Algebraic Structures and Stochastic Differential Equations driven by Levy processes

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by  Charles Curry

Submitted article
Algebraic Structures and Stochastic Differential
Equations driven by L´evy processes
CharlesCurry · KuruschEbrahimi–Fard ·
Simon J.A. Malham · Anke Wiese
6
1
0
2 22ndDecember2015

n
a
J Abstract We define a new numerical integration scheme for stochastic dif-

9 ferential equations driven by L´evy processes that has a leading order error
1 coefficient less than that of the scheme of the same strong order of conver-

gence obtained by truncating the stochastic Taylor series. This holds for all
]
R such equations where the driving processes possess moments of all orders and
P the coefficients are sufficiently smooth. The results are obtained using the
. quasi-shuffle algebra of multiple iterated integrals of independent L´evy pro-
h
cesses. Our findings generalize recent results concerning equations driven by
t
a Wiener processes.
m
[
Keywords efficient integrators · L´evy processes · quasi-shuffle algebra
1
v
0
8 CharlesCurry
8 Maxwell Institute for Mathematical Sciences and School of Mathematical and Computer
4 Sciences,Heriot-WattUniversity,EdinburghEH144AS,UK.
0 Present address: ofCharlesCurry
. DepartmentofMathematical Sciences, NTNU,7491Trondheim,Norway.
1
0
6 KuruschEbrahimi–Fard
1 Instituto de Ciencias Matema´ticas, Consejo Superior de Investigaciones Cient´ıficas, C/Ni-
cola´sCabrera,no.13-15,28049Madrid,Spain.
:
v
i
X SimonJ.A.Malham
Maxwell Institute for Mathematical Sciences and School of Mathematical and Computer
r
a Sciences,Heriot-WattUniversity,EdinburghEH144AS,UK.
AnkeWiese
Maxwell Institute for Mathematical Sciences and School of Mathematical and Computer
Sciences,Heriot-WattUniversity,EdinburghEH144AS,UK.
2 CharlesCurryetal.
1 Introduction
Ourmainresultisthederivationofanewstrongnumericalintegrationscheme
forstochasticdifferentialequationsdrivenbyindependentL´evyprocesseswith
moments of all orders, where the coefficient functions are sufficiently smooth
autonomous vector fields. L´evy processes are a class of stochastic processes,
comprising processes continuous in probability and possessing stationary in-
crements, independent of the past. They are examples of stochastic processes
with a well understood structure that incorporate jump discontinuities.
Our setting is a complete, filtered probability space Ω,F,(F ) ,P ,
t t≥0
assumedtosatisfytheusualhypotheses—seeProtter[34,p.3].DuetotheL´evy
(cid:0) (cid:1)
decomposition of a L´evy process the equations we consider may be written in
the form
d t ℓ t
y =y + V (y )dWi+ V (y )dJi.
t 0 i s s i s− s
i=0Z0 i=d+1Z0
X X
See for example Applebaum [2]. Here the governing autonomous vector fields
V : RN →RN aresupposedto besmoothandingeneralnon-commuting,and
i
the initial data y ∈ L2(Ω,F ,P). The Wi for i = 1,...,d are independent
0 0
Wienerprocesses,andbyconventionwesetW0 =t.TheJi fori=d+1,...,ℓ
t
are purely discontinuous martingales expressible in the form
t
Ji = vQi(dv,ds),
t
Z0 ZR
where Qi(dv,ds) := Qi(dv,ds)−ρi(dv)ds. The Qi are Poisson measures on
R×R with intensity measures ρi(dv)ds, where the ρi are measures on R
+
with ρi(0)=0 and (1∧v2)ρi(dv)<∞. Intuitively, Qi(B,(a,b]) counts the
R
number of jumps of the process Ji taking values in the set B in the time
R
interval (a,b], whilst ρi(B) measures the expected number of jumps per unit
time the process Ji accrues taking values in the set B. We assume that the
Ji possess moments of all orders. L´evy processes have many applications, no-
tablyinmathematicalfinancefortheconstructionofmodelsgoingbeyondthe
Black–Scholes–Merton model in incorporating discontinuities in stock prices,
see Cont & Tankov [8] and Barndorff–Nielsen, Mikosch& Resnick [3] and the
references therein. There are also applications to physical sciences, see for in-
stance Barndorff–Nielsen, Mikosch & Resnick [3]. Pathwise integrals of L´evy
processes have recently been constructed in the framework of rough paths by
Friz & Shekhar [15].
We are concerned with strong integration schemes for such L´evy-driven
stochasticdifferentialequations.Theseschemesaretypicallyderivedfromthe
stochastic Taylorexpansion. This is an expressionfor the flowmapϕ : y 7→
s,t s
y as a sum of iterated integrals with an explicit remainder, which formally
t
can be extended to give an infinite series. See Platen & Bruti-Liberati [32]
for a derivation of the stochastic Taylor expansion for L´evy-driven stochastic
differential equations, as well as Platen [30,31] and Platen & Wagner [33].
SDEsdrivenbyL´evyprocesses 3
For sufficiently smooth vector fields, integration schemes of arbitrary strong
order of convergence may be constructed by truncating the series expansion
to an appropriate number of terms. Platen & Bruti-Liberati [32] show which
terms must be retained to obtain a strong integration scheme with a given
strong order of convergence. We assume hereafter the governing vector fields
are sufficiently smooth for the stochastic Taylor expansion to exist.
Combinatorial algebras are a natural setting for the study of relations
among the iterated integrals appearing in stochastic Taylor expansions. In
this paper we focus on the case when the stochastic Taylor expansion can
be written in separated form. This means that each term in the stochastic
Taylor expansion can be decomposed as the product of a multiple Itˆo inte-
gral, containing stochastic information only, and a composition of associated
differential operators, containing geometric information only. In this instance
the algebra of integrand-free multiple integrals of the driving processes is of
interest alongside the algebra of the associated differential operators. Curry,
Ebrahimi–Fard, Malham & Wiese [10] proved that the algebra of multiple
integrals of a minimal family of semimartingales is isomorphic to the combi-
natorialquasi-shuffle algebra of words.A set of L´evy processes represents one
example. Indeed Gaines [19] considered the case of multiple iterated integrals
of Wiener processes and Li & Liu [22] considered multiple iterated integrals
ofWienerprocessesandstandardPoissonprocesses.Thequasi-shufflealgebra
is an extension of the shuffle algebra introduced abstractly, and comprehen-
sively, by Hoffman [17]. The significance of the shuffle algebra was cemented
in the work of Eilenberg & Maclane [14], Schu¨tzenberger [38] and Chen [6].
See Reutenauer [36] for more details.
One advantage of abstracting to the quasi-shuffle algebra is that we can
immediately identify the minimum set of iterated integrals that need to be
simulated to implement a given accurate strong scheme. This optimizes the
total computation time which is dominated by the strong simulation of the
iteratedintegrals.Radford[35]provedthatthe shuffle algebrais generatedby
Lyndonwords.Thiswasextendedtothequasi-shufflealgebrabyHoffman[17],
though it had already been established by Gaines [19] for the case of Wiener
processes and Li & Liu [22] for the case of Wiener processes and standard
Poissonprocesses,whileSussmann[40]hadconsideredaHallbasis.Hencethe
setofiteratedintegralsweneedtosimulateatanygivenorderareidentifiedby
Lyndon words. Having thus optimized the total computation time associated
with a scheme of a givenstrongorder,the focus turns to how to minimize the
coefficient of the leading order error. This problem was considered by Clark
[7] and Newton [28,29] for drift-diffusion equations driven by a single Wiener
process; also see Kloeden & Platen [20, Section 13.4]. They derived integra-
tion schemes that are asymptotically efficient in the sense that the coefficient
of the leading order error is minimal among all schemes. For Stratonovich
stochastic differential equations, Castell & Gaines [4,5] constructed integra-
tion schemes using the Chen-Strichartz exponential Lie series expansion (see
Strichartz [39]). In the strong order one case for one driving Wiener process,
4 CharlesCurryetal.
and the strong order one-half case for two or more driving Wiener processes,
their schemes are asymptotically efficient in the sense of Clark and Newton.
Malham & Wiese [25] considered the problem of designing efficient inte-
grators for Stratonovich drift-diffusion equations driven by multiple indepen-
dentWienerprocesses.Here,efficientintegratorsofagivenorderhaveleading
order error which is always less than that for the corresponding stochastic
Taylor scheme, independent of the governing vector fields. They considered
a class of strong integration schemes we shall hereafter call map-truncate-
invert schemes. These are constructed as follows. Given an invertible map
f: Diff(RN) → Diff(RN), we expand the series f(ϕ), truncate according to
a chosen grading and then apply f−1. The case f = log corresponds to the
exponential Lie series integrator of Castell & Gaines. Malham & Wiese [25]
then proved that an integration scheme based on the map f =sinhlog is effi-
cient.These resultswereobtainedinthe absenceofadrifttermandextended
toincorporatedriftinEbrahimi–Fard,Lundervold,Malham,Munthe–Kaas&
Wiese [11].
Herein, we study map-truncate-invertschemes for Itoˆ equations driven by
L´evy processes.We must consider the algebraic structure of iterated integrals
with respect to L´evy processes, for which we rely on recent results of Curry,
Ebrahimi–Fard, Malham & Wiese [10]. In this paper, we:
1. ShowthestochasticTaylorseriesexpansionfortheflowmapcanbewritten
in separated form, i.e. as a series of terms, each of which is decomposable
into a multiple Itoˆ integral and a composition of associated differential
operators;
2. Describe the class of map-truncate-invert schemes for L´evy-driven equa-
tions and give an algebraic framework for encoding and comparing such
schemes.TheseresultsgeneralizethoseinMalham&Wiese[25]andEbrahimi–
Fard et al. [11];
3. Provetruncationsaccordingtothewordlengthgradinggivemoreaccurate
schemes than approximations of the same order of convergence obtained
by truncations according to the mean-square grading;
4. Introduce the antisymmetric sign reverse integrator, a new integration
scheme for L´evy-drivenequations,representedas half the difference of the
identity andsignreverseendomorphismsonthe vectorspacegeneratedby
wordsindexingmultipleintegrals.Thisschemeisefficientinthesensethat
its leading order mean-square error is less than that of the corresponding
stochastic Taylor scheme, independent of the governing vector fields;
5. Show how to compute map-truncate-invert schemes in practice, in par-
ticular, how to deal with the inversion stage. We call these direct map-
truncate-invertschemes.Inessencetheirimplementationinvolvescomput-
ing the stochastic Taylor expansion to the given order, and then adding
a polynomial of the already simulated multiple integrals to emulate the
map-truncate-invertscheme.
Thestructureofthispaperisasfollows.In§2,wediscussthestochasticTaylor
expansionforL´evy-drivenequationsgiveninPlaten&Bruti-Liberati[32].We
SDEsdrivenbyL´evyprocesses 5
show when and how we can represent the stochastic Taylor expansion in sep-
arated form. In §3, we introduce map-truncate-invert schemes in the context
of equations driven by L´evy processes. We show how to algebraically encode
them using the quasi-shuffle convolutionendomorphism algebra,thus extend-
ing the encoding for drift-diffusion equations in Ebrahimi–Fard et al. [11]. In
§4 we undertake a local error analysis of integration schemes for L´evy-driven
equations. A component of this analysis is the comparison of two gradings,
the mean-square grading and the word length grading. Our main results are
presented in §5. We prove that the antisymmetric sign reverse integrator, an
extensionofthesinhlogintegratorofMalham&Wiese[25],isefficient.Wealso
introduceourdirectmap-truncate-invertschemes.Wedemonstratethesewith
two numerical experiments in §6. Finally in §7 we derive global convergence
results from local error estimates.
2 Separated stochastic Taylor expansions
A stochastic Taylor expansion is an expression for the flowmap as a sum of
iterated integrals. We write down the stochastic Taylor expansion for L´evy-
drivenequationsandshowhowandwhenitcanbe writteninseparatedform.
Recall the flowmap is defined as the map ϕ : y 7→y . It acts on sufficiently
s,t s t
smooth functions f: RN → RN as the pullback ϕ (f(y )) := f(y ). We set
s,t s t
ϕ :=ϕ .Itoˆ’sformula(seeforexampleApplebaum[2,p.203]orProtter[34,
t 0,t
p. 71]) implies
d t ℓ t
f(y )=f(y )+ V˜ ◦f(y )dWi+ (V˜ ◦f)(y ,v)Qi(dv,ds),
t 0 i s s i s−
i=0Z0 i=d+1Z0 ZR
X X
where for i=1,...,ℓ, the V˜ are operators defined as follows
i
(V ·∇)f, if i=1,...,d,
V˜ ◦f := i
i
(f · +vVi(·) −f(·), if i=d+1,...,ℓ,
and V˜ is the operator d(cid:0)efined by (cid:1)
0
d N
1
V˜ ◦f :=(V ·∇)f + VjVk∂ ∂ f
0 0 2 i i xj xk
i=1j,k=1
X X
ℓ
+ (V˜ ◦f)(·,v)−v(V ·∇)f ρi(dv).
i i
R
i=d+1Z
X (cid:2) (cid:3)
Note for i=d+1,...,ℓ, the V˜ introduce an additional dependence on a real
i
parameter v. The stochastic Taylor expansion is derived by expanding the
integrands in the L´evy-drivenequation using Itoˆ’s formula. This procedure is
repeated iteratively,where the iterations are encoded as follows. Let A be the
alphabet A := {0,1,...,ℓ}. For any such alphabet, we use A∗ to denote the
6 CharlesCurryetal.
free monoid over A—the set of words w = a ...a constructed from letters
1 m
a ∈A. We write 1 for the empty word.For a givenwordw =a ...a define
i 1 m
theoperatorV˜ :=V˜ ◦···◦V˜ .Lets(w)bethe numberoflettersofw from
w a1 am
the subset {d+1,...,ℓ} ⊂ A. For a given integrand g(t,v): R ×Rs(w) →
+
RN, we define the iterated integrals I (t)[g] inductively as follows. We write
w
I1(t)[g]:=g(t), and
t
I (s)[g]dWam, if a =0,1,...,d,
a1...am−1 s m
I (t)[g]:=Z0
w  t I (s )[g(·,v)]Qam(dv,ds), if a =d+1,...,ℓ.
a1...am−1 − m
Z0 ZR
Iterativelyapplyingthechainrulegeneratesthefollowing(seePlaten&Bruti-
Liberati [32]).
Theorem 1 (Stochastic Taylor expansion) For a L´evy-driven equation
the action of the flowmap ϕ on sufficiently smooth functions f: RN → RN
t
can be expanded as follows
ϕ ◦f = I (t) V˜ ◦f .
t w w
w∈A∗
X (cid:2) (cid:3)
Remark 1 (Stochastic Taylor expansion convergence) For integrationschemes
derived from the stochastic Taylor expansion to converge, it suffices that the
terms V˜ ◦ f included in the expansion, and those at leading order in the
w
remainder,satisfy globalLipschitzandlineargrowthconditions(see Platen&
Bruti-Liberati [32]). Hereafter we assume these conditions are satisfied.
Remark 2 (Platen and Bruti–Liberati form) The stochastic Taylor expansion
derived in Platen & Bruti-Liberati [32] is an equivalent though different rep-
resentation. The equivalence of the expansions is seen from the identity
t ℓ t
(V˜ ◦f)(y ,v)Q(dv,ds)= (V˜ ◦f)(y ,v)Qi(dv,ds).
−1 s− i s−
Z0 ZRℓ−d i=d+1Z0 ZR
X
The expression on the left is the encoding employed by Platen and Bruti-
Liberati of the jump terms in the stochastic differential equation while our
encoding is that on the right. The equivalence is explained as follows. On
the left above Q is a compensated Poisson random measure on Rℓ × R .
+
The L´evy-driven equation may be written in this form, where V (x,v) =
−1
ℓ−dv V (x) and Q is the compensationof the Poissonrandommeasure Q
i=1 i i+d
defined such that Q(B,(a,b])= #{∆J1 ∈ B ,...,∆Jℓ ∈B : s ∈ (a,b]} for
aPBorelset B =B ×···×B ⊂Rℓ bousnded1awayfroms theℓorigin,i.e. 0∈/ B¯.
1 ℓ
As independent L´evy processes almost surely never jump simultaneously (see
Cont & Tankov [8, Theorem 5.3]), the measure Q is concentrated on sets of
the form 0×···×0×B ×0×···×0 with intensity measure ρ(0×···×0×
i
B ×0×···×0)dt=ρi(B )dt.Theidentity abovethus follows.The stochastic
i i
SDEsdrivenbyL´evyprocesses 7
Taylor expansion given in Platen & Bruti-Liberati [32] is of the same form as
theexpansionwehavegiven,butwherethealphabetisinstead{−1,0,...,d},
and the operator associated to the letter −1 is the multi-dimensional shift
V˜ : f(y)7→f y+V (y,v) −f(y).
−1 −1
(cid:0) (cid:1)
Given the stochastic Taylor expansionfor the flowmap in Theorem 1, we now
show how we can write it in separatedform. One component of the separated
form are iterated integrals which are free in the sense of having no integrand.
We define these abstractly for an arbitrary given alphabet for the moment,
the reason for this will be apparent presently.
Definition 1 (Free multiple iterated integrals) Given a collection of
stochasticprocesses{Ztai}ai∈A,indexed by agivencountable alphabetA,free
multiple iterated integrals take the form
I (t):= dZa1··· dZam,
w τ1 τm
Z0<τ1<···<τm<t
where w =a ...a are words in A∗.
1 m
We need to augment {W0,W1,...,Wd,Jd+1,...,Jℓ} with a set of extended
driving processes as follows. A key component in the characterization of the
algebra generated by the vector space of free iterated integrals of the driving
processes {W0,W1,...,Wd,Jd+1,...,Jℓ} are the compensated power brack-
ets defined for each i∈{d+1,..., ℓ} and for p≥2 by
t
Ji(p) := vpQi(dv,ds).
t
Z0 ZR
Equivalently we have Ji(p) = [Ji](p) −t vpρi(dv), where [Ji](p) is the pth
t R t
order nested quadratic covariation bracket of Ji. Importantly, the compen-
R t
sated power brackets have the property that if Ji(p) is contained in the linear
span of {t,Ji,Ji(2),...,Ji(p−1)} for some p ≥ 2, then Ji(q) is also in this lin-
t t t
ear span for all q ≥ p. Hence inductively for p ≥ 2, we augment our family
{W0,W1,...,Wd,Jd+1,...,Jℓ} to include the compensated power brackets
Ji(p) aslongastheyarenotcontainedinthelinearspanof t,Ji,Ji(2),...,Ji(p−1) .
t t t
By doing so, we obtain a possibly infinite family of stochastic processes. The
(cid:8) (cid:9)
iterated integrals of this extended family form the algebra generated by the
iterated integrals of our driving processes. See Curry et al. [10] for further
details. In summary we define our extended alphabet as follows.
Definition 2 (Extended alphabet) We define our alphabet A to contain
the letters 0,1,...,ℓ, associated with {W0,W1,...,Wd,Jd+1,...,Jℓ}, and
theadditionallettersi(p) correspondingtoanyJi(p) containedintheextended
family as described above.
8 CharlesCurryetal.
Definition 3 (Separated stochastic Taylor expansion) Theflowmapfor
a L´evy-drivenstochastic differentialequation possessesa separatedstochastic
Taylor expansion if it can be written in the form
ϕ = I V˜ ,
t w w
w∈A∗
X
where{Iw}w∈A∗ arethefreeiteratedintegralsassociatedtotheextendeddriv-
ing processes and V˜ = V˜ ◦···◦V˜ are operators indexed by words that
w a1 am
compose associatively.
Remark 3 (Separable cases) Fori∈{1,...,d}andj ∈{d+1,...,ℓ},consider
the term
t s
I [(V˜ ◦id)(y )]= V y +vV (y ) −V (y ) Qj(dv,dτ) dWi,
ji ji 0 i 0 j 0 i 0 s
Z0 (cid:18)Z0 ZR(cid:16) (cid:0) (cid:1) (cid:17) (cid:19)
in the general stochastic Taylor expansion. The double integral with respect
to the random measure is a sum of compensated power bracketprocesses and
thusseparable,iftheintegrandV y +vV (y ) −V (y )iseitherapolynomial
i 0 j 0 i 0
in the jump size v or it can be expressed as a power series in v. We see how
(cid:0) (cid:1)
this works in detail presently.
Remark 4 (Separated expansion for jump-diffusion equations) In the case of
jump-diffusion equations, i.e. L´evy-driven equations for which all the discon-
tinuous driving processes Ji are standard Poisson processes, the stochastic
Taylor expansion is of the separated form. Indeed, as standard Poisson pro-
cesseshavejumpsofsizeoneonly,theoperatorsV˜ withi=d+1,...,ℓdonot
i
introduce a dependence on an additional parameter. The integrands are thus
constantacrosstherangeofintegration,andhencetheexpansionisseparated.
We can construct a separated stochastic Taylor expansion for the flowmap as
follows. By Taylor expansion of the term f y+vV (y) appearing in the shift
i
V˜ ◦f, we have
i (cid:0) (cid:1)
(V˜ ◦f)(y,v)= vmV˜ ◦f(y),
i i(m)
m≥1
X
where we write V = V1,...,VN T and f = f1,...,fN T. We define the
i i i
operators V˜ by
i(m) (cid:0) (cid:1) (cid:0) (cid:1)
1 ∂mfj
V˜ ◦fj := Vi1···Vik ,
i(m) m! i i ∂yi1···∂yik
Xk≥1i1+..X.+ik=m
where the i ∈ N. The product in Vi1···Vik is multiplication in R. We then
j i i
have
t
I (t)[(V˜ ◦f)(·,v)]= V˜ ◦f dJi(m),
i i i(m) s
m≥1Z0
X (cid:0) (cid:1)
SDEsdrivenbyL´evyprocesses 9
for i=d+1,...,ℓ. Inserting the above into the relation
I (t) V˜ ◦f =I (t) I (·) V˜ ◦(V˜ ◦f)
w w a2...am a1 a1 a2...am
(cid:2) (cid:3) h (cid:2) (cid:3)i
and iterating gives the separatedexpansion, where the operatorsV˜ are those
a
of the stochastic Taylor expansion for a ∈ {0,1,...,d}, and for a = i(m) are
given by the V˜ defined above. We have thus just established the following
i(m)
result.
Theorem 2 (Separated stochastic Taylor expansion: existence) For a
L´evy-driven equation suppose the terms V˜ ◦f in the stochastic Taylor expan-
w
sion for the flowmap are analytic on RN. Then it can be written in separated
form, i.e. as a separated stochastic Taylor expansion.
Remark 5 (Linear vector fields andlinear diffeomorphisms) Theseparatedex-
pansion has a simple form when the governing vector fields are linear with
constant coefficients and we consider the action of the flowmap on homoge-
neous linear diffeomorphisms, say f(y) = Fy, where F = [f ] is an N ×N
ij
matrix. The identity map is a special case of such a linear diffeomorphism
which generates the solution y directly. In this case, the operators in the
t
expansion are given by matrix multiplication. We write V (y) = Aiy, where
i
Ai =[ai ]areconstantN×N matricesandconsidertheactionoftheflowmap
jk
on linear functions. First, for i = 1,...,d by direct computation we find
V˜ ◦f(y) = FAiy. Moreover, for i = d+1,...,ℓ the higher order differen-
i
tial operators V˜ with m ≥ 2 vanish due to the linearity, and we obtain
i(m)
(V˜ ◦f)(y,v) =v(V ·∇)f(y)= vFAiy. The above relation shows in addition
i i
that the term in V˜ ◦f involving an integral over the jump sizes vanishes. As
0
the functions we act on are linear, the second order terms of V˜ also vanish.
0
We therefore have V˜ ◦f(y) = f(V (y)) = FA0y. The operators V˜ act by
0 0 i
matrix multiplication. It follows that the operatorsV˜ act on linear functions
w
by matrix multiplication in the reverse order, V˜a1...ak ◦f(y)=FAak···Aa1y.
Hereafter we assume the existence of a separated Taylor expansion for the
flowmap, and all iterated integrals are free iterated integrals.
3 Convolution algebras and map-truncate-invert schemes
We now introduce a class of numerical integration schemes we call map-
truncate-invertschemesandshowhowtheycanbeencodedalgebraically.The
algebraicstructuresarisenaturallyfromtheproductsofiteratedintegralsand
the composition of operatorsappearing in the separatedstochastic Taylor ex-
pansion. Consider the class of numerical integration schemes obtained from
thestochasticTaylorexpansionbysimulatingtruncationsofthe expansionon
each subinterval of a uniform discretization of a given time domain [0,T].
10 CharlesCurryetal.
Definition 4 (Grading function and truncations) A grading function
g: A∗ → N assigns a positive integer to each non-empty word w ∈ A∗ and
zero to the empty word. A truncation is specified by a grading function and
truncation value n∈N. We write π , π and π for the projections of
g=n g≤n g≥n
A∗ onto the following subsets:
1. π (A∗):={w∈A∗: g(w)=n};
g=n
2. π (A∗):={w∈A∗: g(w)≤n}; and
g≤n
3. π (A∗):={w∈A∗: g(w)≥n}.
g≥n
A numerical integration scheme based on the stochastic Taylor expansion is
thus given by successive applications of the approximate flow
I (t)V˜
w w
w∈πXg≤n(A∗)
across the computational subintervals. Now more generally, consider a larger
class of numerical schemes that are constructedas follows.For a giveninvert-
ible map f: Diff(RN) → Diff(RN), we construct a series expansion for f(ϕ )
t
usingthestochasticTaylorexpansionforϕ .Wetruncatetheseriesandsimu-
t
late the retained iterated Itoˆ integrals across each computational subinterval.
An integration scheme is obtained by computing the inverse map f−1 of the
simulated truncations at each step; see Malham & Wiese [25] and Ebrahimi–
Fard et al. [11]. We now develop an algebraic framework for studying such
map-truncate-invert schemes. The starting point is the quasi-shuffle algebra
of Hoffman [17]. This gives an explicit description of the algebra of iterated
integrals of the extended driving processes. Let RA denote the R-linear span
of A.
Definition 5 (Quasi-shuffle product) For a given alphabet A, suppose
[·, ·]: RA ⊗RA → RA is a commutative, associative product on RA. The
quasi-shuffleproductonRhAi,whichis commutative,isgeneratedinductively
as follows: if 1 is the empty word then u∗1=1∗u=u and
ua∗vb=(u∗vb)a+(ua∗v)b+(u∗v)[a,b],
forallwordsu,v ∈A∗ andlettersa,b∈A.Hereuadenotestheconcatenation
of u and a.
Remark 6 (Word-to-integral isomorphism) The word-to-integralmapµ: w 7→
I is an algebra isomorphism. Here the domain is the vector space RhAi
w
equippedwiththequasi-shuffleproduct.Iteratedintegralsindexedbypolyno-
mialsaredefinedbylinearity,i.e.I =k I +k I ,foranytwoconstants
kuu+kvv u u v v
k ,k ∈ R and words u,v ∈ A∗. This was proved in Curry et al. [10], it had
u v
already been established by Gaines [19] for drift-diffusions and by Li & Liu
[22] for jump-diffusions.
Remark 7 (Shuffle product) The quasi-shuffle product is a deformation of the
shuffleproductonthevectorspaceRhAiofpolynomialsinthenon-commuting