “ exp p x i q and by Lemma have Sig p X q ,n “ exp p x q ¨ ¨ ¨ exp p x n q and therefore Ω “ log p exp p x q ¨ ¨ ¨ exp p x n qq — BCH p x , . . . , x n q . A computation based on Theorem , but now applied without jumps, reveals the general form
BCH p x , . . . , x n q “ x n ` n ´ ÿ k “ ż Ψ p exp p θ ad x k q ˝ exp p ad x k ` q ˝ ¨ ¨ ¨ ˝ exp p ad x n qqp x k q d θ “ ÿ i x i ` ÿ i ă j r x i , x j s ` ÿ i ă j pr x i , r x i , x j ss ` r x j , r x j , x i ssq´ ÿ i ă j ă k r x j , r x i , x k ss ´ ÿ i ă j r x i , r x j , r x i , x j sss ¨ ¨ ¨ NIFIED SIGNATURE CUMULANTS AND GENERALIZED MAGNUS EXPANSIONS 17
The flexibility of our Theorem is then nicely illustrated by the fact that this n -fold BCH formula isan immediate consequence of ( ) , applied to a piecewise constant càdlàg path p X t : 0 ď t ď n q with X ¨ ´ X i ´ ” x i on r i ´ , i q . Diamond relations for multivariate cumulants.
As in Section we write S for the symmet-ric algebra over R d , and S , S for those elements with scalar component 0 ,
1, respectively. Recall theexponential map exp : S Ñ S with global defined inverse log. Following Definition the diamondproduct for S -valued semimartingales ˜ X , ˜ Y is another S -valued semimartingale given by p ˜ X ˛ ˜ Y q t p T q “ E t ` x ˜ X c , ˜ Y c y t,T ˘ “ ÿ p E t x ˜ X w , ˜ Y w y t,T q ˆ e w ˆ e w , with summation over all w , w P x W d , provided all brackets are integrable. This trivially adapts to S N -valued semimartingales, N P N ě , in which case all words have length less equal N , the summationis restricted accordingly to | w ` | w | ď N . Theorem 5.3. (i) Let Ξ “ p , Ξ p q , Ξ p q , ... q be an F T -measurable random variable with values in S p R d q , componentwise in L . Then K t p T q – log E t exp p Ξ q satisfy the following functional equation, for all ď t ď T , K t p T q “ E t Ξ ` p K ˛ K q t p T q ` J t p T q (5.3) with jump component, J t p T q “ E t ˜ ÿ t ă u ď T ` e ∆ K u ´ ´ ∆ K u ˘¸ “ E t ˜ ÿ t ă u ď T ˆ p ∆ K u q ` p ∆ K u q ` ¨ ¨ ¨ ˙¸ . Furthermore, if N P N ě , and Ξ “ p Ξ p q , ..., Ξ p N q q is F T -measurable with graded integrability condition } Ξ p n q } L N { n ă 8 , n “ , ..., N, (5.4) then the identity ( ) holds for the truncated signature cumulant K p ,N q – p log E t p Sig p X p ,N q q t,T qq ď t ď T with values in S p N q p R d q . Remark 5.4.
Identity ( ) is reminiscent of generalized Riccati equations for affine jump diffusions.The relation is, in a nutshell, that ( ) reduces to a PIDE system when the involved processes havea Markov structure. (We will make this point explicit in Section below, even in the fully non-commutative setting.) These PIDEs reduce to generalized Riccati under appropriate (affine linear)structure of the characteristics. The framework described here however requires neither Markov nor affinestructure. We will show in Section that such computations also possible in the fully non-commutativesetting, i.e. to obtain signature cumulants.Proof.
We first observe that since Ξ P L , by Doob’s maximal inequality and the BDG inequality, wehave that ˜ X t – E t Ξ is a martingale in H p S q . In particular, thanks to Theorem , the signaturemoments are well defined. According to Section , the signature is then given bySig p ˜ X q t,T “ exp p Ξ ´ E t Ξ q , hence κκκ t p T q “ K t p T q ´ ˜ X t .Projecting Equation ( ) onto the symmetric algebra yields κκκ t p T q “ E t ˜ X t,T ` x ˜ X c y t,T ` x κκκ p T q c y t,T ` x ˜ X c , κκκ p T q c y t,T ` ÿ t ă u ď T ´ e ∆ ˜ X u ` ∆ κκκ u p T q ´ ´ ∆ ˜ X u ´ ∆ κκκ u p T q ¯+ “ E t Ξ ` x K p T q c y t,T ` ÿ t ă u ď T ´ e ∆ K u p T q ´ ´ ∆ K u p T q ¯+ ´ ˜ X t , and eq. ( ) follows upon recalling that p K ˛ K q t p T q “ E t x K p T q c y t,T . The proof of the truncatedversion is left to the reader. (cid:3) As a corollary, we provide a general view on recent results of [
AGR20 , LRV19 , FGR20 ]. Note that wealso include jump terms in our recursion.
Corollary 5.5.
The conditional multivariate cumulants p K t q ď t ď T of a random variable Ξ with valuesin S p R d q , componentwise in L satisfy the recursion K p q t “ E t p Ξ p q q and K p n q t “ E t p Ξ p n q q ` n ÿ k “ ´ K p k q ˛ K p n ´ k q ¯ t p T q ` J p n q t p T q for n ě , (5.5) with J p n q t p T q “ E t ¨˝ ÿ t ă u ď T n ÿ k “ k ! ÿ } ‘ }“ n, | ‘ |“ k ∆ K p ‘ q u p T q ¨ ¨ ¨ ∆ K p ‘ k q u p T q ˛‚ . The analogous statement holds true in the N -truncated setting, i.e. as recursion for n “ , .., N underthe condition ( ). Example 5.6 (Continuous setting) . In case of absence of jumps and higher order information (i.e. J ” , Ξ p q “ Ξ p q “ ... ”
0, this type of cumulant recursion appears in [
LRV19 ] and under optimalintegrability conditions Ξ p q with finite N .th moments, [ FGR20 ]. (This requires a localization argumentwhich is avoided here by directly working in the correct algebraic structure.) ♦ Example 5.7 (Discrete filtration) . As opposite of the previous continuous example, we consider apurely discrete situation, starting from a discretely filtered probability space with filtration p F t : t “ , , . . . , T P N q . For Ξ as in Corollary , a discrete martingale is defined by E t exp p Ξ q , which mayregard as cádlág semimartingale with respect to F t – F r t s , and similar for K t p T q “ log E t exp p Ξ q P S ,i.e. the conditional cumulants of Ξ. Clearly, the continuous martingale part of K p T q vanishes, as doesany diamond product with K p T q . What remains is the functional equation K t p T q “ E t p Ξ q ` J t p T q “ E t p Ξ q ` E t ˆ T ÿ u “ t ` ` e ∆ K u ´ ´ ∆ K u ˘˙ As before, the resulting expansions are of interest. On the first level, trivially, K p q t “ E t p Ξ p q q , whereason the second level we see K p q t p T q “ E t p Ξ p q q ` E t ˆ T ÿ u “ t ` p E u p Ξ p q q ´ E u ´ p Ξ p q qq ˙ which one can recognize, in case Ξ p q “ ‘ u : “ E u Ξ p q . Going further in the recursion yields increasingly non-obvious relations. TakingΞ p q “ Ξ p q “ ... ” K p q t p T q “ E t ˜ T ÿ u “ t ` p ‘ u ´ ‘ u ´ q ` p ‘ u ´ ‘ u ´ qt E u κ p ‘, ‘ q u,T ´ E u ´ κ p ‘, ‘ q u ´ ,T u ¸ It is interesting to note that related identities have appeared in the statistics literature under the name
Bartlett identities , cf. Mykland [
Myk94 ] and the references therein. ♦ Remark on tree representation.
As illustrated in the previous section, in the case where d “ ). If one further specializes the situation, in particular discards all jump, we are froman algebraic perspective in the setting of Friz, Gatheral and Radoiçić [ FGR20 ] which give a tree seriesexpansion of cumulants using binary trees. This representation follows from the fact that the diamondproduct of semimartingales is commutative but not associative. As an example (with notations takenfrom Section ), in case of a one-dimensional continuous martingale, the first terms are K t p T q “ ` ` ` ` ` ¨ ¨ ¨ This expansion is organized (graded) in terms of the number of leaves in each tree, and each leafrepresents the underlying martingale.In the deterministic case, tree expansions are also known for the Magnus expansion [
IN99 ] and theBCH formula [
CM09 ]. These expansions also in terms of binary trees, but this time they are also
NIFIED SIGNATURE CUMULANTS AND GENERALIZED MAGNUS EXPANSIONS 19 required to be non-planar to account for the non-commutativity of the Lie algebra. As an example(with the notations of Section ), we haveΩ t p T q “ ` ` ` ` ¨ ¨ ¨ In this expansion, the nodes represent the underlying vector field and edges represent integration andapplication of the Lie bracket, coming from the ad operator.Since our functional equation and the associated recursion puts both contexts into a single commonframework. We suspect that our general recursion, Corollary and thereafter, allows for a sophisticatedtree representation, at least in absence of jumps, and propose to return to this question in future work.6.
Applications
Brownian and stopped Brownian signature cumulants.
Time dependent Brownian motion.
Let B be a m -dimensional standard Brownian motion definedon a portability space p Ω , F , P q with the canonical filtration p F t q t ě and define the continuous (Gaussian)martingale X “ p X t q ď t ď T by X t “ ż t σ p u q d B u , ď t ď T, with σ P L pr , T s , R m ˆ d q . An immediate application of Theorem shows that the integrabilitycondition X “ p , X, , . . . q P H is trivially satisfied. The Brownian signature cumulants κκκ t p T q “ log p E t p Sig p X q t,T qq satisfies the functional equation, with a p t q – σ p t q σ p t q T P Sym p R d b R d q ,κκκ t p T q “ ż Tt H p ad κκκ u p T qqp a p u qq d u, ď t ď T. (6.1)Therefore the tensor levels are precisely given by the Magnus expansion, starting with κκκ p q t p T q “ , κκκ p q t p T q “ ż Tt a p u q d u, and the general term κκκ p n ´ q t p T q ” , κκκ p n q t p T q “ ÿ | ‘ |ě , } ‘ }“ n HMag p X , κκκ ; ‘ q t,T “ ÿ } ‘ }“ n ´ B k k ! ż Tt ad κκκ p ¨ l q u ¨ ¨ ¨ ad κκκ p ¨ l k q u p a p u qq d u. Note that κκκ t p T q is Lie in Sym p R d b R d q Ă T , but, in general, not a Lie series. In the special case X “ B , i.e. m “ d and identity matrix σ “ I d “ ř di “ e ii P Sym p R d b R d q , all commutators vanishand we obtain what is known as Fawcett’s formula [ Faw02 , FH20 ]. κκκ t p T q “ p T ´ t q I d . Example 6.1.
Consider B , B two Brownian motions on the filtered space p Ω , F , P q , with correlationd x B , B y t “ ρ d t for some fixed constant ρ P r´ , s . Suppose that K , K : r , Ñ R are twokernels such that K i p t, ¨q P L pr , t sq for all t P r , T s , and set X it – X i ` ż t K i p t, s q d B is , i “ , X , X . Note that neither process is a semimartingale in general. However,for each T ą
0, the process ξ it p T q – E t r X iT s is a martingale and we have ξ it p T q “ X i ` ż t K i p T, s q d B is , that is, p ξ , ξ q is a time-dependent Brownian motion as defined above. In particular, one sees that a p t q “ ˜ ş t K p T, u q d u ρ ş t K p T, u q K p T, u q d uρ ş t K p T, u q K p T, u q d u ş t K p T, u q d u ¸ . Equation ( ) and the paragraph below it then give an explicit recursive formula for the signaturecumulants, the first of which are given by κκκ p q t p T q “ ,κκκ p q t p T q “ ˜ ş Tt ş u K p T, r q d r d u ρ ş Tt ş u K p T, r q K p T, r q d r d uρ ş Tt ş u K p T, r q K p T, r q d r d u ş Tt ş u K p T, r q d r d u ¸ ,κκκ p q t p T q “ ,κκκ p q t p T q “ ÿ i,j,i ,j “ «ż Tt ż Tu p a ij p u q a i j p r q ´ a i j p u q a ij p r qq d r d u ff e iji j . We notice that in the particular case when K “ K ” K , the matrix a has the form a p t q “ ż t K p T, u q d u ˆ ˆ ρρ ˙ . Therefore, we have a p t q b a p t q ´ a p t q b a p t q “ t, t P r , T s . Hence, in this case, our recursionshows that for any ρ P r´ , s , κκκ p q t p T q “ , κκκ p q t p T q “ ż Tt ż u K p T, r q d r d u ˆ ˆ ρρ ˙ , and κκκ p n q t p T q “ ď t ď T and n ě ♦ Brownian motion up to the first exit time from a domain.
Let B “ p B t q t ě be a d -dimensionalBrownian motion defined on a filtered probability space p Ω , F , P q with the canonical filtration p F t q t ě and a possibly random starting value B . Assume that there is a family of probability measures t P x u x P R d on p Ω , F q such that P x p B “ x q “ E x the expectation with respect to P x .Further let Γ Ă R d be a bounded domain and define the stopping time τ Γ of the first exit of B fromthe domain Γ, i.e. τ Γ “ inf t t ě | B t P Γ c u . In [
LN15 ] Lyons–Ni exhibit an infinite system of partial differential equations for the expected signatureof the Brownian motion until the exit time as a functional of the starting point. The following resultcan be seen as the corresponding result for the signature cumulant, which follows directly from theexpansion in Theorem . Recall that a boundary point x P B
Γ is called regular if and only if P x ` inf t t ą | B t P Γ c u “ ˘ “ . (6.2)The domain Γ is called regular if all points on the boundary are regular. For example domains withsmooth boundary are regular and see [ KS98 , Section 4.2.C] for a further characterization of regularity.
Corollary 6.2.
Let Γ Ă R d be a regular domain, such that sup x P Γ E x p τ n Γ q ă 8 , n P N ě . (6.3) The signature cumulant κκκ t “ log p E p Sig p B q t ^ τ Γ ,τ Γ qq of the Brownian motion B up to the first exit fromthe domain Γ has the following form κκκ t “ t t ă τ Γ u F p B t q , t ě , where F “ ř | w |ě e w F w with F w P C p Γ , R q X C p Γ , R q is the unique bounded classical solution to theelliptic PDE ´ ∆ F p x q “ d ÿ i “ H p ad F p x qq ´ e ii ` Q p ad F p x qqpB i F p x q b q ` e i G p ad F p x qqpB i F p x qq ¯ , (6.4) for all x P Γ with the boundary condition F | B Γ ” .Proof. Define the martingale X “ pp , B t ^ τ Γ , , . . . qq t ě P S p T q and note that |x X y | “ τ Γ . It thenfollows from the integrability of τ Γ that X P H p T q and thus by Theorem that p Sig p X q ,t q t ě P H p T q . This implies that the signature cumulant κκκ t p T q – log p E t p Sig p X q t,T qq is well defined for all NIFIED SIGNATURE CUMULANTS AND GENERALIZED MAGNUS EXPANSIONS 21 ď t ď T ă 8 and furthermore under (component-wise) application of the dominated convergencetheorem that it holds κκκ t “ lim T Ñ8 κκκ t p T q “ lim T Ñ8 log p E t p Sig p X q t,T qq “ log p E t p Sig p B q t ^ τ Γ ,τ Γ qq , t ě . Again by X P H p T q it follows that Theorem applies to the martingale p X t q ď t ď T for any T ą κκκ p T q satisfies the functional equation ( ). It is well known that all martingaleswith respect to the filtration p F t q ď t ď T are continuous, and therefore it is easy to see that also κκκ p T q P S c p T q . Therefore ( ) simplifies to the following equation κκκ t p T q “ t t ă τ Γ u E t ż τ Γ ^ Tt H p ad κκκ u qp I d q d u ` ż τ Γ ^ Tt H p ad κκκ u q ˝ Q p ad κκκ u qp d (cid:74) κκκ, κκκ (cid:75) u q` ż τ Γ ^ Tt H p ad κκκ u q ˝ p Id d G p ad κκκ u qqp d (cid:74) X, κκκ (cid:75) u q + , (6.5)where we have already used the martingality of X and the explicit form of the quadratic variation x X y t “ I d p t ^ τ Γ q with I d “ ř di “ e ii P p R d q b . It follows that κκκ p q ” κκκ p T q p q ” τ Γ and the strong Markov property of Brownian motion that κκκ p q t “ I d lim T Ñ8 E t ` t t ă τ Γ u p τ Γ ^ T ´ t q ˘ “ I d t t ă τ Γ u E x p τ Γ q| x “ B t , t ě . Now note that the function u p x q – E x p τ Γ q for x P Γ is in C p Γ , R q X C p Γ , R q and solves the Poissonequation ´p { q ∆ u “ g with boundary condition u | B Γ “ g ”
1. Indeed, since Γ is regularand g is bounded and differentiable, this follows from Theorem 9.3.3 (and the remark thereafter)in [ Øks14 ]. Moreover from the assumption ( ) we immediately see that u is bounded on Γ and itfollows from Theorem 9.3.2 in [ Øks14 ] that u is the unique bounded classical such solution. Thus wehave shown that the statement holds true up to the second tensor level with F p q ” F p q “ I d u under the usual notation F p n q “ ř | w |“ n e w F w .Now assume that the statement of the corollary holds true up to the tensor level p N ´ q for some N ě
3. Then, for any n, k ă N we have by applying Itô’s formula (cid:114) κκκ p n q , κκκ p k q (cid:122) t “ d ÿ i “ ż t ^ τ Γ pB i F p n q p B u qq b pB i F p k q p B u qq d u, t ě , and (cid:114) X , κκκ p n q (cid:122) t “ d ÿ i “ ż t ^ τ Γ e i b pB i F p n q p B u qq d u, t ě . Further define the function G p N q by the projection under π N of the right hand side of ( ) multipliedby the factor 1 {
2. Then applying Theorem to X p ,N q on the probability space p Ω , F , P x q we seethat it follows from the estimate ( ) that there exists a constant c ą x P Γ E x "ż τ Γ ˇˇ G p N q p B u q ˇˇ d u * ď c sup x P Γ ||| X p ,N q ||| H ,N p P x q “ c sup x P Γ E x p τ N Γ q ă 8 Therefore it follows, from projecting ( ) to level N and using the dominated convergence theorem topass to the T Ñ 8 limit, that κκκ p N q is of the form κκκ p N q t “ t t ă τ Γ u F p N q p B t q with F p N q p x q – E x "ż τ Γ G p N q p B u q d u * , x P Γ . Furthermore, by the assumption it also holds that G w P C p Γ q for all w P W d , | w | “ N . Therefore wecan conclude again with Theorem 9.3.3 in [ Øks14 ] that F w P C p Γ , R q X C p Γ , R q solves the Poissonequation with data g “ G w for all words w with | w | “ N . The statement then follows by induction. (cid:3) Example 6.3.
For n P t , . . . , d u , let D n be the open unit ball in R n and define the (regular) domainΓ “ D n ˆ R d ´ n Ă R d . Further note that it holds τ Γ “ inf t t ě | B t R Γ u “ inf t t ě | |p B t , . . . , B nt q| ě u . Hence we readily see that τ Γ satisfies the condition ( ). Applying Corollary it follows thatthe signature cumulant of the Brownian motion B up to the exit of the domain Γ is of the form κκκ t “ t t ă τ Γ u F p B t q , where F satisfies the PDE ( ). Recall that F p q ” ´ ∆ F p q p x q “ I d , x P Γ; F p q | B Γ ” . The unique bounded solution the above Poisson equation is given by F p q p x q “ I d ˜ ´ n ÿ i “ x i ¸ , x P Γ . More generally, we see that the Poisson equation ∆ u “ ´ g on Γ with zero boundary condition, where g : Γ Ñ R is a polynomial in the first n -variables, has a unique bounded solution u which is also apolynomial of the first n -variables of degree deg p u q “ deg p g q ` p ´ ř ni “ x i q (seeLemma 3.10 in [ LN15 ]). Hence it follows inductively that each component of F p n q is a polynomial ofdegree n with the factor p ´ ř ni “ x i q . The precise coefficients of the polynomial can be obtained asthe solution to a system of linear equations recursively derived from the forcing term in ( ). This issimilar to [ LN15 , Theorem 3.5], however we note that a direct conversion of the latter result for theexpected signature to signature cumulants is not trivially seen to yield the same recursion and requirescombinatorial relations as studied in [
BO20 ]. ♦ Lévy and diffusion processes.
Let X P S p R d q and throughout this section assume that thefiltration p F t q ď t ď T is generated by X . Denote by ε a the Dirac measure at point a P R d , the randommeasure µ X associated to the jumps of X is an integer-valued random measure of the form µ X p ω ; d t, d x q – ÿ s ě t ∆ X s p ω q‰ u ε p s, ∆ X s p ω qq p d t, d x q . There is a version of the predictable compensator of µ X , denoted by ν , such that the R d -valuedsemimartingale X is quasi-left continuous if and only if ν p ω, t t u ˆ R d q “ ω P Ω, see [
JS03 ,Corollary II.1.19]. In general, ν satisfies p| x | ^ q ˚ ν P A loc , i.e. locally of integrable variation. Thesemimartingale X admits a canonical representation (using the usual notation for stochastic integralswith respect to random measures as introduced e.g. in [ JS03 , II.1]) X “ X ` B p h q ` X c ` p x ´ h p x qq ˚ µ X ` h p x q ˚ p µ X ´ ν q , (6.6)where h p x q “ x | x |ď is a truncation function (other choice are possible.) Here B p h q is a predictable R d -valued process with components in V and X c is the continuous martingale part of X .Denote by C the predictable R d b R d -valued covariation process defined as C ij – x X i,c , X j,c y .Then the triplet p B p h q , C, ν q is called the triplet of predictable characteristics of X (or simply the characteristics of X ). In many cases of interest, including the case of Lévy and diffusion processesdiscussed in the subsection below, we have differential characteristics p b, c, K q such thatd B t “ b t p ω q d t, d C t “ c t p ω q d t, ν p d t, d x q “ K t p d x ; ω q d t, where b is a d -dimensional predictable process, c is a predictable process taking values in the set ofsymmetric non-negative definite d ˆ d -matrices and K is a transition kernel from p Ω ˆ R ` , B d q into p R d , B d q . We call such a process Itô semimartingale and the triplet p b, c, K q its differential (or local ) characteristics . This extends mutatis mutandis to an T N (and then T ) valued semimartingale X , withlocal characteristics p b , c , K q .While every Itô semimartingale is quasi-left continuous it is in general not true that κκκ is continuous(with the notable exception of time-inhomogeneous Lévy processes discussed below) and therefore thereis no significant simplification of the functional equation ( ) in these general terms. The followingexample illustrates this point in more detail. Example 6.4.
Take X P S p R d q and then d “
1, so that we are effectively in the symmetric setting.In this case y exp p κκκ t p T qq “ E t p y exp p X T ´ X t qq , in the power series sense of enlisting all moments withfactorial factors. These can also be obtained by taking higher order derivatives at u “ E t p e u p X T ´ X t q q ,now with the classical calculus interpretation of the exponential. The important class of affine modelssatisfies E t p e uX T ´ uX t q “ exp p φ p T ´ t, u q ` p Ψ p T ´ t, u q ´ u q X t q In the Levy-case, we have the trivial situation Ψ p¨ , u q ” u , but otherwise p φ, Ψ q solve (generalized)Riccati equations and are in particular continuous in T ´ t . We see that, in non-trivial situations, thelog of E t p e uX T ´ uX t q and any of its derivatives will jump when X jumps. In particular, κ t p T q will not NIFIED SIGNATURE CUMULANTS AND GENERALIZED MAGNUS EXPANSIONS 23 be continuous in t , even if X is quasi-left continuous. Let us note in this context that, in the generalnon-commutative setting and directly from definition of κκκ ,exp p κκκ t ´ q “ E t ´ p exp p ∆ X t q exp p κκκ t qq “ E t ´ p exp κκκ t q where the second equality holds true under the assumption of quasi-left continuity of X . If we assumefor a moment F t ´ “ F t , then we could conclude that κκκ t ´ “ κκκ t and hence (right-continuity is clear)that κκκ t is continuous in t . Since we know that this fails beyond Lévy processes, if follows that such leftcontinuity of filtrations is not a good assumption, at least not beyond Lévy processes. ♦ The case of time-inhomogeneous Lévy processes.
We consider now a d -dimensional time-inhomogeneousLévy processes of the form X t “ ż t b p u q d u ` ż t σ p u q d B u ` ż p ,t s ż | x |ď x p µ X ´ ν qp d s, d x q ` ż p ,t s ż | x |ą x µ X p d s, d x q , (6.7)for all 0 ď t ď T , with b P L pr , T s , R d q , σ P L pr , T s , R m ˆ d q , B a d -dimensional Brownian motion, µ X is an independent inhomogeneous Poisson random measure with the intensity measure ν on r , T s ˆ R d ,such that ν p d t, d x q “ K t p d x q d t with Lévy measures K t , i.e. K t pt uq “
0, and ż T ż R d p| x | ^ q K t p d x q d t ă 8 , and measurability of t ÞÑ K t p A q P r , , any measurable A Ă R d . Consider further the condition ż T ż R d | x | N | x |ą K t p d x q ă 8 , (6.8)for some integer N P N ě . The Brownian case ( ) then generalizes as follows. Corollary 6.5.
Let X be an inhomogenous Lévy process of the form ( ) , such that the family ofLévy measures t K t u t ą satisfy the moment condition ( ) for all N P N ě . Then X P H p R d q andthe signature cumulant κκκ t – log p E t p Sig p X q t,T qq satisfies the following integral equation κκκ t “ ż Tt H p ad κκκ u qp y p u qq d u, ă t ď T, (6.9) where a p t q “ σ p t q σ p t q T P R d b R d Ă T and y p t q : “ b p t q ` a p t q ` ż R d p exp p x q ´ ´ x | x |ď q K t p d x q P T . (6.10) In case the Lévy measures t K t u t ą satisfy the condition ( ) only up to some finite level N P N ě , wehave X P H N and the identity ( ) holds for the truncated signature cumulant in T N . Remark 6.6.
Corollary extends a main result of [
FS17 ], where a Lévy-Kintchin type formula wasobtained for the expected signature of Lévy processes with triplet p b, a, K q . Now this is an immediateconsequence of ( ) , with all commutators vanishing in time-homogeneous case, and explicit solution κκκ t p T q “ p T ´ t q ˆ b ` a ` ż R d p exp p x q ´ ´ x | x |ď q K p d x q ˙ . Proof.
Assume that the Lévy measures t K t u t ą satisfy the condition ( ) for some N P N ě . We willfirst show that X P H N p R d q . Note that the decomposition ( ) naturally yields a semimartingaledecomposition X “ M ` A , where the local martingale M and the adapted bounded variation process A are defined by M “ ż ¨ σ p u q d B u ` p x | x |ď q ˚ p µ X ´ ν q , A “ ż ¨ b p u q d u ` p x | x |ą q ˚ µ X . Regarding the integrability of the 1-variation of A we have first note that it holds | A | ´ var; r ,T s “ ż T | b p u q| d u ` p| x | | x |ą q ˚ µ XT . Define the increasing, piecewise constant process V – p| x | | x |ą q ˚ µ X . Since b is deterministic andintegrable over the interval r , T s it suffices to show that V T has finite N th moment. To this end, notethat it holds E p V T q “ ż T ż | x |ą | x | K t p d x q d t ă 8 . Further it holds for any n P t , . . . , N u that V nT “ ÿ ă t ď T ` V nt ´ V nt ´ ˘ “ ÿ ă t ď T n ´ ÿ k “ ˆ nk ˙ V ks ´ p ∆ V s q n ´ k and by definition ∆ V t “ | ∆ X t | | ∆ X t |ą . Now let n “ k P t , . . . , n ´ u then we have E ˜ ÿ ă t ď T V ks ´ p ∆ V s q n ´ k ¸ “ E ˜ż T ż | x |ą V ks ´ | x | n ´ k K t p d x q d t ¸ ď E ` V kT ˘ ż T ż | x |ą | x | n ´ k K t p d x q d t ă 8 . It then follows inductively that E p V nT q is finite for all n “ , . . . , N and hence that the 1-variation of A has finite N -th moment.Concerning the integrability of the quadratic variation of M , let w P t , . . . , d u , then it is well knownthat (see e.g. [ JS03 , Ch. II Theorem 1.33]) @ p x w | x |ď q ˚ p µ X ´ ν q D T “ p x w q | x |ď ˚ ν T , where x M y denotes the dual predictable projection (or compensator) of r M s . Further using that thecompensated martingale p x | x |ď q ˚ p µ X ´ ν q is orthogonal to continuous martingales, we have x M w y T “ ż T a ww p t q d t ` ż T ż | x |ď p x w q K t p d x q d t ă 8 . Now let q P r , , then from Theorem 8.2.20 in [ CE15 ] we have the following estimation E pr M w s qT q ď c E ˆ x M w y qT ` sup ď t ď T p ∆ M wt q q ˙ ď c px M w y qT ` q ă 8 , where c ą q .We have shown that X “ p , X, , . . . , q P H ,N and it follows from Theorem that the signaturecumulant κκκ t “ log p E t p Sig p X q t,T qq satisfies the functional equation ( ). On the other hand, it followsfrom the condition ( ) that y in ( ) is well defined. Now define r κκκ “ p r κκκ t q ď t ď T by the identity( ). Noting that r κκκ is deterministic and has absolutely continuous components it is easy to see that r κκκ also satisfies the functional equation ( ) for the semimartingale X . It thus follows that κκκ and r κκκ areidentical. (cid:3) Markov jump diffusions.
The generator of a general Markov jump diffusion X is given by L f p x q “ ÿ i b i p x qB i f p x q ` ÿ i,j a ij p x qB i B j f p x q ` ż R d ˆ f p x ` y q ´ f p x q ´ | y |ď ÿ i y i B i f p x q ˙ K p x, d y q , (6.11)where the summations are over i, j P t , . . . , d u , b : R d Ñ R d and a : R d Ñ R d b R d (symmetric, positivedefinite) are bounded Lipschitz, K is a Borel transition kernel from R d into R d with K p¨ , t uq ” x P R d ż R d p| y | ^ q K p x, d y q ă 8 . Note that (the law of) X is the unique solution to the martingale problem associated to L . That said,the extensions to Markov processes with differential characteristics p b p t, x q , a p t, x q , K p t, x, d y qq , withassociated local Lévy generators [ Str75 ] is mostly notational. For the construction of general jumpdiffusions and their semimartingale characteristics see e.g. [
JS03 , Ch. III.2.c] and [
Jac79 , XIII.3].The expected signature of X was seen in [ FS17 ] (in [
Ni12 ] for the continuous case) to satisfy a systemof (linear) partial integro-differential equations (PIDEs). Passage to signatures cumulants amounts totake the logarithm, which represents a non-commutative Cole–Hopf transform, with resulting quadraticnon-linearity, if viewed as T -valued PIDE, resolved thanks to the graded structure so that again a NIFIED SIGNATURE CUMULANTS AND GENERALIZED MAGNUS EXPANSIONS 25 system of (linear) PIDEs arises. In the proof of the following corollary we will show how this PIDE canbe derived from Theorem . Corollary 6.7.
Let X be a d -dimensional Markov diffusion with generator given by ( ) , where thetransition kernel K have uniformly bounded moments of all orders, i.e. sup x P R d ˆż R d | y | n K p x, d y q ˙ ă 8 , n P N . (6.12) Then X P H and the signature cumulant is of the form κκκ t p T q “ v p t, X t ; T q “ v p t, X t q , where v “ ř w v w e w is the unique solution with v w P C , b pr , T s ˆ R d ; R q for all w P W d of the followingpartial integro-differential equation ´ rB t ` L s v “ H p ad v q " b ` a ` ÿ i,j a ij Q p ad v qpB i v b B j v q ` ÿ i,j a ij e j G p ad v qB i v * ` ż R d " H p ad v q ´ exp p y q exp p v ˝ τ y q exp p´ v q ´ ´ | y |ď y ¯ ´ ` v ˝ τ y ´ v ˘* K p¨ , d y q , (6.13) on r , T s ˆ R d with terminal condition v p T, ¨q ” , where τ y p t, x q “ p t, x ` y q .Proof. First note that X has the semimartingale characteristics p B, C, ν q where (see [ Jac79 , XIII.3])d B t “ b p X t ´ q d t, d C t “ a p X t ´ q d t, ν p d t, d x q “ d tK p X t ´ , d x q , with respect to the truncation function h p x q “ | x |ď . Further denote by µ X the random measureassociated with the jumps of X and recall the canonical representation ( ). We can easily verify thatthe boundedness of b and a , and the moment condition ( ) implies that X “ p , X, , . . . q P H (compare also with the proof of Corollary ). It then follows from Theorem that κκκ p T q “p E t p Sig p X q t,T qq ď t ď T is the unique solution to the functional equation ( ).Now assume that v is the (unique) solution to the above PIDE with v w P C , b pr , T s ˆ R d ; R q for all w P W d (this is really an infinite-dimensional system of linear PIDEs, solved inductively upon projectionto the linear span of e w with | w | ď ‘ , for ‘ P N ě , see that standard results as found in [ CT04 , Section12.2] and references therein apply). Then define ˜ κκκ P S p T q by ˜ κκκ t – v p t, X t q for all 0 ď t ď T and notethat ˜ κκκ t ´ “ v p t, X t ´ q . We are going to show that also ˜ κκκ also satisfies the functional equation ( ).Since X solves the martingale problem with generator L and v is sufficiently regular it holds˜ κκκ t “ ´ E t ` v p T, X T q ´ v p t, X t q ˘ “ E t ˆ ´ ż Tt rB t ` L s v p u, X u ´ q d u ˙ . (6.14)On the other hand, we can plug in ˜ κκκ into the right-hand side of ( ). We then obtain for the firstintegral inside the conditional expectation ż p ,t s H p ad ˜ κκκ u ´ qp d X u q “ ż p ,t s H p ad ˜ κκκ u ´ qp d B u ` d X cu q ` W ˚ p µ X ´ ν q t ` W ˚ µ Xt , where W t p y q – H p ad ˜ κκκ t ´ qp h p y qq , and W t p y q – H p ad ˜ κκκ t ´ qp y ´ h p y qq , for all 0 ď t ď T and y P R d . Similarly we have ÿ ă u ď t " H p ad ˜ κκκ u ´ q ´ exp p ∆ X u q exp p ˜ κκκ u q exp p´ ˜ κκκ u ´ q ´ ´ ∆ X u ¯ ´ ∆˜ κκκ u * “ J ˚ µ Xt , where 0 ď t ď T and y P R d J t p y q – " H p ad v q ´ exp p y q exp p v ˝ τ y q exp p´ v q ´ ´ y ¯ ´ p v ˝ τ y ´ v q * p t, X t ´ q . Finally for the quadratic variation terms with respect to continuous parts we have U t : “ ż t H p ad ˜ κκκ u ´ q ! d x X y u ` ` Id d G p ad ˜ κκκ u ´ q ˘` d (cid:74) X c , ˜ κκκ c (cid:75) u ˘ ` Q p ad ˜ κκκ u ´ q ` d (cid:74) ˜ κκκ c , ˜ κκκ c (cid:75) u ˘) “ ÿ i,j ż t a ij H p ad v q " e ij ` p Id d G p ad v qqp e i b B j v q ` Q p ad v qpB i v b B j v q * p t, X u ´ q d u “ : ż t H p ad v qp u p u, X u ´ qq d u. Provided that we can show the following integrability property holds for all words w P W d E ˆ ż T ˇˇ(cid:32) H p ad ˜ κκκ u ´ qp b p X u ´ q ` u p u, X u ´ qq ( w ˇˇ d u ` ` | W w | ` | W w | ` | J w | ˘ ˚ ν T ˙ ă 8 , (6.15)it follows that E t " ż p t,T s H p ad ˜ κκκ u ´ qp d X u q ` U t,T ` J ˚ µ Xt,T * “ E t " ż Tt H p ad ˜ κκκ u ´ qp d B u q ` U t,T ` p J ´ W q ˚ ν t,T * “ E t " ż p t,T s ˆ H p ad v q ` b p X u ´ q ` u p u, X u ´ q ˘ ` ż R d p J u p y q ´ W u p y qq K p X u ´ , d y q ˙ d u * “ E t ˆ ´ ż Tt rB t ` L s v p u, X u ´ q d u ˙ . where in the last line we have used v satisfies the PIDE. Since the above left-hand side is preciselythe right-hand side of the functional equation ( ), it follows together with ( ) that ˜ κκκ satisfies thefunctional equation ( ).Note that in case the integrability condition ( ) is satisfied for all words w P W d with | w | ď n for some length n P N ě it follows that the above equality holds up to the projection with π p ,n q . Forwords with | w | “ ) is an immediate consequence of X P H . It then followsinductively, by the same arguments as in the proof of Claim that ( ) is indeed satisfied for allwords w P W d .Since κκκ p T q is the unique solution to ( ) it then follows that ˜ κκκ ” κκκ p T q . (cid:3) Affine Volterra processes.
For i “ , K i be an integration kernel such that K i p t, ¨q P L pr , t sq for all 0 ď t ď T and let V i be the solution to the Volterra integral equation V it “ V i ` ż t K i p t, s q a V is d W is , ď t ď T, with V i ą
0, where W and W are uncorrelated standard Brownian motions which generate thefiltration p F t q ď t ď T . Note that in general V i is not a semimartingale. In particular this is not the casewhen K i is a power-law kernel of the form K p t, s q „ p t ´ s q H ´ { for some H P p , { q , which is theprototype of a rough affine volatility model (see e.g. [ KLP18 ]). However, a martingale ξ i p T q is naturallyassociated to V i by ξ it p T q “ E t p V iT q , ď t ď T. In the financial context, ξ i p T q is the central object of a forward variance model (see e.g. [ GKR19 ]). It wasseen in [
FGR20 ] that the iterated diamond products of ξ p T q are of a particularly simple form and easilytranslated to a system of convolutional Riccati equations of the type studied in [ AJLP19 ], [
GKR19 ] forthe cumulant generating function. We are interested in the signature cumulant of the two dimensionalmartingale X “ p ξ p T q , ξ p T qq . NIFIED SIGNATURE CUMULANTS AND GENERALIZED MAGNUS EXPANSIONS 27
Corollary 6.8.
It holds that X “ p , ξ p T q e ` ξ p T q e , , . . . q P H and the signature cumulant κκκ t p T q “ log E t p Sig p X q t,T q is the unique solution to the functional equation: for all ď t ď Tκκκ t p T q “ ´ E t ˜ ÿ i “ , ż Tt H p ad κκκ u qp e ii q K i p T, u q V iu d u ` ż Tt H p ad κκκ u q ˝ Q p ad κκκ u qp d p κκκ ˛ κκκ q u p T qq` ÿ i “ , ż Tt H p ad κκκ u q (cid:32) e i G p ad κκκ u q ` d p ξ i p T q ˛ κκκ q u p T q ˘(¸ . Proof.
Regarding the integrability statement it suffices to check that V iT has moments of all order for i “ ,
2. This is indeed the case and we refer to [
AJLP19 , Lemma 3.1] for a proof. Hence we can applyTheorem and we see that κκκ satisfies the functional equation ( ). As described in Section this equation can be reformulated with brackets replaced by diamonds. Further note that, due to thecontinuity, jump terms vanish and, due to the martingality, the Itô integrals with respect to X havezero expectation. The final step to arrive at the above form of the functional equation is to calculatedthe brackets x ξ i p T q , ξ j p T qy . From the definition ξ i p T q and V i we have for all 0 ď t ď Tξ it p T q “ E t ˜ V i ` ż t K i p T, s q a V is d W is ` ż Tt K i p t, s q a V is d W is ¸ “ V i ` ż t K i p T, s q a V is d W is . Therefore and due to the independence we have x ξ p T q , ξ p T qy “ x ξ i p T q , ξ i p T qy t “ K i p T, t q V it d t . (cid:3) The recursion for the signature cumulants from Corollary are easily simplified in analogy to theabove corollary. In the rest of this section we are going to demonstrate explicit calculations for thefirst four levels. Clearly, due to the martingality the first level signature cumulants are identically zero κκκ p q p T q ”
0. In the second level we start to observe the type of simplifications that appear due to theaffine structure κκκ p q t p T q “ ÿ i “ , e ii p ξ i p T q ˛ ξ i p T qq t p T q “ ÿ i “ , e ii E t ˜ż Tt K i p T, u q V iu d u ¸ “ ÿ i “ , e ii ż Tt K i p T, u q ξ it p u q d u, where ξ it p u q “ E t p V iu q for all 0 ď t ď u ď T . The third level is of the same form κκκ p q t p T q “ ÿ i “ , e i p ξ i p T q ˛ κκκ p q p T qq t p T q“ ÿ i “ , e iii ż Tt ˜ż Tu K i p T, s q K i p T, u q K i p s, u q d s ¸ ξ i p u q d u, where we have used that for any suitable h : r , T s Ñ R it holds for all 0 ď t ď T ż Tt h p u q ξ it p u q d u “ ż T h p u q V i d u ´ ż t h p u q V iu d u ` ż t ˜ż Tu h p s q K i p s, u q d s ¸a V iu d W iu . The fourth level starts to reveal some of the structure that is not visible in the commutative setting κκκ p q t p T q “ ÿ i “ , r e ¯ i ¯ i , e ii s ż Tt ˜ż Tu K ¯ i p T, s q ξ ¯ i d s ¸ K i p T, u q ξ it p u q d u ` e iiii ż Tt h i p T, u q ξ it p u q d u + , where t i, ¯ i u “ t , u and h i is defined by h i p T, u q “ ˜ż Tu K i p T, s q K i p u, s q d s ¸ ` ż Tu ˜ż Ts K i p T, r q K i p T, s q K i p r, s q d r ¸ K i p T, s q K i p s, u q ds, ď u ď T. Proofs
For ease of notation we introduce a norm on the space of tensor valued finite variation process, whichcould have been introduced already in Section , was however not needed until now. Let q P r , and A P V pp R d q b n q for some n P N ě then we define } A } V q : “ } A } V q pp R d q b n q : “ ››› | A | ´ var; r ,T s ››› L q . It is easy to see that it holds } A } H q ď } A } V q and this inequality can be strict.Further for an element A P T b T we introduce the following notation A “ ÿ w ,w P W d A w ,w e w b e w , A w ,w P R , and for l , l P N ě A p l ,l q “ ÿ | w |“ l , | w |“ l e w w b A w ,w P p R d q b l b p R d q b l Ă T b T . Next we will proof two well known lemmas translated to the setting of tensor valued semimartingales.
Lemma 7.1 (Kunita-Watanabe inequality) . Let X P S pp R d q b n q and Y P S pp R d q b n q then thefollowing estimate holds a.s. |x X c , Y c y| ´ var; r ,T s ` ÿ ă t ď T | ∆ X t ∆ Y t | ď ÿ | w |“ n b r X w s T ÿ | w |“ m b r Y w s T ď c b |r X s T | b |r Y s T | , where c ą is a constant that only depends on d , m and n .Proof. From the definition of the quadratic variation of tensor valued semimartingales in Section we have |x X c , Y c y| ´ var; r ,T s ` ÿ ă s ď T | ∆ X s ∆ Y s | ď ÿ | w |“ n, | w |“ m ż T | d x X w c , Y w c y s | ` ÿ ă s ď T | ∆ X w s ∆ Y w s |ď ÿ | w |“ n, | w |“ m b r X w s T b r Y w s T ď d p n ` m q{ d ÿ | w |“ n r X w s T d ÿ | w |“ m r Y w s T ď d n ` m |r X s T ||r Y s T | , where the first estimate follows form the triangle inequality, the second estimate from the (scalar)Kunita-Watanabe inequality [ Pro05 , Ch. II, Theorem 25] and the last two estimates follow from thestandard estimate between the 1-norm and the 2-norm on p R d q b m – R d m . (cid:3) In order to proof the next well known lemma (Emery’s inequality) we need the following technical
Lemma 7.2.
Let A P V pp R d q b n q , Y P D pp R d q b l q , Z P D pp R d q b m q then it holds ˇˇˇˇˇż p , ¨s Y s ´ d A s Z s ´ ˇˇˇˇˇ ´ var; r ,T s ď ż p ,T s | Y s ´ Z s ´ || d A s | where the integration with respect to | d A | denotes the integration with respect to the increasing one-dimensional path p| A | ´ var; r ,t s q ď t ď T . Further, let Y P D pp R d q b l q , Z P D pp R d q b m q and let p A t q ď t ď T be a process taking values in p R d q b n b p R d q b n such that A w ,w P V for all w , w P W d with | w | “ n and | w | “ n . Then it holds ˇˇˇˇˇż p , ¨s p Y s ´ Id Y s ´ q d p Z s ´ Id Z s ´ qp d A s q ˇˇˇˇˇ ´ var; r ,T s ď ż p ,T s | YY ZZ | s ´ | d m p A q s | , where p Y Id Y qp A q “ YAY is the left- respectively right-multiplication by Y respectively Y . NIFIED SIGNATURE CUMULANTS AND GENERALIZED MAGNUS EXPANSIONS 29
Proof.
Let 0 ď s ď t ď T then it holds ˇˇˇˇˇż p s,t s Y u ´ d A u Z u ´ ˇˇˇˇˇ ď ż p s,t s | Y u ´ Z u ´ || d A u | . Indeed, as it follows e.g. from [
You36 , Theorem on Stieltjes integrability], we can approximate theintegral in the left-hand side by Riemann sums. Then for a partition p t i q i “ ,...,k of the interval r s, t s wehave ˇˇˇˇˇ k ´ ÿ i “ Y t i ´ p A t i ` ´ A t i q Z t i ´ ˇˇˇˇˇ ď k ´ ÿ i “ ˇˇ Y t i ´ p A t i ` ´ A t i q Z t i ´ ˇˇ ď k ´ ÿ i “ | Y t i ´ Z t i ´ | ˇˇ A t i ` ´ A t i ˇˇ , where the last inequality follows from the fact that for homogeneous tensors x P p R d q b m and y P p R d q b n it holds that | xy | “ | yx | ď | x || y | . Regarding the 1-variation we then have ˇˇˇˇˇż p , ¨s Y s ´ d A s Z s ´ ˇˇˇˇˇ ´ var; r ,T s “ sup ď t 﨨¨ď t k ď T k ÿ i “ ˇˇˇˇˇż p t i ,t i ` s Y s ´ d A s Z s ´ ˇˇˇˇˇ ď sup ď t 﨨¨ď t k ď T k ÿ i “ ż p t i ,t i ` s | Y s ´ Z s ´ || d A s |“ ż p ,T s | Y s ´ Z s ´ || d A s | . Regarding the second statement we see that for any 0 ď s ď t ď T we have ˇˇˇˇˇż p s,t s p Y u ´ Id Y u ´ q d p Z u ´ Id Z u ´ qp d A u q ˇˇˇˇˇ ď ż p s,t s | YY ZZ | u ´ | d m p A q u | Indeed, we approximate the integral in the right-hand side again by a Riemann sum. Then for apartition p t i q i “ ,...,k of the interval r s, t s we have ˇˇˇˇˇ k ´ ÿ i “ p Y t i ´ Id Y t i ´ q d p Z t i ´ Id Z t i ´ qp A t i ` ´ A t i q ˇˇˇˇˇ ď k ´ ÿ i “ ˇˇˇˇˇˇ ÿ | w |“ m, | w |“ m Y t i ´ e w Y t i ´ Z t i ´ e w Z t i ´ p A t i ` ´ A t i q ˇˇˇˇˇˇ ď k ´ ÿ i “ ˇˇ Y t i ´ Y t i ´ Z t i ´ Z t i ´ ˇˇˇˇ m p A t i ` q ´ m p A t i q ˇˇ , where the last inequality follows from the definition of the norm on (homogeneous) tensors and thedefinition of the multiplication map m . We conclude analogously to the proof of the first statement. (cid:3) Lemma 7.3 (Emery’s inequality) . Let X P S pp R d q b n q , Y P D pp R d q b l q and Z P D pp R d q b m q then for p, q P r , and { r “ { p ` { q it holds ›››››ż p , ¨s Y s ´ d X s Z s ´ ››››› H r pp R d q bp l ` n ` m q q ď c } YZ } S q pp R d q bp l ` m q q } X } H p pp R d q b n q , where c ą is a constant that only depends on d and m . Proof.
Let X “ X ` M ` A be a semimartingale decomposition with M “ A “
0. Then it followsby definition of the H r -norm and the above Lemma ›››››ż p , ¨s Y s ´ d X s Z s ´ ››››› H r ď ››››››ˇˇˇˇˇż p ,T s p Y s ´ Id Z s ´ q d d (cid:74) M , M (cid:75) s ˇˇˇˇˇ { ` ˇˇˇˇˇż p , ¨s Y s ´ d A s Z s ´ ˇˇˇˇˇ ´ var; r ,T s ›››››› L r ď ››››››ˇˇˇˇˇż p ,T s ˇˇ p Y s ´ Z s ´ q ˇˇ | d r M s s | ˇˇˇˇˇ { ` ż p ,T s | Y s ´ Z s ´ || d A s | ›››››› L r ď ›››› sup ď s ď T | Y s Z s | ´ |r M s| { ´ var; r T s ` | A | ´ var; r ,T s ¯›››› L r ď c } YZ } S q ››› |r M s T | ` | A s | ´ var; r ,T s ››› L p , where we have used the generalized Hölder inequality and the Kunita-Watanabe inequality (Lemma )to get to the last line. Taking the infimum of over all semimartingale decomposition M ` A yields thestatement. (cid:3) The following technical lemma will be used in the proof of both Theorem and Theorem . Lemma 7.4.
Let X , Y P S p T N q , N P N ě , q P r , and assume that there exists a constant c ą such that } Y p n q } H qN { n ď c ÿ } ‘ }“ n } X p l q } H qN { l ¨ ¨ ¨ } X p l j q } H qN { lj , n “ , . . . , N, where the summation is over ‘ “ p l , . . . , l j q P p N ě q j , j P N ě , } ‘ } “ l ` ¨ ¨ ¨ ` l j . Then there exists aconstant C ą , depending only on c and N , such that ||| Y ||| H q,N ď C ||| X ||| H q,N . Proof.
Note that for any n P t , . . . , N u it holds ˆ ÿ } ‘ }“ n } X p l q } H qN { l ¨ ¨ ¨ } X p l j q } H qN { lj ˙ { n ď ÿ } ‘ }“ n p} X p l q } { l H qN { l q l { n ¨ ¨ ¨ p} X p l j q } { l j H qN { lj q l j { n ď ÿ } ‘ }“ n ˆ l n } X p l q } { l H qN { l ` ¨ ¨ ¨ ` l n } X p l j q } { l j H qN { lj ˙ ď c n n ÿ i “ } X p i q } { i H qN { i , where c n ą n and the second inequality follows from Young’sinequality for products. Hence by the above estimate and the assumption we have ||| Y ||| H q,N “ N ÿ n “ } Y p n q } { n H qN { n ď c { n c n N ÿ n “ n ÿ i “ } X p i q } { i H qN { i ď C ||| X ||| H q,N , where C ą c and N . (cid:3) Proof of Theorem . Proof.
Denote by S “ p Sig p X q ,t q ď t ď T the signature process. We will first proof the upper inequality,i.e. that there exists a constant C ą d , N and q such that ||| S ||| H q,N ď C ||| X ||| H q,N . (7.1)According to Lemma it is sufficient to show that for all n P t , . . . , N u it holds c n } S p n q } H qN { n ď ÿ } ‘ }“ n } X p l q } H qN { l ¨ ¨ ¨ } X p l j q } H qN { lj “ : ρ n X (7.2)where c n ą q , d and n ). Note that it holds ρ n X ď ÿ } ‘ }“ n ρ l j X } X p l j ´ q } H qN { lj ´ ¨ ¨ ¨ } X p l q } H qN { l ď c ρ n X (7.3) NIFIED SIGNATURE CUMULANTS AND GENERALIZED MAGNUS EXPANSIONS 31 where c is a constant depending only on n . We are going to proof ( ) inductively.For n “ S p q “ X p q ´ X p q “ X p q P H qN and therefore the estimate follows immediately.Now, assume that ( ) holds for all tensor levels up to some level n ´ n P t , . . . , N u . We willdenote by c , c ą n , d and q . Then we have from ( ) S p n q t “ ÿ } ‘ }“ n, | ‘ |ď ż t S p l q u ´ d X p l q u ` ÿ } ‘ }“ n, ď| ‘ |ď ż t S p l q u ´ d x X p l q c , X p l q c y u ` ÿ } ‘ }“ n, | ‘ |ě ÿ ď u ď t S p l j q u ´ ∆ X p l j ´ q u ¨ ¨ ¨ ∆ X p l q u p j ´ q ! . For the first term in the above right-hand side we have by Emery’s inequality (Lemma ) the followingestimate ›››››› ÿ } ‘ }“ n, | ‘ |ď ż ¨ S p l q u ´ d X p l q u ›››››› H qN { n ď c ÿ } ‘ }“ n, | ‘ |ď } S p l q } S qN { l } X p l q } H qN { l ď c ρ n X where the last inequality follows from the induction claim and ( ). Further, from the Kunita-Watanabeinequality (Lemma ) and the generalized Hölder inequality, it follows that for all l , l P N ě with l ` l ď n we have ››› x X p l q c , X p l q c y ››› V qN {p l ` l q ď c } X p l q } H qN { l } X p l q } H qN { l Then we have again by Emery’s inequality, the induction base and ( ) that it holds ›››››› ÿ } ‘ }“ n, ď| ‘ |ď ż t S p l q u ´ d x X p l q c , X p l q c y u ›››››› H qN { n ď c ρ n X . Finally we have for the summation term ›››››› ÿ } ‘ }“ n, | ‘ |ě ÿ ď u ď t S p l j q u ´ ∆ X p l q u ¨ ¨ ¨ ∆ X p l k q u p j ´ q ! ›››››› H qN { n ď ÿ } ‘ }“ n, | ‘ |ě ››››› ÿ ď u ď t ˇˇˇ S p l k q u ´ ˇˇˇˇˇˇ ∆ X p l k ´ q u ˇˇˇ ¨ ¨ ¨ ˇˇˇ ∆ X p l q u ˇˇˇˇˇˇ ∆ X p l q u ∆ X p l q u ˇˇˇ››››› L qN {| w | ď c ÿ } ‘ }“ n, | ‘ |ě } S p l k q } S qN { l } X p l k ´ q } S qN { lk ´ ¨ ¨ ¨ } X p l q } S qN { l ››››› ÿ ď u ď t ˇˇˇ ∆ X p l q u ∆ X p l q u ˇˇˇ››››› L qN {p l ` l q ď c ρ n X with the last inequality follows again by the Kunita-Watanabe inequality, the induction basis and ( ).Thus we have shown that ( ) holds for all n P t , . . . , N u .Now we will proof the lower inequality, i.e. that there exists a constant c ą d , N and q such that c ||| X ||| H q,N ď ||| S ||| H q,N . (7.4)Therefore define ¯ X n – p , X p q , . . . , X p n q , , . . . , q P H q,N and note that it holds ||| ¯ X ||| H q,N “ } X p q } H qN “ } S p q } H qN ď ||| S ||| H q,N . Now assume that it holds ||| ¯ X n ´ ||| H q,N ď c ||| S ||| H q,N for some n P t , . . . , N u . It follows from the definition of the signature that S p n q t “ X p n q ,t ` Sig p ¯ X n ´ q p n q ,t (7.5)and further we have from the upper bound ( ), which was already proven above, that ||| Sig p ¯ X n ´ q , ¨ ||| H q,N ď C ||| ¯ X n ´ ||| H q,N ď Cc ||| S ||| H q,N . (7.6) Then we have ||| ¯ X n ||| H q,N “ ||| ¯ X n ´ ||| H q,N ` } X p n q } { n H qN { n ď c ||| S ||| H q,N ` } S p n q } { n H qN { n ` } Sig p ¯ X n ´ q p n q , ¨ } { n H qN { n ď c ||| S ||| H q,N ` ||| S ||| H q,N ` ||| Sig p ¯ X n ´ q , ¨ ||| H q,N ď c ||| S ||| H q,N , where we have used ( ) in the second line and ( ) in the last line. Therefore, noting that ¯ X N “ X ,the inequality ( ) follows by induction. (cid:3) Proof of Theorem . We prepare the proof of Theorem with a few more lemmas.
Lemma 7.5.
Let N P N ě then we have the following directional derivatives of the truncated exponentialmap exp N : T N Ñ T N pB w exp N qp x q “ G p ad x qp e w q exp N p x q “ exp N p x q G p´ ad x qp e w q , x P T N , pB w B w exp N qp x q “ r Q p ad x qp e w b e w q exp N p x q , x P T N , for all words w, w P W d with ď | w | , | w | ď N , where G is defined in ( ) and for r Q p ad x qp a b b q “ G p ad x qp b q G p ad x qp a q ` ż τ r G p τ ad x qp b q , e τ ad x p a qs d τ “ N ÿ n,m “ p ad x q n p b qp n ` q ! p ad x q m p a qp m ` q ! ` N ÿ n,m “ rp ad x q n p b q , p ad x q m p a qsp n ` m ` qp n ` q ! m ! , x , a, b P T N . Proof.
For all w P W d with 0 ď | w | ď N and x P T N the expression exp N p x q w is a polynomial in thetensor components p x v q ď| v |ď| w | . Therefore the map exp N : T N Ñ T N is smooth and in particular thefirst and second order partial derivatives exist in all directions. For a proof of the explicit form of the firstorder partial derivatives we refer to [ FV10 , Theorem 7.23]. For the second order derivatives we followthe proof of [
KPP20 , Lemma A.1]. Therefore let x P T N and w, w arbitrary with 1 ď | w | , | w | ď N .Then we have by the definition of the partial derivatives in T N and the product rule B w pB w exp N p x qq “ dd t ´ G p ad x ` te w qp e w q exp N p x ` te w q ¯ˇˇˇ t “ “ dd t G p ad x ` te w qp e w q ˇˇˇ t “ exp N p x q ` G p ad x qp e w q G p ad x qp e w q exp N p x q . From [
FV10 , Lemma 7.22] it holds exp N p ad x qp y q “ exp N p x q y exp N p´ x q for all x , y P T N and it followsfurther by representing G in integral form thatdd t G p ad x ` te w qp e w q ˇˇˇ t “ “ dd t ˆ ż exp N p τ ad x ` te w qp e w q d τ ˙ˇˇˇˇ t “ “ ż dd t ´ exp N p τ p x ` te w qq e w exp N p´ τ p x ` te w qq ¯ˇˇˇ t “ d τ “ ż τ G p ad τ x qp e w q exp N p τ x q e w exp N p´ τ x q d τ ´ ż τ exp N p τ x q e w exp N p´ τ x q G p ad τ x qp e w q d τ “ ż τ r G p τ ad x qp e w q , exp N p τ ad x qp e w qs d τ. NIFIED SIGNATURE CUMULANTS AND GENERALIZED MAGNUS EXPANSIONS 33
Then the proof is finished after noting that ż τ G p τ ad x qp e w q exp p τ ad x qp e w q d τ “ ż t τ N ÿ n,m “ p τ ad x q n p n ` q ! p τ ad x q m m ! d τ “ ż t N ÿ n,m “ p ad x q n p n ` q ! p ad x q m m ! τ ` m ` n d τ “ N ÿ n,m “ p ad x q n p ad x q m p n ` q ! m ! p n ` m ` q . (cid:3) Note that the operator Q defined in ( ) differs from the operator ˜ Q defined above. However wehave the following Lemma 7.6.
Let N P N ě and x P T N , then it holds r Q p ad x qp A q “ Q p ad x qp A q , for all A P T N b T N with symmetric coefficients A w ,w “ A w ,w for all w , w P W d .Proof. Let N P N ě and x P T N be arbitrary. Then from the bilinearity of ˜ Q p ad x q and the symmetryof A we have, with summation over all words w , w with 1 ď | w | , | w | ď N , r Q p ad x qp A q “ ÿ w ,w A w ,w N ÿ n,m “ ˆ p ad x q n p e w qp n ` q ! p ad x q m p e w qp m ` q ! ` rp ad x q n p e w q , p ad x q m p e w qsp n ` m ` qp n ` q ! m ! ˙ “ ÿ w ,w A w ,w ˆ N ÿ n,m “ p ad x q n p e w qp n ` q ! p ad x q m p e w qp m ` q ! ` p ad x q n p e w qp ad x q m p e w q ´ p ad x q m p e w qp ad x q n p e w qp n ` m ` qp n ` q ! m ! ˙ “ ÿ w ,w A w ,w ˆ N ÿ n,m “ p ad x q n p e w qp n ` q ! p ad x q m p e w qp m ` q ! ` p ad x q n p e w qp ad x q m p e w qp n ` m ` qp n ` q ! m ! ´ p ad x q n p e w qp ad x q m p e w qp m ` n ` qp m ` q ! n ! ˙ “ ÿ w ,w A w ,w N ÿ n,m “ p m ` q p ad x q n p e w qp ad x q m p e w qp n ` q ! p m ` q ! p n ` m ` q p A q “ Q p ad x qp A q . (cid:3) The following two applications of Itô’s formula in the non-commutative setting will be a key ingredientin the proof of Theorem . Lemma 7.7 (Itô’s product rule) . Let X , Y P S p T N q for some N P N ě , then it holds X t Y t ´ X Y “ ż p ,t s p d X u q Y u ` ż p ,t s X u p d Y u q ` m p (cid:74) X , Y (cid:75) ,T q , ď t ď T. Proof.
The statement is an immediate consequence of the one-dimensional Itô’s product rule for càdlàgsemimartingales (e.g. [
Pro05 , Ch. II, Corollary 2]) and the definition of the outer bracket and themultiplication map in Section . (cid:3) Lemma 7.8.
Let X P S p T N q for some N P N ě , then it holds exp N p X t q´ exp N p X q “ ż p ,t s G p ad X u ´ qp d X u q exp N p X u ´ q ` ż t Q p ad X u ´ qp d (cid:74) X c , X c (cid:75) u q exp N p X u ´ q` ÿ ă u ď t ´ exp N p X u q ´ exp N p X u ´ q ´ G p ad X u ´ qp ∆ X u q exp N p X u ´ q ¯ , for all ď t ď T . Proof.
As discussed in the proof of Lemma , it is clear that the map exp N : T N Ñ T N is smooth.Further T N is isomorphic to R D with D “ d ` ¨ ¨ ¨ ` d N and we can apply the multidimensional Itô’sformula for càdlàg semimartingales (e.g. [ Pro05 , Ch. II, Theorem 33]) to obtainexp N p X t q ´ exp N p X q “ ÿ ď| w |ď N ż p ,t s pB w exp N qp X u ´ q d X wu ` ÿ ď| w | , | w |ď N ż t pB w B w exp N qp X u ´ q d x X w c , X w c y u ` ÿ ă u ď t ˆ exp N p X u q ´ exp N p X u ´ q ´ ÿ ď| w |ď N pB w exp N qp X u ´ qp ∆ X wu q ˙ for all 0 ď t ď T . From Lemma we then have for the first integral term ÿ ď| w |ď N ż p ,t s pB w exp N qp X u ´ q d X wu “ ÿ ď| w |ď N ż p ,t s G p ad X u ´ qp e w q exp N p X u ´ q d X wu “ ż p ,t s G p ad X u ´ qp d X u q exp N p X u ´ q , and analogously ÿ ď| w |ď N pB w exp N qp X u ´ qp ∆ X wu q “ ÿ ď| w |ď N G p ad X u ´ qp ∆ X u q exp N p X u ´ q . Moreover, from Lemma and the definition of the outer bracket in Section ÿ ď| w | , | w |ď N ż p ,t s pB w B w exp N qp X u ´ q d x X w c , X w c y u “ ż t ˜ Q p ad X u ´qp d (cid:74) X c , X c (cid:75) u q exp N p X u ´ q . Finally, the outer bracket (cid:74) X c , X c (cid:75) t P T N b T N is symmetric in the sense of Lemma and thereforewe can replace ˜ Q with Q in the above identity. (cid:3) Lemma 7.9.
Let X P S p T q and let A P V p T q . For all k P N ě and ‘ “ p l , . . . , l k q P p N ě q k it holds ˇˇˇˇż t ´ ad X p l q u ¨ ¨ ¨ ad X p l k q u ¯´ d A p l q u ¯ˇˇˇˇ ď k ´ ż t ˇˇˇ X p l q u ¨ ¨ ¨ X p l k q u ˇˇˇˇˇˇ d A p l q u ˇˇˇ , for all ď t ď T . Furthermore, let p A t q ď t ď T be a process taking values in T b T such that A w ,w P V for all w , w P W d . Then it holds for all ď t ď T ˇˇˇˇż t ´ ad X p l q u ¨ ¨ ¨ ad X p l m q u d ad X p l m ` q u ¨ ¨ ¨ ad X p l k q u ¯´ d A p l ,l q u ¯ˇˇˇˇ ď k ´ ż t ˇˇˇ X p l q u ¨ ¨ ¨ X p l k q u ˇˇˇˇˇˇ d m ´ A p l ,l q ¯ˇˇˇ . Proof.
Recall from ( ) that we expand iterated adjoined operations into a sum of left- and righttensor multiplications and apply Lemma . Note again that for homogeneous tensors x and y it holds | xy | “ | yx | . Therefore the statement follows by counting the terms in the expansion. (cid:3) Lemma 7.10.
Let N P N ě , ∆ x , y , ∆ y P T N and define the function f : r , s ˆ r , s Ñ T N , p s, t q ÞÑ f p s, t q “ exp N p s ∆ x q exp N p y ` t ∆ y q exp N p´ y q . Then f p , q “ and the first order partial derivatives of f at p s, t q “ p , q are given by pB s f q| p s,t q“p , q “ ∆ x , pB t f q| p s,t q“p , q “ G p ad y qp ∆ y q . Further the following explicit bound for the second order partial derivatives holds sup ď s,t ď ˇˇˇ p ∇ f p n q q| p s,t q ˇˇˇ ď c n ÿ } ‘ }“ n, | ‘ |ě ´ˇˇˇ ∆ x p l q ˇˇˇ ` ˇˇˇ ∆ y p l q ˇˇˇ¯´ˇˇˇ ∆ x p l q ˇˇˇ ` ˇˇˇ ∆ y p l q ˇˇˇ¯ z l ¨ ¨ ¨ z l , for all n P t , . . . , N u , where c n ą is a constant depending only on n , ‘ “ p l , . . . , l k q P p N ě q k with | ‘ | “ k and z l – max t| ∆ x p l q | , | y p l q | , |p y ` ∆ y q p l q |u for all l P t , . . . , N ´ u . NIFIED SIGNATURE CUMULANTS AND GENERALIZED MAGNUS EXPANSIONS 35
Proof.
The tensor components of f p s, t q are polynomial in s and t and it follows that f is smooth.From Lemma we have that the first order partial derivatives of f are given by pB s f q| p s,t q “ G p ad s ∆ x qp ∆ x q exp N p s ∆ x q exp N p y ` t ∆ y q exp N p´ y q , pB t f q| p s,t q “ exp N p s ∆ x q G p ad y ` t ∆ y qp ∆ y q exp N p y ` t ∆ y q exp N p´ y q . Evaluating at s “ t “ n P t , . . . , N u . Then it follows fromLemmas and that we can bound the second order derivatives as followssup ď s,t ď pB ss f p n q q| p s,t q ď sup ď s,t ď ˇˇ π p n q ` Q p ad s ∆ x qpp ∆ x q b q exp N p s ∆ x q exp N p y ` t ∆ y q exp N p´ y q ˘ˇˇ ď c n ÿ } ‘ }“ n, | ‘ |ě | ∆ x p l q || ∆ x p l q | z l ¨ ¨ ¨ z l k , sup ď s,t ď pB tt f p n q q| p s,t q ď sup ď s,t ď ˇˇ π p n q ` exp N p s ∆ x q Q p ad y ` t ∆ y qpp ∆ y q b q exp N p y ` t ∆ y q exp N p´ y q ˘ˇˇ ď c n ÿ } ‘ }“ n, | ‘ |ě | ∆ y p l q || ∆ y p l q | z l ¨ ¨ ¨ z l k , and sup ď s,t ď pB st f p n q q| p s,t q ď sup ď s,t ď ˇˇ π p n q ` G p ad s ∆ x qp ∆ x q exp N p s ∆ x q G p ad y ` t ∆ y q exp N p y ` t ∆ y q exp N p´ y q ˘ˇˇ ď c n ÿ } ‘ }“ n, | ‘ |ě | ∆ x p l q || ∆ y p l q | z l ¨ ¨ ¨ z l k , where c n , c n , c n ą n and the second statement of the lemmafollows. (cid:3) Lemma 7.11.
For all N P N ě and all x P T N it holds H p ad x q ˝ G p ad x q “ Id . where G and H are defined in ( ) . Hence, the identity also holds for all x P T .Proof. Recall the exponent generating function of the Bernoulli numbers, for z near 0, H p z q “ ÿ n “ B k k ! z k “ ze z ´ , G p z q “ ÿ k “ k ` z k “ e z ´ z . Therefore H p z q G p z q ” z in a neighbourhood of zero. Repeated differentiation in z then yields the following property of the Bernoulli numbers n ÿ k “ B k k ! 1 p n ´ k ` q ! “ , n P N ě . Hence the statement of the lemma follows by projecting H p ad x q ˝ G p ad x q to each tensor level. (cid:3) We are now ready to give the
Proof of Theorem . Since π p ,N q Sig p X q “ Sig p X p ,N q q for any X P S p T q and all truncation levels N P N ě , it suffices to show that the identities ( ) and ( ) hold for the signature cumulant of anarbitrary X P H ,N . Recall from Theorem that this implies that ||| Sig p X q||| H ,N ă 8 and thusthe truncated signature cumulant κκκ “ p E t p Sig p X q t,T qq ď t ď T P S p T N q is well defined. Throughoutthe proof we will use the symbol " (cid:46) " to denote an inequality that holds up to a multiplication of theright-hand side by a constant that may depend only on d and N . Recall the definition of the signature in the Marcus sense from Section . Projecting ( ) to thetruncated tensor algebra, we see that the signature process S “ p Sig p X q ,t q ď t ď T P S p T N q satisfiesthe integral equation S t “ ` ż p ,t s S u ´ d X u ` ż t S u d x X c y u ` ÿ ă u ď t S u ´ ` exp N p ∆ X u q ´ ´ ∆ X u ˘ , (7.7)for 0 ď t ď T . Then by Chen’s relation ( ) we have E t p S T exp N p κκκ T qq “ E t p Sig p X q ,T q “ S t E t p Sig p X q t,T q “ S t exp N p κκκ t q , ď t ď T. It then follows from the above identity and the integrability of S T that the process S exp N p κκκ q is a T N -valued martingale in the sense of Section . On the other hand, we have by applying Itô’s productrule in Lemma S t exp N p κκκ t q ´ “ ż p ,t s p d S u q exp N p κκκ u ´ q ` ż p ,t s S u ´ p d exp N p κκκ u qq ` m ´ (cid:74) S c , exp N p κκκ q c (cid:75) ,t ¯ ` ÿ ă u ď t ∆ S u ∆ exp N p κκκ u q Further, by applying the Itô’s rule for the exponential map from Lemma to the T N -valuedsemimartingale κκκ and using ( ), we have the following form of the continuous covariation term m ´ (cid:74) S c , exp N p κκκ c q (cid:75) ,t ¯ “ m ¨˝ (cid:116) ż p , ¨s S u ´ d X cu , ż p , ¨s G p ad κκκ u ´ qp d κκκ cu q exp N p κκκ u ´ q (cid:124) ,t ˛‚ “ ż p ,t s S u ´ p Id d G p ad κκκ u ´ qqp d (cid:74) X c , κκκ c (cid:75) u q exp N p κκκ u ´ q and for the jump covariation term ÿ ă u ď t ∆ S u ∆ exp N p κκκ u q “ ÿ ă u ď t S u ´ ` exp N p ∆ X u q ´ ˘` exp N p κκκ u q exp N p´ κκκ u ´ q ´ ˘ exp N p κκκ u ´ q . From the above identities and again with Lemma and ( ) we have S t exp N p κκκ t q ´ “ ż p ,t s S u ´ d p L u ` κκκ u q exp N p κκκ u ´ q , ď t ď T, (7.8)where L P S p T N q is defined by L t “ X t ` x X c y t ` ÿ ă u ď t ` exp N p ∆ X u q ´ ´ ∆ X u ˘ ` ż p ,t s p G ´ Id qp ad κκκ u ´ qp d κκκ u q` ż t Q p ad κκκ u ´ qp d (cid:74) κκκ c , κκκ c (cid:75) u q ` ÿ ă u ď t ´ exp N p κκκ u q exp N p´ κκκ u ´ q ´ ´ G p ad κκκ u ´ qp ∆ κκκ u q ¯ ` ż p ,t s p Id d G p ad κκκ u ´ qqp d (cid:74) X c , κκκ c (cid:75) u q` ÿ ă u ď t ` exp N p ∆ X u q ´ ˘` exp N p κκκ u q exp N p´ κκκ u ´ q ´ ˘ “ X t ` x X c y t ` ż p ,t s p G ´ Id qp ad κκκ u ´ qp d κκκ u q ` V t ` C t ` J t , (7.9)with V , C , J P V p T q given by V t “ ż t Q p ad κκκ u ´ qp d (cid:74) κκκ c , κκκ c (cid:75) u q , C t “ ż p ,t s p Id d G p ad κκκ u ´ qqp d (cid:74) X c , κκκ c (cid:75) u q , J t “ ÿ ă u ď t ` exp N p ∆ X u q exp N p κκκ u q exp N p´ κκκ u ´ q ´ ´ ∆ X u ´ G p ad κκκ u ´ qp ∆ κκκ u q ˘ . Note that we have explicitly separated the identity operator Id from G in the above definition of L .Since left-hand side in ( ) is a martingale and since S t and exp p κκκ t q have the multiplicative left-respectively right-inverse S ´ t and exp p´ κκκ t q respectively for all 0 ď t ď T , it follows that L ` κκκ is a NIFIED SIGNATURE CUMULANTS AND GENERALIZED MAGNUS EXPANSIONS 37 T N -valued local martingale. Let p τ k q k ě be a sequence of increasing stopping times with τ k Ñ T a.s.for k Ñ 8 , such that the stopped process p L t ^ τ k ` κκκ t ^ τ k q ď t ď T is a true martingale. Using furtherthat κκκ T “ κκκ t ^ τ k “ E t (cid:32) L T ^ τ k ,t ^ τ k ( , ď t ď T, k P N ě . (7.10)The estimate ( ) below shows that L has sufficient integrability in order to use the dominatedconvergence theorem to pass to the k Ñ 8 limit in the above identity ( ), which yields precisely theidentity ( ) and hence concludes the first part of the proof of Theorem . Claim 7.12.
It holds that ||| L ||| H ,N (cid:46) ||| X ||| H ,N . (7.11) Proof of Claim . According to Lemma it suffices to show that for all n P t , . . . , N u it holds } L p n q } H qN { n (cid:46) ÿ } ‘ }“ n } X p l q } H qN { l ¨ ¨ ¨ } X p l j q } H qN { lj — ρ n X , where the summation above (and in the rest of the proof) is over multi-indices ‘ “ p l , . . . , l j q P p N ě q j ,with | ‘ | “ j and } ‘ } “ l ` ¨ ¨ ¨ ` l j . For M P M loc p T N q and A P V p T N q define ρ n M , A – ÿ } ‘ }“ n ζ l { N p M p l q , A p l q q ¨ ¨ ¨ ζ l j { N p M p l j q , A p l j q q ă 8 , n “ , . . . , N, where for any q P r , ζ q p M p l q , A p l q q : “ ››› |r M p l q s T | { ` | A p l q | ´ var; r ,T s ››› L q . Note that it holds ρ n M , A ď ÿ } ‘ }“ n ´ ρ l M , A ¨ ¨ ¨ ρ l j M , A ¯ (cid:46) ρ n M , A . (7.12)Furthermore, it follows from the definition of the H q -norm that ρ n X “ inf X “ M ` A ρ n M , A , (7.13)where the infimum is taken over all semimartingale decomposition of X .Now fix M P M loc p T N q and A P V p T N q arbitrarily, such that X “ M ` A and ρ n M , A ă 8 for all n P t , . . . , N u (such a decomposition always exists since X P H ,N ). In particular it holds that M isa true martingale. Next we will proof the following Claim 7.13.
For all n P t , . . . , N u it holds } L p n q } H N { n (cid:46) ρ n M , A , (7.14)and further there exits a semimartingale decomposition κκκ p n q “ κκκ p n q ` m p n q ` a p n q , with m p n q P M pp R d q b n q and a p n q P V pp R d q b n q such that } a p n q } V N { n ă ρ n M , A and in case n ď N ´ ζ n { N p m p n q , a p n q q (cid:46) ρ n M , A . (7.15) Proof of Claim . We are going to proof inductively over n P t , . . . , N u . Let n “ L p q “ X p q , and therefore } L p q } H N ď ζ N p M p q , A p q q “ ρ M , A . Using that M p q is a martingale we can identify a semimartingale decomposition of κκκ p q by m p q t – E t ´ A p q T ¯ ´ E ´ A p q T ¯ , a p q t – ´ A t , ď t ď T. In case N ě
2, we further have from the BDG-inequality and the Doob’s maximal inequality that ››› m p q ››› H N (cid:46) ››› m p q ››› S N (cid:46) ››› m p q T ››› L N “ ››› E ´ A p q T ¯ ´ A p q T ››› L N (cid:46) ››› A p q ››› V N { n (cid:46) ρ M , A and this shows the second part of the induction claim.Now assume that N ě ) and ( ) holds true up level n ´ n P t , . . . , N u . Note that L p n q has the following decomposition L p n q “ ! M p n q ` N p n q ) ` " A p n q ` x X c y p n q ` B p n q ` V p n q ` C p n q ` J p n q * , (7.16) where N p n q P M loc pp R d q b n q and B p n q P V pp R d q b n q are defined by N p n q “ π p n q ż p ,t s p G ´ Id qp ad κκκ u ´ qp d m u q , B p n q “ π p n q ż p ,t s p G ´ Id qp ad κκκ u ´ qp d a u q with a “ π p ,N q p a p q ` ¨ ¨ ¨ ` a p n ´ q q P V p T N q and m “ π p ,N q p m p q ` ¨ ¨ ¨ ` m p n ´ q q P M p T N q .From Lemma and the generalized Hölder inequality we have ››› x X c y p n q ››› V N { n ď n ÿ i “ ›››A M p i q c , M p n ´ i q c E››› V N { n (cid:46) n ÿ i “ ››› M p i q ››› H N { i ››› M p n ´ i q ››› H N {p n ´ i q (cid:46) ρ n M , A . (7.17)It follows from ( ) and the induction basis that for all l P t , . . . , n ´ u it holds that } κκκ p l q } H N { l “ } κκκ p l q ´ κκκ p l q } H N { l (cid:46) ρ l M , A , (7.18)and further that κ l ˚ T – sup ď t ď T | κκκ p l q t | , } κκκ p l q } S N { l “ } κ l ˚ T } L N { l ď | κκκ p l q | ` } κκκ p l q ´ κκκ p l q } S N { l (cid:46) ρ l M , A . (7.19)From the definition and linearity of Q p ad x q ( x P T N ), Lemmas and we have the followingestimate ˇˇˇ V p n q ˇˇˇ ´ var; r ,T s (cid:46) ÿ } ‘ }“ n, | ‘ |ě j ÿ m “ ˇˇˇˇż ¨ ´ ad κκκ p l q u ´ ¨ ¨ ¨ ad κκκ p l m q u ´ d ad κκκ p l m ` q u ´ ¨ ¨ ¨ ad κκκ p l j q u ´ ¯´ d (cid:114) m p l q c , m p l q c (cid:122) u ¯ˇˇˇˇ ´ var; r ,T s (cid:46) ÿ } ‘ }“ n, | ‘ |ě ż t ˇˇˇ κκκ p l q u ´ ˇˇˇ ¨ ¨ ¨ ˇˇˇ κκκ p l j q u ´ ˇˇˇ d ˇˇˇA m p l q c , m p l l q c E u ˇˇˇ (cid:46) ÿ } ‘ }“ n, | ‘ |ě κ l ˚ T ¨ ¨ ¨ κ l j ˚ T bˇˇ“ m p l q ‰ T ˇˇbˇˇ“ m p l q ‰ T ˇˇ . It then follows from the generalized Hölder inequality ››› V p n q ››› V N { n (cid:46) ÿ } ‘ }“ n, | ‘ |ě ››› κκκ p l q T ››› S N { l ¨ ¨ ¨ ››› κκκ p l j q T ››› S N { lj ››› m p l q ››› H N { l ››› m p l q ››› H N { l (cid:46) ÿ } ‘ }“ n, | ‘ |ě ρ l M , A ¨ ¨ ¨ ρ l j M , A (cid:46) ρ n M , A , (7.20)where the second inequality follows from the induction basis and the estimates ( ) and ( ), notingthat } ‘ } “ n and | l | ě l , . . . , l j ď n ´
1, and the third inequality follows from ( ).From similar arguments we see that the following two estimate also hold ››› C p n q ››› V N { n (cid:46) ÿ } ‘ }“ n, | ‘ |ě ››››ż ¨ ´ Id d ad κκκ p l q u ´ ¨ ¨ ¨ ad κκκ p l j q u ´ ¯´ d (cid:114) M p l q c , m p l q c (cid:122) u ¯›››› V N { n (cid:46) ÿ } ‘ }“ n, | ‘ |ě ››› κκκ p l q T ››› S N { l ¨ ¨ ¨ ››› κκκ p l j q T ››› S N { lj ››› M p l q ››› H N { l ››› m p l q ››› H N { l (cid:46) ρ n M , A (7.21)and ››› B p n q ››› V N { n (cid:46) ÿ } ‘ }“ n, | ‘ |ě ››› κκκ p l q T ››› S N { l ¨ ¨ ¨ ››› κκκ p l j q T ››› S N { lj ››› a p l q ››› V N { l (cid:46) ρ n M , A . (7.22) NIFIED SIGNATURE CUMULANTS AND GENERALIZED MAGNUS EXPANSIONS 39
For the local martingale N p n q , we use Lemmas and to estimate its quadratic variation as follows ˇˇˇ” N p n q ı T ˇˇˇ “ ˇˇˇˇˇˇ»– ÿ } ‘ }“ n, | ‘ |ě ż p , ¨s k ! ´ ad κκκ p l q u ´ ¨ ¨ ¨ ad κκκ p l k q u ´ ¯´ d m p l q u ¯fifl T ˇˇˇˇˇˇ (cid:46) ÿ } ‘ }“ n, | ‘ |ě ˇˇˇˇˇż p ,T s ´ ad κκκ p l q u ´ ¨ ¨ ¨ ad κκκ p l m q u ´ d ad κκκ p l m ` q u ´ ¨ ¨ ¨ ad κκκ p l k q u ´ ¯´ d (cid:114) m p l q , m p l q (cid:122) u ¯ˇˇˇˇˇ (cid:46) ÿ } ‘ }“ n, | ‘ |ě κ l ˚ T ¨ ¨ ¨ κ l j ˚ T bˇˇ“ m p l q ‰ T ˇˇbˇˇ“ m p l q ‰ T ˇˇ . Then it follows once again by the generalized Hölder inequality and the induction basis that ››› N p n q ››› H N { n (cid:46) ρ n M , A . (7.23)Finally let us treat the term J p n q . First define Z l – sup ă u ď T ´ max !ˇˇˇ ∆ X p l q u ˇˇˇ , ˇˇˇ κκκ p l q u ´ ˇˇˇ , ˇˇˇ κκκ p l q u ˇˇˇ)¯ , l “ t , . . . , n ´ u . and from( ) it follows that for all l “ t , . . . , n ´ u it holds } Z l } L N { l ď } X p l q } S N { l ` } κκκ p l q T } S N { l ď } X p l q } H N { l ` } κκκ p l q T } S N { l (cid:46) ρ l,N M , A . (7.24)Then by Taylor’s theorem and Lemma we have ˇˇˇ J p n q ˇˇˇ ´ var; r ,T s “ ÿ ă u ď T ˇˇˇˇ π p n q ´ exp N p ∆ X u q exp N p κκκ u q exp N p´ κκκ u ´ q ´ ´ ∆ X u ´ G p ad κκκ u ´ qp ∆ κκκ u q ¯ˇˇˇˇ (cid:46) ÿ } ‘ }“ n, || ‘ |ě Z l ¨ ¨ ¨ Z l j ÿ ă u ď T ´ˇˇˇ ∆ X p l q u ˇˇˇ ` ˇˇˇ ∆ κκκ p l q u ˇˇˇ¯´ˇˇˇ ∆ X p l q u ˇˇˇ ` ˇˇˇ ∆ κκκ p l q u ˇˇˇ¯ (cid:46) ÿ } ‘ }“ n, || ‘ |ě Z l ¨ ¨ ¨ Z l j ´bˇˇ“ X p l q ‰ T ˇˇ ` bˇˇ“ κκκ p l q ‰ T ˇˇ¯´bˇˇ“ X p l q ‰ T ˇˇ ` bˇˇ“ κκκ p l q ‰ T ˇˇ¯ . Hence it follows by the generalized Hölder inequality that ››› J p n q ››› V N { n (cid:46) ÿ } ‘ }“ n, || ‘ |ě ›› Z l ›› L N { l ¨ ¨ ¨ ›› Z l j ›› L N { lj ´››› X p l q ››› H N { l ` ››› κκκ p l q ››› H N { l ¯ ¨ ´››› X p l q ››› H N { l ` ››› κκκ p l q ››› H N { l ¯ (cid:46) ρ n M , A (7.25)where the last estimate follows from ( ), ( ) and ( ).Summarizing the estimates ( ), ( ), ( ), ( ) and ( ) we have ››› L p n q ››› H N { n (cid:46) ››› M p n q ››› H N { n ` ››› N p n q ››› H N { n ` ››› A p n q ››› V N { n ` ››› x X c y p n q ››› V N { n ` ››› B p n q ››› V N { n ` ››› V p n q ››› V N { n ` ››› C p n q ››› V N { n ` ››› J p n q ››› V N { n (cid:46) ρ n M , A , (7.26)which proofs the first part of the induction claim ( ). Then it follows form dominated convergencetheorem that projecting ( ) to the tensor level n and passing to the k Ñ 8 limit yields κκκ p n q t “ E t ´ L p n q T,t ¯ , ď t ď T. Since M p n q and N p n q are true martingales (for the latter this follows from ( )), we are able toidentify a decomposition κκκ p n q “ κκκ p n q ` m p n q ` a p n q by a p n q “ ´ " A p n q ` x X c y p n q ` B p n q ` V p n q ` C p n q ` J p n q * m p n q t “ E ´ a p n q T ¯ ´ E t ´ a p n q T ¯ , ď t ď T. Again from the estimates ( ), ( ), ( ) and ( ) it follows that } a p n q } V N { n (cid:46) ρ n M , A and in case n ď N ´ ››› m p n q ››› H N { n (cid:46) ››› m p n q ››› S N { n (cid:46) ››› m p n q T ››› L N { n “ ››› E ´ a p n q T ¯ ´ a p n q T ››› L N { n (cid:46) ››› a p n q ››› V N { n (cid:46) ρ n M , A , which proofs the second part of the induction claim ( ). (cid:4) The estimate ( ) immediately follows from ( ) and ( ), which finishes the proof of Claim . (cid:4) Note that since x X c y , V , C and J are independent of the decomposition X “ M ` A it follows fromtaking the infimum over all such decompositions in the inequality ( ) that ››› x X c y p n q ››› V N { n ` ››› B p n q ››› V N { n ` ››› V p n q ››› V N { n ` ››› C p n q ››› V N { n ` ››› J p n q ››› V N { n (cid:46) ρ n X , (7.27)for all n P t , . . . , N u . The same argument applies to κκκ and the estimate ( ) and we obtain ››› κκκ p n q ››› H N { n (cid:46) ρ n X , (7.28)for all n P t , . . . , N ´ u .Next we are going to show that κκκ satisfies the functional equation ( ). Recall that L ` κκκ P M loc p T N q .From Lemma we have the following equality ż p ,t s H p ad κκκ u ´ qp d p L u ` κκκ u qq “ κκκ t ´ κκκ ` r L t for all 0 ď t ď T , where r L t “ ż p ,t s H p ad κκκ u ´ q " d X u `
12 d x X c y u ` d V u ` d C u ` d J u * . (7.29)From Lemma (Emery’s inequality) and the estimates ( ) and ( ) it follows } r L p n q } H N { n (cid:46) ÿ } ‘ }“ n ››› κκκ p l q ››› S N { l ¨ ¨ ¨ ››› κκκ p l j q ››› S N { lj !››› X p l q ››› H N { l ` ››› x X c y p l q ››› V N { l ` ››› V p l q ››› V N { l ` ››› C p l q ››› V N { l ` ››› J p l q ››› V N { l ) (cid:46) ρ n X , Hence by Lemma it holds ||| r L ||| H ,N (cid:46) ||| X ||| H ,N . (7.30)Now note that we have already shown in Claim that κκκ “ κκκ ` m ` a , where m P M p T N q and a P V p T N q which satisfies that } a p n q } V ă 8 for all n P t , . . . , N u . Together with the above estimateit then follows that κκκ ` r L is indeed a true martingale and therefore κκκ t “ E ´r L T,t ¯ , ď t ď T, which is precisely the identity ( ). (cid:3) References [AGR20] E. Alos, J. Gatheral, and R. Radoičić,
Exponentiation of conditional expectations under stochastic volatility ,Quantitative Finance; SSRN (2017) (2020), no. 1, 13–27.[AJLP19] E. Abi Jaber, M. Larsson, and S. Pulido, Affine volterra processes , Ann. Appl. Probab. (2019) , no. 5,3155–3200.[App09] D. Applebaum, Lévy processes and stochastic calculus , Cambridge university press, 2009.[BCEF20] Y. Bruned, C. Curry, and K. Ebrahimi-Fard,
Quasi-shuffle algebras and renormalisation of rough differentialequations , Bulletin of the London Mathematical Society (2020), no. 1, 43–63.[BCOR09] S. Blanes, F. Casas, J. Oteo, and J. Ros, The magnus expansion and some of its applications , Phys. Rep. (2009), no. 5-6, 151–238.[BO20] P. Bonnier and H. Oberhauser,
Signature cumulants, ordered partitions, and independence of stochasticprocesses , Bernoulli (2020), no. 4, 2727–2757.[CE15] S. Cohen and R. J. Elliott, Stochastic calculus and applications , 2nd ed., Birkhäuser, Basel, 2015.[CF19] I. Chevyrev and P. K. Friz,
Canonical rdes and general semimartingales as rough paths , Ann. Probab. (2019) , no. 1, 420–463. NIFIED SIGNATURE CUMULANTS AND GENERALIZED MAGNUS EXPANSIONS 41 [CFMT11] C. Cuchiero, D. Filipović, E. Mayerhofer, and J. Teichmann,
Affine processes on positive semidefinitematrices , The Annals of Applied Probability (2011), no. 2, 397–463.[Che54] K.-T. Chen, Iterated integrals and exponential homomorphisms† , Proc. London Math. Soc. s3-4 (1954) ,no. 1, 502–512.[CL16] I. Chevyrev and T. Lyons,
Characteristic functions of measures on geometric rough paths , Ann. Probab. (2016) , no. 6, 4049–4082. MR 3572331[CM09] F. Casas and A. Murua, An efficient algorithm for computing the Baker-Campbell-Hausdorff series andsome of its applications , J. Math. Phys. (2009) , no. 3, 033513, 23. MR 2510918[CT04] R. Cont and P. Tankov, Financial modelling with jump processes , 1 ed., Financial Mathematics Series,Chapman & Hall/CRC, 2004.[DFS `
03] D. Duffie, D. Filipović, W. Schachermayer, et al.,
Affine processes and applications in finance , Ann. Appl.Probab. (2003), no. 3, 984–1053.[Est92] A. Estrade, Exponentielle stochastique et intégrale multiplicative discontinues , Ann. Inst. Henri PoincaréProbab. Stat. (1992), no. 1, 107–129.[Faw02] T. Fawcett, Problems in stochastic analysis : connections between rough paths and non-commutativeharmonic analysis , Ph.D. thesis, University of Oxford, 2002.[FGR20] P. K. Friz, J. Gatheral, and R. Radoičić,
Forests, cumulants, martingales , 2020, arXiv:2002.01448[math.PR] .[FH20] P. K. Friz and M. Hairer,
A course on rough paths , 2nd ed., Universitext, Springer International Publishing,2020.[FS17] P. K. Friz and A. Shekhar,
General rough integration, Lévy rough paths and a Lévy–Kintchine-type formula , Ann. Probab. (2017) , no. 4, 2707–2765.[FV06] P. K. Friz and N. B. Victoir, The burkholder-davis-gundy inequality for enhanced martingales , Lecture Notesin Mathematics (2006) .[FV10] ,
Multidimensional stochastic processes as rough paths: Theory and applications , Cambridge Studiesin Advanced Mathematics, Cambridge University Press, 2010.[GKR19] J. Gatheral and M. Keller-Ressel,
Affine forward variance models , Finance Stoch. (2019), no. 3, 501–533.[Hau06] F. Hausdorff, Die symbolische Exponentialformel in der Gruppentheorie , Ber. Verh. Kgl. Sächs. Ges. Wiss.Leipzig., Math.-phys. Kl. (1906), 19–48.[HDL86] M. Hakim-Dowek and D. Lépingle, L’exponentielle stochastique des groupes de Lie , Séminaire de ProbabilitésXX 1984/85, Springer, 1986, pp. 352–374.[IMKNZ05] A. Iserles, H. Munthe-Kaas, S. Nørsett, and A. Zanna,
Lie-group methods , Acta numerica (2005).[IN99] A. Iserles and S. P. Nørsett,
On the solution of linear differential equations in lie groups , Philos. Trans. Roy.Soc. A (1999), no. 1754, 983–1019.[Jac79] J. Jacod,
Calcul stochastique et problèmes de martingales , Lecture Notes in Mathematics, vol. 714, SpringerBerlin Heidelberg, Berlin, Heidelberg, 1979 (eng).[JS03] J. Jacod and A. N. Shiryaev,
Limit theorems for stochastic processes. , Grundlehren der MathematischenWissenschaften [Fundamental Principles of Mathematical Sciences], no. 488, Springer Berlin, 2003.[KLP18] M. Keller-Ressel, M. Larsson, and S. Pulido,
Affine Rough Models , arXiv e-prints (2018), arXiv:1812.08486.[KPP95] T. G. Kurtz, E. Pardoux, and P. Protter,
Stratonovich stochastic differential equations driven by generalsemimartingales , Ann. Inst. Henri Poincaré Probab. Stat. (1995), no. 2, 351–377.[KPP20] K. Kamm, S. Pagliarani, and A. Pascucci, The stochastic magnus expansion , 2020, arXiv:2001.01098[math.PR] .[KRST11] M. Keller-Ressel, W. Schachermayer, and J. Teichmann,
Affine processes are regular , Probab. Theory RelatedFields (2011), no. 3-4, 591–611.[KS98] I. Karatzas and S. Shreve,
Brownian motion and stochastic calculus , 2 ed., Graduate Texts in Mathematics,vol. 113, Springer, New York, NY, 1998.[LN15] T. Lyons and H. Ni,
Expected signature of brownian motion up to the first exit time from a bounded domain , Ann. Probab. (2015) , no. 5, 2729–2762.[LQ11] Y. LeJan and Z. Qian, Stratonovich’s signatures of brownian motion determine brownian sample paths , Probability Theory and Related Fields (2011) .[LRV19] H. Lacoin, R. Rhodes, and V. Vargas,
A probabilistic approach of ultraviolet renormalisation in the boundarysine-gordon model , 2019, arXiv:1903.01394 [math.PR] .[LV04] T. Lyons and N. Victoir,
Cubature on wiener space , Proceedings of the Royal Society of London. Series A:Mathematical, Physical and Engineering Sciences (2004), no. 2041, 169–198.[Lyo14] T. Lyons,
Rough paths, signatures and the modelling of functions on streams , Proceedings of the InternationalCongress of Mathematicians—Seoul 2014. Vol. IV, Kyung Moon Sa, Seoul, 2014, pp. 163–184. MR 3727607[Mag54] W. Magnus,
On the exponential solution of differential equations for a linear operator , Commun. Pure Appl.Math. (1954) , no. 4, 649–673.[Mar78] S. I. Marcus, Modeling and analysis of stochastic differential equations driven by point processes , IEEETrans. Inform. Theory (1978), no. 2, 164–172.[Mar81] , Modeling and approximation of stochastic differential equations driven by semimartingales , Stochastics (1981) , no. 3, 223–245.[McK69] H. P. McKean, Stochastic integrals , AMS Chelsea Publishing Series, no. 353, American Mathematical Society,1969.[Mil72] W. Miller, Jr.,
Symmetry groups and their applications , Pure and Applied Mathematics, vol. 50, AcademicPress, New York-London, 1972.[Myk94] P. A. Mykland,
Bartlett type identities for martingales , Ann. Statist. (1994) , no. 1, 21–38. [Ni12] H. Ni, The expected signature of a stochastic process , Ph.D. thesis, University of Oxford, 2012.[Øks14] B. Øksendal,
Stochastic differential equations: An introduction with applications , 6 ed., Springer Berlin /Heidelberg, Berlin, Heidelberg, 2014 (eng).[Pro05] P. E. Protter,
Stochastic integration and differential equations , 2 ed., Stochastic Modelling and AppliedProbability, Springer-Verlag Berlin Heidelberg, 2005.[Reu03] C. Reutenauer,
Free lie algebras , Handbook of algebra, vol. 3, Elsevier, 2003, pp. 887–903.[Str75] D. W. Stroock,
Diffusion processes associated with lévy generators , Zeitschrift für Wahrscheinlichkeitstheorieund verwandte Gebiete (1975), no. 3, 209–244.[You36] L. C. Young, An inequality of the hölder type, connected with stieltjes integration , Acta Math. (1936)(1936)