Table Of ContentA MILSTEIN-TYPE SCHEME WITHOUT LE´VY AREA TERMS FOR
SDES DRIVEN BY FRACTIONAL BROWNIAN MOTION
0 A. DEYA, A. NEUENKIRCH, S. TINDEL
1
0
Abstract. In this article, we study the numericalapproximationof stochastic differ-
2
ential equations driven by a multidimensional fractional Brownian motion (fBm) with
n
Hurst parameter greater than 1/3. We introduce an implementable scheme for these
a
J equations, which is based on a second order Taylor expansion, where the usual L´evy
9 areatermsarereplacedbyproductsofincrementsofthedrivingfBm. Theconvergence
1 ofourschemeis shownby meansofacombinationofroughpathstechniques anderror
bounds for the discretisation of the L´evy area terms.
]
R
P
.
h
t 1. Introduction and Main Results
a
m
Fractional Brownian motion (fBm in short for the remainder of the article) is a nat-
[ ural generalisation of the usual Brownian motion, insofar as it is defined as a centered
1 Gaussian process B = B ; t R with continuous sample paths, whose increments
t +
v (δB) := B B , s,t { R a∈re ch}aracterised by their variance E[(δB)2 ] = t s 2H.
4 st t − s ∈ + st | − |
4 HeretheparameterH (0,1), whichiscalledHurst parameter, governsinparticular the
∈
3 H¨older regularity of the sample paths of B by a standard application of Kolmogorov’s
3
criterion: fBm has H¨older continuous sample paths of order λ for all λ < H. The par-
.
1
ticular case H = 1/2 corresponds to the usual Brownian motion, so the cases H = 1/2
0
6
0 are a natural extension of the classical situation, allowing e.g. any prescribed H¨older
1 regularity of the driving process. Moreover, fBm is H-self similar, i.e. for any c > 0 the
:
v process cHB ; t R is again a fBm, and also has stationarity increments, that is
t/c +
Xi for any h{ 0 the p∈roces}s B B ; t R is a fBm.
t+h h +
≥ { − ∈ }
r These properties (partially) explain why stochastic equations driven by fBm have
a
received considerable attention during the last two decades. Indeed, many physical sys-
tems seem to be governed by a Gaussian noise with different properties than classical
Brownian motion. Fractional Brownian motion as driving noise is used e.g. in electrical
engineering [12, 13], or biophysics [5, 23, 34]. Moreover, after some controversial discus-
sions (see [3] for a summary of the early developments) fBm has established itself also in
financial modelling, see e.g. [17, 2]. For empirical studies of fractional Brownian motion
in finance see e.g. [8, 39, 7]. All these situations lead to different kind of stochastic
differential equations (SDEs), whose simplest prototype can be formally written as
m t
Y = a+ σ(i)(Y )dB(i), t [0,T], a Rd, (1)
t u u ∈ ∈
i=1 Z0
X
Date: January 19, 2010.
2000 Mathematics Subject Classification. Primary 60H35; Secondary 60H07, 60H10, 65C30.
Key words and phrases. fractional Brownian motion, L´evy area, approximationschemes.
Supported by the DAAD (PPP-Procope D/0707564)and Egide (PHC-Procope 17879TH).
1
2 A. DEYA,A. NEUENKIRCH,S.TINDEL
where σ = (σ(1),...,σ(m)) is a smooth enough function from Rd to Rd×m and B =
(B(1),...,B(m)) is a m-dimensional fBm with Hurst parameter H > 1/3.
At a mathematical level, fractional differential equations of type (1) are typically
handled (for H = 1/2) by pathwise or semi-pathwise methods. Indeed for H > 1/2, the
6
integrals tσ(i)(Y )dB(i), i = 1,...,m, in (1) can be defined using Young integration
0 u u
or fractional calculus tools, and these methods also yield the existence of a unique
R
solution, see e.g. [33, 40]. When 1/4 < H < 1/2, the existence and uniqueness result
for equation (1) can be seen as the canonical example of an application of the rough
paths theory. The reader is referred to [16, 25] for the original version of the rough paths
theory, and to [18] for a (slightly) simpler algebraic setting which will be used in the
current article. In the particular case 1/3 < H < 1/2, the rough path machinery can
be summarised very briefly as follows: assume that our driving signal B allows to define
iterated integrals with respect to itself. Then one can define and solve equation (1) in a
reasonable class of processes.
Once SDEs driven by fBm are solved, it is quite natural (as in the case of SDEs
driven by the usual Brownian motion) to study the stochastic processes they define.
However, even if some progress has been made in this direction, e.g. concerning the law
of the solution [1, 4, 29] or its ergodic properties [20], the picture here is far from being
complete. Moreover, explicit solutions of stochastic differential equations driven by fBm
are rarely known, as in the case of SDEs driven by classical Brownian motion. Thus one
has to rely on numerical methods for the simulation of these equations.
So far, some numerical schemes for equations like (1) have already been studied in the
literature. In the following, we consider uniform grids of the form t = kT/n; 0 k
k
{ ≤ ≤
n for a fixed T > 0. The simplest approximation method is the Euler scheme defined
}
by
Yn = a,
0
m
Yn = Yn + σ(i)(Yn)δB(i) , k = 0,...,n 1.
tk+1 tk tk tktk+1 −
i=1
X
For H > 1/2, the Euler scheme converges to the solution of the SDE (1). See e.g.
in [26], where an almost sure convergence rate n−(2H−1)+ε with ε > 0 arbitrarily small is
established. A detailed analysis of the one-dimensional case is given in [28], where the
exact convergence rate n−2H+1 and the asymptotic error distribution are derived.
However, the Euler scheme is not appropriate to approximate SDEs driven by fBm
when 1/3 < H < 1/2. This is easily illustrated by the following one-dimensional exam-
ple, in which B denotes a one-dimensional fBm: consider the equation
dY = Y dB , t [0,1], Y = 1,
t t t 0
∈
whose exact solution is
Y = exp(B ), t [0,1].
t t
∈
The Euler approximation for this equation at the final time point t = 1 can be written
as
n−1
Yn = (1+(δB) ).
1 k/n,(k+1)/n
k=0
Y
A MILSTEIN-TYPE SCHEME FOR SDES DRIVEN BY FBM 3
So for n N sufficiently large and using a Taylor expansion, we have
∈
n−1 n−1
1
Yn = exp log(1+(δB) ) = exp B (δB) 2 +ρ ,
1 k/n,(k+1)/n 1 − 2 | k/n,(k+1)/n| n
(cid:16)Xk=0 (cid:17) (cid:16) Xk=0 (cid:17)
a.s.
where lim ρ = 0 for H > 1/3. Now it is well known that
n→∞ n
n−1
(δB) 2 a.s.
k/n,(k+1)/n
| | −→ ∞
k=0
X
for H < 1/2 as n , which implies that lim Yn a=.s. 0. This is obviously incompat-
→ ∞ n→∞ 1
ible with a convergence towards Y = exp(B ). In the case H = 1/2 this phenomenon
1 1
is also well known: here the Euler scheme converges to the Itˆo solution and not to the
Stratonovich solution of SDE (1).
To obtain a convergent numerical method Davie proposed in [9] a scheme of Milstein
type. For this, assume that all iterated integrals of B with respect to itself are collected
into a m m matrix B2, i.e. set
×
t u
B2(i,j) = dB(i)dB(j), 0 s < t T, 1 i,j m.
st v u ≤ ≤ ≤ ≤
Zs Zs
The matrix B2 (respectively its elements) is (are) usually called L´evy area. Davie’s
scheme is then given by
Yn = a, (2)
0
m m
Yn = Yn + σ(i)(Yn)δB(i) + (i)σ(j)(Yn)B2 (i,j), k = 0,...,n 1,
tk+1 tk tk tktk+1 D tk tktk+1 −
i=1 i,j=1
X X
with the differential operator (i) = d σ(i)∂ . (Recall that we use the notation
D l=1 l xl
(i) (i) (i)
δB = B B for s,t [0,T].) This scheme is shown to be convergent as long as
st t − s ∈ P
H > 1/3 in [9], with an almost sure convergence rate of n−(3H−1)+ε for ε > 0 arbitrarily
small. Thisresulthasthenbeenextendedin[16]toanabstractroughpathwitharbitrary
regularity, under further assumptions on the higher order iterated integral of the driving
signal.
As the classical Milstein scheme for SDEs driven by Brownian motion, the Milstein-
type scheme (2) is in general not a directly implementable method. Indeed, unless the
commutativity condition
(i)σ(j) = (j)σ(i), i,j = 1,...,m,
D D
holds, the simulation of the iterated integrals B2 (i,j) is necessary. However, the law
tktk+1
of these integrals is unknown, so that they can not be simulated directly and have to be
approximated.
In this article we replace the iterated integrals by a simple product of increments, i.e.
we use the approximation
1
B2 (i,j) δB(i) δB(j) . (3)
tktk+1 ≈ 2 tktk+1 tktk+1
4 A. DEYA,A. NEUENKIRCH,S.TINDEL
This leads to the following simpler Milstein-type scheme: Set Zn = a and
t0
m m
1
Zn = Zn + σ(i)(Zn)δB(i) + (i)σ(j)(Zn)δB(i) δB(j) (4)
tk+1 tk tk tktk+1 2 D tk tktk+1 tktk+1
i=1 i,j=1
X X
for k = 0,...,n 1. Moreover, for t (t ,t ), define
k k+1
− ∈
t t
Zn = Zn + − k δZn , (5)
t tk T/n tktk+1
i.e. if t [0,T] is not a discretisation point,(cid:0)then(cid:1) Zn is defined by piecewise linear
∈ t
interpolation. This scheme is now directly implementable and is still convergent.
Theorem 1.1. Assume that σ C3(Rd;Rd×m) is bounded with bounded derivatives.
∈
Let Y be the solution to equation (1) and Zn the Milstein approximation given by (4)
and (5). Moreover, let 1/3 < γ < H. Then, there exists a finite and non-negative
random variable η such that
H,γ,σ,T
Y Zn η log(n) n−(H−γ) (6)
γ,∞,T H,γ,σ,T
k − k ≤ · ·
for n > 1. p
Here denotes the κ-Ho¨lder norm of a function f : [0,T] Rl, i.e.
κ,∞,T
k·k →
f(t) f(s)
f = sup f(t) + sup | − |. (7)
κ,∞,T
k k | | t s κ
t∈[0,T] s,t∈[0,T]
| − |
Remark 1.2. Notethatthealmostsureestimate(6)cannotbeturnedintoanL1-estimate
for Y Zn . This is a common consequence of the use of the rough paths method,
γ,∞,T
k − k
which exhibits non-integrable (random) constants, as a careful examination of the proof
of Theorem 2.6 would show. See also [16] for further details.
Our strategy to prove the above Theorem consists of two steps. First we determine
the error between Y and its Wong-Zakai approximation
m t
Zn = a+ σ(i)(Zn)dB(i),n, t [0,T], a Rd, (8)
t u u ∈ ∈
i=1 Z0
X
where
t t
Bn = B + − k (δB) , t [0,T],
t tk T/n tktk+1 ∈
(cid:18) (cid:19)
i.e. B in equation (1) is replaced with its piecewise linear interpolation. (For a survey
on Wong-Zakai approximations for standard SDEs see e.g. [36].) Here, we denote the
L´evy area corresponding to Bn by Bn. Using the Lipschitzness of the Itˆo map of Y,
i.e. the solution of equation (1) depends continuously in appropriate H¨older norms on
B and the L´evy-area B, and error bounds for the difference between B and Bn resp. B
and Bn, we obtain
Y Zn η(1) log(n) n−(H−γ),
k − kγ,∞,T ≤ H,γ,σ,T · ·
where η(1) is a finite and non-negative randopm variable.
H,γ,σ,T
In the second step we analyse the difference between Zn and Zn. The second order
A MILSTEIN-TYPE SCHEME FOR SDES DRIVEN BY FBM 5
Taylor scheme with stepsize T/n for classical ordinary differential equations applied to
the Wong-Zakaiapproximation (8) gives our simplified Milstein scheme (4). So to obtain
the error bound
Zn Zn η(2) log(n) n−(H−γ),
k − kγ,∞,T ≤ H,γ,σ,T · ·
we can proceed in a similar way as for the nupmerical analysis of classical ordinary
differential equations. We first determine the one-step error and then control the error
propagation using a global stability result with respect to the initial value for differential
equations driven by rough paths. The latter can be considered as a substitute for
Gronwall’s lemma in this context.
Combining both error bounds then gives Theorem 1.1.
Remark 1.3. For H = 1/2 the scheme (2) corresponds to the classical Milstein scheme
for Stratonovich SDEs driven by Brownian motion, while our scheme (4) corresponds to
the so called simplified Milstein scheme. See e.g. [22].
Remark 1.4. At the price of further computations, which are simpler than the ones in
this article, our convergence result can be extended to an equation with drift, i.e. to
t m t
Y = a+ b(Y )du+ σ(i)(Y )dB(i), t [0,T], a Rd,
t u u u ∈ ∈
Z0 i=1 Z0
X
where b : Rd Rd is a C3 function and where the other coefficients satisfy the as-
→ b
sumptions of Theorem 1.1. Indeed, the equation above can be treated like our original
(0)
system (1) by adding a component B = t to the fractional Brownian motion. The
t
additional iterated integrals of B(0) with respect to B(j) for j = 1,...,m are easier
to handle than B2(i,j) for i,j 1,...,m , since they are classical Riemann-Stieltjes
∈ { }
integrals. For sake of conciseness we do not include the corresponding details.
Remark 1.5. Theorem 1.1 requires σ to be bounded. However, if σ C3(Rd;Rd×m) is
∈
neither boundednorhasboundedderivatives butequation(1)hasstill auniquepathwise
solution in the sense of Theorem 2.6 below, then the assertion of Theorem 1.1 is still
valid. This follows from a standard localisation procedure, see e.g. [21], and applies in
particular to affine-linear coefficients.
Remark 1.6. The error bound of Theorem 1.1 is sharp. To see this, consider the most
simple equation
dY(1) = dB(1), t [0,T], Y = a R,
t t ∈ 0 ∈
for which our approximation obviously reduces to Zn = Bn. Then, due to results of
Hu¨sler, Piterbarg and Seleznjev ([14]) for the deviation of a Gaussian process from its
linear approximation, one can prove that
lim P(ℓ(n) Y Zn < ) = 0,
γ,∞,T
n→∞ ·k − k ∞
if
liminf ℓ(n) log(n) n−(H−γ) = . (9)
n→∞ · · ∞
For further details see Section 4.3. p
6 A. DEYA,A. NEUENKIRCH,S.TINDEL
Remark 1.7. If the Wong-Zakai approximation is discretised with an arbitrary numerical
scheme for ODEs of at least second order (e.g. Heun, Runge-Kutta 4), then the arising
scheme for equation (1) satisfies the same error bound as the proposed modified Milstein
scheme. So, the strategy of our proof is in fact an instruction for the construction of
arbitrary implementable and convergent numerical schemes for SDEs driven by fBm.
Remark 1.8. Instead of replacing the L´evy terms in Davie’s scheme by the ”rough”
approximation (3) one coulddiscretise these terms very finely using theresults contained
in [31], where (exact) convergence rates for approximations of the L´evy area are derived.
However, it is well known that already for SDEs driven by Brownian motion such a
scheme is rarely efficient, if the convergence rate of the scheme is measured in terms of
its computational cost. For a survey on the complexity of the approximation of SDEs
driven by Brownian motion, see e.g. [27].
The γ-Ho¨lder norm, which appears in Theorem 1.1 since the Itˆo-map of Y is only
Lipschitz in appropriate H¨older norms with 1/3 < γ < H and thus is natural in the
rough path setting, is not typical for measuring the error of approximations to stochastic
differential equations. A more standard criterion would be the error with respect to the
supremum norm, i.e.
Y Zn = sup Y Zn .
k − k∞,T | t − t |
t∈[0,T]
The error (in the supremum norm) of the piecewise linear interpolation of fractional
Brownian motion is of order log(n)n−H, see [14]. Moreover, for the iterated inte-
T u (1) (2)
gral dB dB the proposed Milstein-type scheme leads to the trapezoidal type
0 0 v u p
approximation
R R
n−1
1
(1) (1) (2) (2)
B +B B B .
2 tk tk+1 tk+1 − tk
k=0
X(cid:0) (cid:1)(cid:0) (cid:1)
The Lp-error for this approximation is of order n−2H+1/2, see [31].
Based on these two findings, our guess for the rate of convergence in supremum norm
is that
Y Zn η log(n) n−H +n−2H+1/2
∞,T H,σ,T
k − k ≤ · ·
holds under the assumptions of Theorem 1.1. This conjecture is also supported by the
p (cid:0) (cid:1)
numerical examples we give in Section 4.
The remainder of this article is structured as follows: In Section 2 we recall some
basic facts on algebraic integration and rough differential equations. The proofs of
Theorem 1.1 and Remark 1.6 are given in Section 3 and 4. Finally, Section 5 contains
the mentioned numerical examples.
2. Algebraic integration and differential equations
In this section, we recall the main concepts of algebraic integration, which will be
essential todefinethegeneralizedintegralsinoursetting. Namely, westatethedefinition
of the spaces of increments, of the operator δ, and its inverse called Λ (or sewing map
accordingtotheterminologyof[15]). Wealsorecallsomeelementarybutusefulalgebraic
relations on the spaces of increments. The interested reader is sent to [18] for a complete
account on the topic, or to [11, 19] for a more detailed summary.
A MILSTEIN-TYPE SCHEME FOR SDES DRIVEN BY FBM 7
2.1. Increments. The extended integral we deal with is based on the notion of incre-
ments, together with an elementary operator δ acting on them.
The notion of increment can be introduced in the following way: for two arbitrary real
numbers ℓ > ℓ 0, a vector space V, and an integer k 1, we denote by ([ℓ ,ℓ ];V)
2 1 k 1 2
≥ ≥ C
the set of continuous functions g : [ℓ ,ℓ ]k V such that g = 0 whenever t = t
1 2 → t1···tk i i+1
for some i 0,...,k 1 . Such a function will be called a (k 1)-increment, and
∈ { − } −
we will set ([ℓ ,ℓ ];V) = ([ℓ ,ℓ ];V). To simplify the notation, we will write
∗ 1 2 k≥1 k 1 2
C ∪ C
(V), if there is no ambiguity about [ℓ ,ℓ ].
k 1 2
C
The operator δ is an operator acting on k-increments, and is defined as follows on
(V):
k
C
k+1
δ : (V) (V), (δg) = ( 1)ig , (10)
Ck → Ck+1 t1···tk+1 − t1···tˆi···tk+1
i=1
X
where tˆ means that this particular argument is omitted. Then a fundamental property
i
of δ, which is easily verified, is that δδ = 0, where δδ is considered as an operator from
(V) to (V). We will denote (V) = (V) Kerδ and (V) = (V) Imδ.
k k+2 k k k k
C C ZC C ∩ BC C ∩
Somesimpleexamples ofactionsofδ, which willbetheoneswewillreallyusethrough-
out the article, are obtained by letting g (V) and h (V). Then, for any
1 2
∈ C ∈ C
t,u,s [ℓ ,ℓ ], we have
1 2
∈
(δg) = g g and (δh) = h h h . (11)
st t s sut st su ut
− − −
Our future discussions will mainly rely on k-increments with k = 2 or k = 3, for
which we will use some analytical assumptions. Namely, we measure the size of these
increments by H¨older norms defined in the following way: for f (V) let
2
∈ C
f
f = sup | st| and µ(V) = f (V); f < .
k kµ t s µ C2 { ∈ C2 k kµ ∞}
s,t∈[ℓ1,ℓ2] | − |
Using this notation, we define in a natural way
µ(V) = f (V); δf < ,
C1 { ∈ C1 k kµ ∞}
and recall that we have also defined a norm at equation (7). In the same way,
κ,∞,T
k·k
for h (V), we set
3
∈ C
h
sut
h = sup | | , (12)
k kγ,ρ u s γ t u ρ
s,u,t∈[ℓ1,ℓ2] | − | | − |
h = inf h ; h = h , 0 < ρ < µ ,
k kµ k ikρi,µ−ρi i i
( )
i i
X X
where the last infimum is taken over all sequences h , i N (V) such that
i 3
{ ∈ } ⊂ C
h = h and over all choices of the numbers ρ (0,µ). Then is easily seen to be
i i i ∈ k·kµ
a norm on (V), and we define
3
P C
µ(V) := h (V); h < .
C3 { ∈ C3 k kµ ∞}
Eventually, let 1+(V) = µ(V), and note that the same kind of norms can be
considered on thCe3 spaces ∪µ>(1VC)3, leading to the definition of the spaces µ(V) and
ZC3 ZC3
1+(V). In order to avoid ambiguities, we denote in the following by [ ; κ] the
ZC3 N · Cj
8 A. DEYA,A. NEUENKIRCH,S.TINDEL
κ-Ho¨lder norm on the space , for j = 1,2,3. For ζ (V), we also set [ζ; 0(V)] =
Cj ∈ Cj N Cj
sup ζ .
s∈[ℓ1;ℓ2]jk skV
The operator δ can be inverted under some H¨older regularity conditions, which is
essential for the construction of our generalized integrals.
Theorem 2.1 (The sewing map). Let µ > 1. For any h µ(V), there exists a
unique Λh µ(V) such that δ(Λh) = h. Furthermore, ∈ ZC3
∈ C2
1
Λh [h; µ(V)]. (13)
k kµ ≤ 2 2µ N C3
−
This gives rise to a continuous linear map Λ : ZC3µ(V) → C2µ(V) such that δΛ = idZC3µ(V).
Proof. The original proof of this result can be found in [18]. We refer to [11, 19] for two
simplified versions.
(cid:3)
The sewing map creates a first link between the structures we just introduced and the
problem of integration of irregular functions:
Corollary 2.2 (Integration of small increments). For any 1-increment g (V) such
2
∈ C
that δg 1+, set h = (id Λδ)g. Then, there exists f (V) such that h = δf and
∈ C3 − ∈ C1
n
(δf) = lim g ,
st titi+1
|Πst|→0
i=0
X
where the limit is over any partition Π = t = s,...,t = t of [s,t] whose mesh tends
st 0 n
{ }
to zero. The 1-increment δf is the indefinite integral of the 1-increment g.
We also need some product rules for the operator δ. For this recall the following
convention: for g ([ℓ ,ℓ ]Rl,d) and h ([ℓ ,ℓ ];Rd,p) let gh be the element of
n 1 2 m 1 2
([ℓ ,ℓ ];Rl,p∈) dCefined by ∈ C
n+m−1 1 2
C
(gh) = g h (14)
t1,...,tm+n−1 t1,...,tn tn,...,tm+n−1
for t ,...,t [ℓ ,ℓ ]. With this notation, the following elementary rule holds true:
1 m+n−1 1 2
∈
Proposition2.3. Let g ([ℓ ,ℓ ];Rl,d) and h ([ℓ ,ℓ ];Rd). Thenghis an element
2 1 2 1 1 2
of ([ℓ ,ℓ ];Rl) and δ(g∈h)C= δgh gδh. ∈ C
2 1 2
C −
2.2. Random differential equations. One of the main appeals of the algebraic inte-
gration theory is that differential equations driven by a γ-Ho¨lder signal x can be defined
and solved rather quickly in this setting. In the case of an H¨older exponent γ > 1/3, the
required structures are just the notion of controlled processes and the L´evy area based
on x.
Indeed, let us consider an equation of the form
m
dy = σ(y )dx = σ(i)(y )dxi, t [0,T], y = a, (15)
t t t t t ∈ 0
i=1
X
where a is a given initial condition in Rd, x is an element of γ([0,T]; Rm), and σ is a
smooth enough function from Rd to Rd,m. Then it is natural (Cs1ee [35] for further expla-
nations) that the increments of a candidate for a solution to (15) should be controlled
by the increments of x in the following way:
A MILSTEIN-TYPE SCHEME FOR SDES DRIVEN BY FBM 9
Definition 2.4. Let z be a path in κ(Rd) with 1/3 < κ γ. We say that z is a weakly
controlled path based on x if z = a,C1with a Rd, and δz≤ κ(Rd) has a decomposition
0 ∈ ∈ C2
δz = ζδx+r, that is, for any s,t [0,T],
∈
(δz) = ζ (δx) +r , (16)
st s st st
with ζ κ(Rd,m) and r 2κ(Rd).
∈ C1 ∈ C2
The space of weakly controlled paths will be denoted by x (Rd), and a process z
Qκ,a ∈
x (Rd) can be considered in fact as a couple (z,ζ). The space x (Rd) is endowed
Qκ,a Qκ,a
with a natural semi-norm given by
[z; x (Rd)] (17)
N Qκ,a
= [z; κ(Rd)]+ [ζ; 0(Rd,m)]+ [ζ; κ(Rd,m)]+ [r; 2κ(Rd)],
N C1 N C1 N C1 N C2
where the quantities [g; κ] have been defined in Section 2.1. For the L´evy area
N Cj
associated to x we assume the following structure:
Hypothesis 1. The path x : [0,T] Rm is γ-H¨older continuous with 1 < γ 1
→ 3 ≤
and admits a so-called L´evy area, that is, a process x2 2γ(Rm,m), which satisfies
δx2 = δx δx, namely ∈ C2
⊗ (δx2) (i,j) = [δxi] [δxj] ,
sut su ut
for any s,u,t [0,T] and i,j 1,...,m .
(cid:2) (cid:3)
∈ ∈ { }
To illustrate the idea behind the construction of the generalized integral assume that
the paths x and z are smooth and also for simplicity that d = m = 1. Then the
Riemann-Stieltjes integral of z with respect to x is well defined and we have
t t t
z dx = z (x x )+ (z z )dx = z (δx) + (δz) dx
u u s t s u s u s st su u
− −
Zs Zs Zs
for ℓ s t ℓ . If z admits the decomposition (16) we obtain
1 2
≤ ≤ ≤
t t t t
(δz) dx = (ζ (δx) +ρ )dx = ζ (δx) dx + ρ dx . (18)
su u s su su u s su u su u
Zs Zs Zs Zs
Moreover, if we set
t
2
(x ) := (δx) dx , ℓ s t ℓ ,
st su u 1 2
≤ ≤ ≤
Zs
then it is quickly verified that x2 is the associated L´evy area to x. Hence we can write
t t
2
z dx = z (δx) +ζ (x ) + ρ dx .
u u s sz s st su u
Zs Zs
Now rewrite this equation as
t t
2
ρ dx = z dx z (δx) ζ (x ) (19)
su u u u s st s st
− −
Zs Zs
and apply the increment operator δ to both sides of this equation. For smooth paths z
and x we have
δ zdx = 0, δ(zδx) = δzδx,
−
(cid:18)Z (cid:19)
10 A. DEYA,A. NEUENKIRCH,S.TINDEL
by Proposition 2.3. Hence, applying these relations to the right hand side of (19), using
the decomposition (16), the properties of the L´evy area and again Proposition 2.3, we
obtain
2 2
δ ρdx = (δz) (δx) +(δζ) (x ) ζ (δx )
su ut su ut s sut
−
(cid:20) (cid:18)Z (cid:19)(cid:21)sut
2
= ζ (δx) (δx) +ρ (δx) +(δζ) (x ) ζ (δx) (δx)
s su ut su ut su ut s su ut
−
2
= ρ (δx) +(δζ) (x ) .
su ut su ut
So in summary, we have derived the representation
2
δ ρdx = ρ (δx) +(δζ) (x ) .
su ut su ut
(cid:20) (cid:18)Z (cid:19)(cid:21)sut
As we are dealing with smooth paths we have δ ρdx 1+ and thus belongs to
∈ ZC3
the domain of Λ due to Proposition 2.1. (Recall that δδ = 0.) Hence, it follows
(cid:0)R (cid:1)
t
2
ρ dx = Λ ρδx+δζx ,
su u st
Zs
(cid:0) (cid:1)
and inserting this identity into (18) we end up with
t
2 2
z dx = z (δx) +ζ (x ) +Λ ρδx+δζx .
u u s st s st st
Zs
Since in addition (cid:0) (cid:1)
2 2
ρδx+δζx = δ(zδx+ζx ),
−
we can also write this as
2
z dx = (id Λδ)(zδx+ζx ).
u u
−
Z
Thus we have expressed the Riemann-Stieltjes integral of z with respect to x in terms
of the sewing map Λ, of the L´evy area x2 and of increments of z resp. x. This can now
be generalized to the non-smooth case. Note that Corollary 2.2 justifies the use of the
notion integral.
In the following, we denote by A∗ the transposition of a vector resp. matrix, and by
A A = Tr(A A∗) the inner product of two vectors or two matrices A and A .
1 · 2 1 2 1 2
Proposition 2.5. For fixed 1 < κ γ, let x be a path satisfying Hypothesis 1. Further-
more, let z x ([ℓ ,ℓ ];Rm3) such≤that the increments of z are given by (16). Define
zˆ by zˆ = αˆ∈wQitκh,ααˆ 1R2and
ℓ1 ∈
(δzˆ) = (id Λδ)(z∗δx+ζ x2) (20)
st − · st
for ℓ1 ≤ s ≤ t ≤ ℓ2. Then J(z∗d(cid:2)x) := zˆ is a well-define(cid:3)d element of Qxκ,αˆ([ℓ1,ℓ2];R)
and coincides with the usual Riemann integral, whenever z and x are smooth functions.
Moreover, the H¨older norm of (z∗dx) can be estimated in terms of the H¨older norm
J
of the integrator z. (For this and also for a proof of the above Proposition, see e.g. [18].)
This allows to use a fixed point argument to obtain the existence of a unique solution
for rough differential equations.
Theorem 2.6. For fixed 1 < κ < γ, let x be a path satisfying Hypothesis 1, and let
3
σ C3(Rd;Rd,m) be bounded with bounded derivatives. Then we have:
∈
