## Abstract

We consider stochastic differential systems driven by continuous semimartingales and governed by non-commuting vector fields. We prove that the logarithm of the flowmap is an exponential Lie series. This relies on a natural change of basis to vector fields for the associated quadratic covariation processes, analogous to Stratonovich corrections. The flowmap can then be expanded as a series in compositional powers of vector fields and the logarithm of the flowmap can thus be expanded in the Lie algebra of vector fields. Further, we give a direct explicit proof of the corresponding Chen–Strichartz formula which provides an explicit formula for the Lie series coefficients. Such exponential Lie series are important in the development of strong Lie group integration schemes that ensure approximate solutions themselves lie in any homogeneous manifold on which the solution evolves.

## 1. Introduction

We are concerned with Itô stochastic differential systems driven by continuous semimartingales and governed by non-commuting vector fields of the following form:
*t*∈[0,*T*] for some *T*>0. Here, the solution process *Y*_{t} is *i*=1,…,*d*, the *V*_{i} which are sufficiently smooth and in general non-commuting. Our goal herein is to compute the logarithm of the flowmap for such a system, i.e. the exponential series for the flowmap, and establish that it is a Lie series. The exponential series for the flowmap for stochastic differential systems driven by general continuous semimartingales was derived in Ebrahimi-Fard *et al.* [1]. What we achieve that is new in this paper is we:

(i) Establish the abstract algebraic structures that underlie the flowmap and computation of functions of the flowmap in the context of general continuous semimartingales;

(ii) Show by a suitable change of coordinates, the exponential series is a Lie series, and give a direct explicit proof of the corresponding Chen–Strichartz formula which provides an explicit formula for the Lie series coefficients.

The key idea that underlies establishing the exponential series as a Lie series is to express the flowmap in Fisk–Stratonovich form for which the standard rules of calculus apply [2]. A crucial integral ingredient in this step is that the Fisk–Stratonovich formulation of the flowmap can be expanded in a basis of terms involving solely compositions of vector fields—without any second-order partial differential operators. We can then compute the logarithm of the Fisk–Stratonovich representation of the flowmap. This can be accomplished in principle via the classical Chen–Strichartz formula using the shuffle relations satisfied by multiple Fisk–Stratonovich integrals. We subsequently convert the multiple Fisk–Stratonovich integrals back into multiple Itô integrals. This procedure thus generates an Itô exponential Lie series. Our proof uses that the logarithm is a Lie element and the Dynkin–Friedrichs–Specht–Wever theorem to expand the logarithm in Lie polynomials of the vector fields (e.g. Theorem 1.4 and Lemma 3.8 in [3]). It is otherwise self-contained. That the logarithm of the flowmap is in fact an exponential Lie series is important for example, for the development of strong stochastic Lie group integration methods. See Malham & Wiese [4] for the development of such methods for Stratonovich stochastic differential systems driven by Wiener processes, for example those based on the Castell–Gaines numerical simulation approach [5,6].

The development of exponential solution series for deterministic systems originates with the work of Magnus [7] and Chen [8] in the 1950s, and more recently with Strichartz [9]. Its development and early application to stochastic systems is represented by the work of Azencott [10], Ben Arous [11], Castell & Gaines [5,6] and Baudoin [12]. Also see Fliess [13] and Lyons [14] for its development in control and in the theory of rough paths, respectively. The shuffle product was cemented in firm foundations by the work of Eilenberg & Mac Lane [15] and Schützenberger [16], also in the 1950s. The quasi-shuffle product is a natural extension of the shuffle product, for example to multiple Itô integrals. For a selective insight into its recent development in this context, see Gaines [17,18], Hoffman [19], Ebrahimi-Fard & Guo [20], Hoffman & Ihara [21] and Curry *et al.* [22].

Our paper is structured as follows. In §2, we derive the Itô chain rule and flowmap for systems driven by continuous semimartingales. Then in §3, we establish the abstract algebraic structures that underpin the flowmap and its logarithm. We endeavour to keep the connection to the stochastic differential system of interest and provide illustrative examples. We define the transformation to Stratonovich form we require in §4 and prove that the exponential series is a Lie series. Our direct explicit derivation of the Chen–Strichartz coefficients is provided in §5. Lastly, we provide some concluding remarks in §6.

## 2. Itô chain rule and flowmap

Consider the Itô stochastic differential system given in the introduction of the form
*i*=1,…,*d*, the *usual conditions* of completeness and right-continuity. We assume without loss of generality that the *X*^{i} are chosen such that the quadratic covariations [*X*^{i},*X*^{j}]=0 for all *i*≠*j*. The *V*_{i} are associated governing vector fields which we assume are sufficiently smooth and in general non-commuting. We suppose the solution process *Y*_{t}, which is *V*_{i} for each *i*=1,…,*d* act as first-order partial differential operators on any function *V*_{i}⋅∂)*f*(*Y*) or *V*_{i}○*f*○*Y* .

### Definition 2.1 (Flowmap)

We define the flowmap *φ*_{t} as the map prescribing the transport of the initial data *f*○*Y*_{0} to the solution *f*○*Y*_{t} at time *t* for any smooth function *f* on *φ*_{t}: *f*○*Y*_{0}↦*f*○*Y*_{t}.

The solution *Y*_{t}=*φ*_{t}○id○*Y*_{0} corresponds to the choice *f*=id, the identity map. The Itô chain rule is the key to developing the Taylor series expansion for the solution *Y*_{t} about the initial data. The Itô chain rule implies that for any function *f*(*Y*_{t}) satisfies
*i*=1,…,*d*, the terms [*X*^{i},*X*^{i}] represent the quadratic variation of *X*^{i}. At this stage it makes sense to extend, first our set of driving continuous semimartingales to include these quadratic variations, and second, our governing vector fields to include the associated second-order partial differential operators shown above. Thus, for *i*=1,…,*d*, we set *D*_{i}:=*V*_{i}⋅∂ and
*d*,[1,1],…,[*d*,*d*]}. Iterating this chain rule produces the formal Taylor series expansion for the solution and thus flowmap given by
*w* that can be constructed from the alphabet *I*_{w}=*I*_{w}(*t*) while the geometric information is encoded through the composition of partial differential operators *D*_{w}. For a word *w*=*a*_{1}⋯*a*_{n}, these terms are *D*_{w}:=*D*_{a1}○⋯○*D*_{an} and
*I*_{u} and *I*_{v} generates a sum over all multiple Itô integrals generated by the quasi-shuffle of the words *u* and *v*; we define this product precisely, presently. The algebra on the right should be endowed with a concatenation product, to reflect the fact that the composition of two differential operators *D*_{u} and *D*_{v} generates the differential operator equivalent to that represented by the concatenation of the words *u* and *v*. In the next section, we define these underlying concatenation and quasi-shuffle algebras, their corresponding Hopf algebras and the algebras associated with endomorphisms on them. These algebras prove useful in the following sections, they keep our proofs direct and succinct.

## 3. Quasi-shuffle Hopf algebras and endomorphisms

Our exposition here is based on Reutenauer [3], Hoffman [19] and Hoffman & Ihara [21]. Let

### Definition 3.1 (Bilinear form)

We define the bilinear form

This is equivalent to the scalar product given in Reutenauer [3, p. 17] and Hoffman [19, p. 57]. For this scalar product, the free monoid

### Example 3.2

Consider a minimal family of general semimartingales {*X*^{1},…,*X*^{d}} in the sense outlined in Curry *et al.* [22]. We do not restrict ourselves here to continuous semimartingales. However, a collection of independent continuous semimartingales, or a collection of independent Lévy processes, is a minimal family. We can construct a countable alphabet *d*. In addition, inductively for *n*≥2, we assign a distinct new letter for each nested quadratic covariation process [*X*^{k1},[*X*^{k2},[…[*X*^{kn−1},*X*^{kn}]…]]] with *k*_{i}∈{1,…,*d*} for *i*=1,…,*n*, ‘provided it is not in the linear span of {*X*^{1},…,*X*^{d}} and previously constructed ones’. Due to commutativity, the order of the letters in the *n*-fold nested bracket is irrelevant and associativity means that we can render all *n*-fold nested brackets to the canonical form of left to right bracketing shown, or more conveniently [*X*^{k1},…,*X*^{kn}]. We denote the new distinct letters by [*k*_{1},…,*k*_{n}]. Hence our underlying countable alphabet *d* and all possible nested brackets [⋅,⋅] generated in this manner. For convenience, we set [*X*^{i}]≡*X*^{i} and thus also [*i*]≡*i* on

We use *u* and *v* are words in

### Definition 3.3 (Quasi-shuffle product)

For words *u*,*v* and letters *a*,*b* the quasi-shuffle product * on *u**1=1**u*=*u*, where ‘1’ represents the empty word, and

Endowed with this product *ua* Ш *vb*=(*u* Ш *vb*)*a*+(*ua* Ш *v*)*b*.

### Example 3.4

The quasi-shuffle of the words 12 and 34 is given by 12*34=1234+3412+1342+3142+1324+3124+ 1[2,3]4 + [1,3]42 + 3[1,4]2 + [1,3]24 + 13[2,4] + 31[2,4] + [1,3][2,4].

### Example 3.5

A minimal family of semimartingales generates a quasi-shuffle algebra. This is proved in Curry *et al.* [22].

### Definition 3.6 (Deconcatenation and de-quasi-shuffle coproducts)

We define the deconcate- nation coproduct

The finiteness condition on

### Definition 3.7 (Convolution products)

Suppose *X* and *Y* are two linear endomorphisms on the Hopf quasi-shuffle algebra *X***Y* by the formula *X***Y* :=quas○(*X*⊗*Y*)○Δ, where ‘quas’ denotes the quasi-shuffle product on

### Remark 3.8

We use the same notation for the quasi-shuffle convolution product as for the underlying product, no confusion should arise from the context. There is also a concatenation convolution product conc○(*X*⊗*Y*)○Δ′ on

In other words, since deconcatenation Δ splits any word *w* into the sum of all two-partitions *u*⊗*v* with *u* or *v* are the empty word 1, we see that

### Remark 3.9

In our context, the significance of this tensor algebra is that it is the natural abstract setting for the flowmap.

Any endomorphism *ν* on the algebra

### Definition 3.10 (Augmented ideal projector)

We use *ν* given above, we see that

We observe we can apply a power series function such as the logarithm function to the element *w*| represents the length of the word *w* and *c*_{k}:=(−1)^{k−1}(1/*k*) for all *k*th quasi-shuffle convolution power of the augmented ideal projector *u*_{1}⋯*u*_{k}=*w* in the sum in the penultimate line are all non-empty. Note that words cannot be deconcatenated further than all the letters it contains, and thus *w* has length less than *k*. We conclude the action of the logarithm function power series on

### Lemma 3.11 (Logarithm convolution power series)

*The logarithm of* *is given by*
*where*
*We will often abbreviate*

We also note that equivalently, the embedding

### Definition 3.12 (Adjoint endomorphisms)

Two endomorphisms *X* and *Y* are adjoints if the images of
*X*^{†} to denote the adjoint of *X*.

This coincides with *X* and *X*^{†} being adjoints in the sense 〈*X*^{†}(*u*),*v*〉=〈*u*,*X*(*v*)〉 for all

There is a natural isomorphism between the Hopf shuffle and quasi-shuffle algebras discovered by Hoffman [19] which will play an important role here; also see [23] for a theoretical perspective. To describe the isomorphism succinctly, we need to introduce the notion of composition action on words. For any natural number *n*, we use *n*, i.e. the set of all tuples of natural numbers whose sum is *n*. A given composition λ in *n* components so λ=(λ_{1},…,λ_{ℓ}) and λ_{1}+⋯+λ_{ℓ}=*n*. For such a λ, we define the following simple multi-index functions, |λ|:=ℓ as well as

### Definition 3.13 (Composition action)

For a given word *w*=*a*_{1}⋯*a*_{n} and composition λ=(λ_{1},…,λ_{ℓ}) in *w* to be
*w*=*a*_{1}⋯*a*_{n}, the notation [*w*] denotes the *n*-fold nested bracket described above.

### Definition 3.14 (Hoffman exponential)

We define the map *w* is
*w* is given by

Hoffman [19] proved the exponential map is an isomorphism from the Hopf shuffle algebra

### Example 3.15

In the case of the word *w*=*a*_{1}*a*_{2}*a*_{3}, the Hoffman exponential is given by

The adjoint of the Hoffman exponential *δ* as coproduct. Then for any letter *a* from the alphabet *a* from the alphabet as above is

## 4. Exponential Lie series for continuous semimartingales

We show the logarithm of the flowmap for a system of stochastic differential equations, driven by a set of *d* continuous semimartingales *V*_{i}, for *i*=1,…,*d*, can be expressed as a Lie series. We begin by emphasizing that for a given set of orthogonal continuous semimartingales, the only non-zero quadratic variations are those of the form [*X*^{i},*X*^{i}] for *i*=1,…,*d*. In particular, all third-order [*X*^{i},*X*^{i},*X*^{i}] and thus higher order variations are zero. Hence, the generator [⋅,⋅], for example in the underlying quasi-shuffle algebra, is nilpotent of degree 3.

### Remark 4.1

Examples of continuous (local) martingales are Brownian motion, time-changed Brownian motion and stochastic integrals of Brownian motion. The representation results of Doob, of Dambis and of Dubins and Schwarz, and of Knight (see Theorems 3.4.2, 3.4.6 and 3.4.13 in [24]) show that these examples are fundamental. Hence important examples for continuous semimartingales are those just mentioned to which a continuous finite variation process, i.e. the difference of two real-valued continuous increasing processes, is added.

We saw in §2 that the Itô flowmap has the form *d* as well as the letters [1,1],…,[*d*,*d*]. We proposed there to represent the Itô flowmap by the abstract expression *I*_{w}=*I*_{w}(*t*) with respect to the semimartingales {*X*^{1},…,*X*^{d}} or any nested quadratic variation processes generated from them, and the constant random variable 1.

### Definition 4.2 (Itô word-to-integral map)

We denote by *μ*: *w*↦*I*_{w} assigning each word *I*_{w}.

The Itô word-to-integral map is a quasi-shuffle algebra homomorphism, i.e. we have *μ*(*u***v*)=*μ*(*u*)*μ*(*v*) for any *D*_{i}.

### Definition 4.3 (Itô word-to-partial differential operator map)

We denote by *D*_{i}. Recall the operators *D*_{i} are given, for *i*=1,…,*d*, by *D*_{i}:=*V*_{i}⋅∂ and

The map *D*_{w} in *V*_{i}⋅∂ and second-order partial differential operators *i*=1,…,*d*. The question is, how can we express the logarithm of the flowmap in Lie polynomials or in particular, in Lie brackets of vector fields? The natural resolution is to use the Fisk–Stratonovich representation of the flowmap. From the stochastic analysis perspective, the procedure is as follows.

### Definition 4.4 (Fisk–Stratonovich integral)

For continuous semimartingales *H* and *Z*, the Fisk–Stratonovich integral is defined as

### Lemma 4.5 [Itô to Fisk–Stratonovich conversion)

*For i*=1,…,*d and any function* *for the integral term in the Itô chain rule we have*
*and*

### Proof.

To establish the first result, we set *H*=(*V*_{i}⋅∂)*f*(*Y*) and *Z*=*X*^{i} in the definition for Fisk–Stratonovich integrals above. For the second result, we use the Itô chain rule in §2 to substitute for (*V*_{i}⋅∂)*f*(*Y*_{t}) into the quadratic covariation bracket on the left. Then using that the bracket is nilpotent of degree 3 for continuous semimartingales, and zero if any argument is constant, establishes the result. ▪

Substituting the product rule

### Corollary 4.6 (Fisk–Stratonovich chain rule)

*For any function* *we have*

We emphasize the differential operator in the second term on the right is a vector field. In addition, the usual rules of calculus apply to Fisk–Stratonovich integrals. As in §2 for the Itô case, we extend the driving continuous semimartingales and governing vector fields as follows. For *i*=1,…,*d*, we set *X*^{[i,i]}:=[*X*^{i},*X*^{i}] as before, however we now set *w* constructed from the alphabet *J*_{w}=*J*_{w}(*t*) are defined over the same simplex as for the multiple Itô integrals but with each nested integration interpreted in the Fisk–Stratonovich sense. The basis terms *V*_{w} are compositions of vector fields from the alphabet *ν* : *w*↦*J*_{w}, in a similar manner to that for the Itô word-to-integral map. Here, *X*^{1},…,*X*^{d}} and any quadratic variation processes [*X*^{i},*X*^{i}] generated from them, and the constant random variable 1. Note the alphabet *d* and [1,1],…,[*d*,*d*]. Let *V*_{i} and *V*_{[i,i]} for *i*=1,…,*d*.

### Definition 4.7 (Word-to-vector field map)

We denote by *V*_{i} and each letter *V*_{[i,i]}, for *i*=1,…,*d*.

The Fisk–Stratonovich word-to-integral map *ν* is a shuffle algebra homomorphism, while the word-to-vector field map

The Fisk–Stratonovich and Itô representations for the flowmap must coincide and so must their logarithms. The Fisk–Stratonovich integrals *J*_{w} satisfy the usual rules of calculus [2], while the basis terms *V*_{w} are compositions of vector fields. Hence, the Chen–Strichartz formula applies to the Fisk–Stratonovich representation for the flowmap [9]. Recall from lemma 3.11 the quasi-shuffle convolution logarithm of the identity

### Theorem 4.8 (Chen–Strichartz Lie series)

*The logarithm of the flowmap has the series representation*
*where* *and*
*Here for any word* *, the basis terms V*_{[w]L}:=[*V*_{a1},[*V*_{a2},…,[*V*_{an−1},*V*_{an}*]*_{L}*⋯ ]*_{L}*]*_{L}*, where [⋅,⋅]*_{L} *is the Lie bracket, are Lie polynomials. The non-negative integers d(σ) denote the number of descents in the permutation σ and |σ| denotes its length, i.e. if* *then |σ|=|w|.*

### Proof.

First, since the Fisk–Stratonovich multiple integrals satisfy the usual rules of calculus, the underlying product is the shuffle product Ш; this is a special case of the quasi-shuffle product for which the quadratic variation generator [⋅,⋅] is identically zero. Hence, we can emulate the derivation of the quasi-shuffle convolution given in §3 to show that
*c*_{σ} shown, is proved in §5 for

Second, we express the logarithm of the Fisk–Stratonovich flowmap in terms of Lie polynomials as shown. The crucial observation here is that the adjoint of the shuffle convolution logarithm *δ*) which we denote by *k*≥1, the adjoint of *k*th concatenation convolution power of *p*]_{L} denote left to right Lie bracketing of the word 1⋯*p* so that [1⋯*p*]_{L}:=[1,[2,[…[*p*−1,*p*]…]]]_{L}. The element (1/*p*)[1⋯*p*]_{L} of *p*. We denote the Dynkin idempotent as
*P* in the concatenation Hopf algebra *θP*=*P* [3], theorem 1.4. The image of the Eulerian idempotent is contained in the free Lie algebra associated with *X* and *Y* are adjoint endomorphisms, and *Z* is an endomorphism on the concatenation Hopf algebra, then we have
*Z*=*θ*. ▪

The Hoffman exponential map

### Proposition 4.10 (Itô to Fisk–Stratonovich: Hoffman exponential)

*For continuous semimartin- gales, for any word w in* *we have* *where explicitly we have*
*The nilpotency of the generator* [⋅,⋅] *implies the compositions* *with a non-zero contribution only contain letters* 1 *and* 2 *and so Γ*(λ)=2^{Σ(λ)−|λ|}. *Thus the set* [[*w*]] *consists of the words we can construct from w by successively replacing any neighbouring pairs ii in w by* [*i*,*i*].

### Proof.

Using the definition of the Fisk–Stratonovich integral, for any word *w*=*a*_{1}⋯*a*_{n} we have
*J*_{a1⋯an−1} and so forth, generates the result. ▪

With this in hand, we deduce the following main result of this section.

### Corollary 4.11 (Itô Lie series)

*Let c*_{σ} *denote the coefficient of σ*^{−1}(*w*) *in the expression for* *above. We can express the Chen–Strichartz Lie series in terms of multiple Itô integrals as follows*:
*or equivalently, by resummation of the series*,
*Here for any word* *the set* ]]*w*[[ *consists of w and all words we construct from w by successively replacing any letter* [*i*,*i*] *with ii, for* *i*=1,…,*d*.

Since the Fisk–Stratonovich and Itô representations for the flowmap must coincide, the expression above must coincide with the logarithm of the Itô representation for the flowmap. The key fact distinguishing the Itô from the Fisk–Stratonovich representation for the flowmap is that the word-to-vector field map *X*^{i},*X*^{i}]. The Itô word-to-integral map *i*↦*X*^{i} and [*i*,*i*]↦[*X*^{i},*X*^{i}]. The former is a quasi-shuffle homomorphism and the latter a shuffle homomorphism and consequently *V*_{i}⋅∂)(*V*_{i}⋅∂)=(*V*_{i}⊗*V*_{i}): ∂^{2}+(*V*_{i}⋅∂*V*_{i})⋅∂ underlie the following result. Recall the definition of the Hoffman exponential and logarithm map adjoints in §3.

### Theorem 4.12 (Itô and Fisk–Stratonovich map relations)

*The two word-to-integral maps μ and ν and the word-to-vector field and word-to-partial differential operator maps* *and* *are related as follows:*

### Proof.

The first relation follows directly from *ii* denotes the concatenation of *i* with *i* and *i*=1,…,*d* and

Some immediate consequences of this result are as follows. First, algebraically, we observe

### Remark 4.13

From the algebraic combinatorial computations above, we observe: (i) In the first computation above the transformation from the Itô to Fisk–Stratonovich flowmaps was instigated by the transformation of coordinates *d*,*d*] as well as 1,…,*d* in the alphabet appears to be natural, especially in the context of using the quasi-shuffle machinery provided by Hoffman [19]. Indeed, this is also the case for stochastic differential equations driven by Wiener processes for which it is usual to replace the quadratic variation terms by the corresponding drift term; (iv) The flowmap satisfies the linear equation *D*_{i} are constant, the solution is the well-known Doléans-Dade exponential; a representation of it in terms of iterated integrals was derived in Jamshidian [26].

## 5. Quasi-shuffle Chen–Strichartz formula

In this section, we do not make any nilpotency assumptions on *n*-fold nested brackets as in §4. We derive an explicit formula for the coefficients of the quasi-shuffle convolution logarithm of the identity endomorphism on *et al.* [27] and generalized to linear matrix valued systems in Ebrahimi-Fard *et al.* [1]. Using the notion of surjections instead of permutations and quasi-descents, we can closely follow the development given in Reutenauer [3]. We begin by outlining the theory of surjections and quadratic covariation permutations, which we hereafter call ‘quasi-permutations’, as well as their action on words. We denote the symmetric group of order *p* by *σ*^{−1} records the following information: ‘The letter *i* is at position *σ*^{−1}(*i*) in *σ*’. We exploit the corresponding result for surjections herein.

We shall denote the set of surjective maps from the set of natural numbers {1,…,*p*} to the set of natural numbers {1,…,*q*} with *q*≤*p* by

### Definition 5.1 (Quasi-permutations)

We denote by

Henceforth we record quasi-permutations simply as *ρ* or as a pair (λ,*ρ*), where λ is a composition and *ρ* a permutation. Given any quasi-permutation in

### Example 5.2

Consider the set of all quasi-permutations

Hence, by analogy with permutations, quasi-permutations play the role of generalized permutations, while the corresponding surjections play the role of the inverse permutations by recording the positions of the letters in the corresponding quasi-permutations. Hence, we have the corresponding statement to that above and crucial fact about surjections: each surjection *ζ* corresponding to a given quasi-permutation *σ* records the information:

### Example 5.3

The surjection 3221 from

With each surjection, we can associate a quasi-descent set.

### Definition 5.4 (Quasi-descent sets)

Given any surjection *ζ*) to be the list of the indices *k*∈{1,…,*p*−1} for which *ζ*(*k*+1)≤*ζ*(*k*).

For the particular subset *k* would correspond to the classical descent indices. Just as there is an intimate relation between shuffles and descents, there is also one between quasi-shuffles and quasi-descents. The following first key result underlies the whole of this section.

### Lemma 5.5 (Quasi-descents and quasi-shuffles)

*The set of surjections* *satisfying* Des(*ζ*)⊆{*q*} *for q*<*p, is identical to the set of surjections satisfying ζ*(1)<⋯<*ζ*(*q*) *and ζ*(*q*+1)<⋯<*ζ*(*p*).

### Proof.

We observe that for any surjection *ζ*)⊆{*q*} for some natural number *q*<*p*, then discounting the case when the quasi-descent set is empty, by definition we must have *ζ*(*k*)≥*ζ*(*k*+1)⇒*k*=*q* ⇔ *k*≠*q*⇒*ζ*(*k*)<*ζ*(*k*+1). The latter condition is equivalent to that in the statement of the lemma. ▪

The second key result we establish in this section is a natural consequence.

### Corollary 5.6 (Quasi-descents and quasi-shuffles)

*Let q*<*p be natural numbers. If we factorize the word* 1⋯*p*=*u*_{1}*u*_{2} *with* |*u*_{1}|=*q* *and* |*u*_{2}|=*p*−*q*, *then the quasi-shuffle product of u**_{1}*u*_{2} *is given by*
*where σ*(*ζ*) *denotes the unique quasi-permutation associated with a given surjection ζ. The first sum is over all* *satisfying the inequalities shown, and the second sum is over all* *such that* Des(*ζ*)⊆{*q*}.

### Proof.

Recall the definition of the quasi-shuffle product and its generation through the formula
*u*_{1} and *u*_{2} is equivalent to the prescription that it is the sum over all quasi-permutations whose corresponding surjections satisfy the set of inequalities *ζ*(1)<⋯<*ζ*(*q*) and *ζ*(*q*+1)<⋯<*ζ*(*p*). This establishes the first result. With this in hand, the result of the quasi-descent and quasi-shuffle conditions lemma 5.5 implies the equivalence to the second result. ▪

The following generalization is then immediate and represents the quasi-shuffle analogue of Lemma 3.13 in Reutenauer [3, p. 65].

### Corollary 5.7 (Multiple quasi-shuffles and quasi-descents)

*Let p*_{1},…,*p*_{k} *be positive integers of sum p and S*={*p*_{1},*p*_{1}+*p*_{2},…,*p*_{1}+⋯+*p*_{k−1}} *be a subset of* {1,…,*p*−1}. *If we factorize the word* 1⋯*p*=*u*_{1}⋯*u*_{k} *with* |*u*_{i}|=*p*_{i} *for i*=1,…,*k, then we have*

Much like the symmetric group action on words, there is an analogous quasi-permutation action on words. Recall that we can decompose any quasi-permutation in *σ*=λ○*ρ*, into its composition

### Definition 5.8 (Quasi-permutation action)

We define the action of *σ*=λ○*ρ* and word *w*=*a*_{1}⋯*a*_{p} by *σw*:=λ○(*a*_{ρ(1)}⋯*a*_{ρ(p)}).

We can now construct the quasi-shuffle logarithm of the identity. We start with the quasi-shuffle convolution powers of the augmented ideal projector

### Corollary 5.9 (Convolution powers and descents)

*For any k*≥1 *and word w, we have*

### Proof.

For any word *w*, the quantity *k*-partitions of *w*, say *v*_{1}⋯*v*_{k}, quasi-shuffled together. Hence, we have

### Corollary 5.10 (Quasi-shuffle convolution logarithm on words)

*The action of the quasi-shuffle convolution logarithm on any word w is as follows*:

### Proof.

By direct computation using corollary 5.9, we find

The following characterization of the quasi-shuffle convolution logarithm is the generalization of the standard shuffle convolution logarithm. We shall need the following integral identity for non-negative integers *d* and *r* which is proved for example in Reutenauer [3, p. 69]:

### Corollary 5.11 (Quasi-shuffle convolution logarithm endomorphism)

*The quasi-shuffle convolution logarithm* *acts on* 1⋯*p as follows*:
*where d*(*ζ*) *denotes the number of quasi-descents in ζ*.

### Proof.

We observe from corollary 5.10 that *σ*(*ζ*). Hence, we directly compute the coefficient of an arbitrary quasi-permutation *σ*(*ζ*) in *ζ* has quasi-descent indicies *p*_{1},…,*p*_{k} so that *d*(*ζ*)=*k*. To compute this coefficient, we therefore have to determine the number of subsets *S*⊆{1,…,*p*−1} which contain *p*_{1},…,*p*_{k}. Note the coefficient itself only depends on the size of such sets. These subsets have possible size |*S*|=*k* through to |*S*|=*p*−1. Starting with the case |*S*|=*k*, there is of course only one set of this size containing *p*_{1},…,*p*_{k}, the set of these integers themselves. Now consider the case |*S*|=*k*+1. Then an extra ‘quasi-descent’ can be placed in total of *p*−1−*k* possible positions, or equivalently in *p*−1−*k* choose 1 ways. When |*S*|=*k*+2, there are *p*−1−*k* choose 2 ways, and so forth so that in general, when |*S*|=*k*+*i*, there are *p*−1−*i* choose *i* possible ways. Hence the coefficient above equals

### Remark 5.12

This is equivalent to the quasi-shuffle logarithm given in Ebrahimi-Fard *et al.* [1, theorem 6.2]. We included an explicit derivation here for completeness.

## 6. Concluding remarks

There has been a recent surge in the development of quasi-shuffle algebras and stochastic Taylor solution formulae in the context of semimartingales, on the theoretical and practical level. See Platen & Bruti–Liberati [28], Marcus [29], Friz & Shekhar [30] and Hairer & Kelly [31] for contemporary references. For example, Li & Liu [32] considered systems driven by both Wiener and Poisson processes. We have shown that the Chen–Strichartz flowmap solution formula which is well known for Stratonovich stochastic differential systems driven by Wiener processes extends to systems driven by general continuous semimartingales. We demonstrated it is in fact a Lie series, and this property holds irrespective of whether we consider the system in the Itô or Stratonovich sense. We also give and prove an explicit formula for the Lie series coefficients. Curry *et al.* [33] have developed so-called efficient simulation schemes (see [34,35]) for such systems driven by Lévy processes. This involves the antisymmetric sign reverse endomorphism rather than the quasi-shuffle logarithm endomorphism.

## Authors' contributions

The research presented herein was a joint and equal effort by all authors.

## Competing interests

We have no competing interests.

## Funding

K.E.F. is supported by Ramón y Cajal research grant no. RYC-2010-06995 from the Spanish government and acknowledges support from the Spanish government under project MTM2013-46553-C3-2-P. This research also received support from a grant by the BBVA Foundation. F.P. acknowledges support from the grant no. ANR-12-BS01-0017, Combinatoire Algébrique, Résurgence, Moules at Applications.

## Acknowledgements

The authors thank one of the referees whose insightful suggestions helped to improve the paper.

- Received June 24, 2015.
- Accepted November 4, 2015.

- © 2015 The Author(s)