A note on Malliavin fractional smoothness for Lévy processes and approximation

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by  Christel Geiss

A NOTE ON MALLIAVIN FRACTIONAL
SMOOTHNESS FOR LE´VY PROCESSES AND
APPROXIMATION
CHRISTEL GEISS, STEFAN GEISS, AND EIJA LAUKKARINEN
2
1
0 Abstract. Assume a L´evy process (Xt)t∈[0,1] that is an L2-
2 martingale and let Y be either its stochastic exponential or X

n itself. For certain integrands ϕ we investigate the behavior of
a
N
J
1 ϕtdXt− vk−1(Ytk −Ytk−1) ,
(cid:13)(cid:13)Z(0,1] kX=1 (cid:13)(cid:13)L2
R] where vk−1 i(cid:13)(cid:13)s Ftk−1-measurable, in dependence o(cid:13)(cid:13)n the fractional
smoothness in the Malliavin sense of ϕ dX . A typical situa-
(0,1] t t
P
tion where these techniques apply occurs if the stochastic integral
. R
h is obtained by the Galtchouk-Kunita-Watanabe decomposition of
at some f(X1). Moreover, using the example f(X1) = 1(K,∞)(X1)
m we showhow fractionalsmoothnessdepends onthe distribution of
the L´evy process.
[

1
v
9
1. Introduction
8
3
0 We consider the quantitative Riemann approximation of stochastic
. integrals driven by L´evy processes and its relation to the fractional
1
0 smoothness in the Malliavin sense. Besides the interest on its own,
2
the problem is of interest for numerical algorithms and for Stochastic
1
v: Finance. To explain the latter aspect, assume a price process (St)t∈[0,1]
given under der martingale measure by a diffusion
i
X
t
r
S = s + σ(S )dW ,
a t 0 r r
Z0
where W is the Brownian motion and where usual conditions on σ are
imposed. For a polynomially bounded Borel function f : R R we
→
obtain a representation
1
(1) f(S ) = V + ϕ dS
1 0 t t
Z0
where (ϕ ) is a continuous adapted process which can be obtained
t t∈[0,1)
via the gradient of a solution to a parabolic backward PDE related to
ThesecondandthirdauthoraresupportedbytheProject133914oftheAcademy
of Finland.
1
2 CHRISTEL GEISS,STEFAN GEISS, ANDEIJA LAUKKARINEN
σ with terminal condition f. The process (ϕ ) is interpreted as
t t∈[0,1)
a trading strategy. In practice one can trade only finitely many times
which corresponds to a replacement of the stochastic integral in (1) by
the sum N ϕ (S S ) with 0 = t < t < < t = 1. The
k=1 tk−1 tk − tk−1 0 1 ··· N
error
P
1 N
(2) ϕ dS ϕ (S S )
t t − tk−1 tk − tk−1
Z0 k=1
X
caused by this replacement is often measured in L and has been stud-
2
ied by various authors, for example by Zhang [21], Gobet and Temam
[11], S. Geiss [8], S. Geiss and Hujo [9] and C. Geiss and S. Geiss [7].
For results concerning L with p (2, ) we refer to [20], the weak
p
∈ ∞
convergence is considered in [10] and [19] and by other authors. In par-
ticular, if S is the Brownian motion or the geometric Brownian motion,
S. Geiss and Hujo investigated in [9] the relation between the Malliavin
fractional smoothness of f(S ) and the L -rate of the discretization er-
1 2
ror (2).
It is natural to extend these results to L´evy processes. A first step was
done by M. Brod´en and P. Tankov [5] (see Remark 4.11). The aim of
this paper is to extend results of [9] into the following directions:
(a) The Brownian motion and the geometric Brownian motion are gen-
eralized to L´evy processes (X ) that are L -martingales and their
t t∈[0,1] 2
Dol´eans-Dade exponentials S = (X),
E
S = 1+ S dX ,
t u− u
Z(0,t]
respectively. For certain stochastic integrals
F = ϕ dX
s− s
Z(0,1]
and for Y X, (X) we study the connection of the Malliavin frac-
∈ { E }
tional smoothness of F (introduced by the real interpolation method)
and the behavior of
N
(3) aopt(F;(t )N ) = inf F v (Y Y ) ,
Y k k=0 − k−1 tk − tk−1
(cid:13) (cid:13)
(cid:13) Xk=1 (cid:13)L2
(cid:13) (cid:13)
where the infimum is taken(cid:13) over -measurable v(cid:13) such that
(cid:13) Ftk−1 (cid:13)k−1
Ev2 (Y Y )2 < and where 0 = t < < t = 1 is a
k−1 tk − tk−1 ∞ 0 ··· N
deterministic time-net.
(b) In contrast to [9], where the reduction of the stochastic approxima-
tion problem to a deterministic one is based on Itˆo’s formula and was
FRACTIONAL SMOOTHNESS, LE´VY PROCESSES AND APPROXIMATION 3
done in [8, 7], we prove an analogous reduction in Theorems 3.3 and
3.4 by techniques based on the Itˆo chaos decomposition.
(c) One more principal difference to [9] is the fact that L´evy pro-
cesses do in general not satisfy the representation property and there-
fore there are F L that cannot be approximated by sums of
2
∈
the form N v (Y Y ) in L . As a consequence we have
k=1 k−1 tk − tk−1 2
to use the (orthogonal) Galtschouk-Kunita-Watanabe projection that
P
projects L onto the subspace I(X) of stochastic integrals λ dX
2 (0,1] s s
with E 1 λ 2ds < that can be defined in our setting as the L -
0 | s| ∞ R 2
closure of
R
(4)
N
0 = a < < a = 1
0 N
v (X X ) : v L ( ), ···
ak−1 ak − ak−1 ak−1 ∈ 2 Fak−1 N = 1,2,...
( )
k=1
X
to deal with our approximation problem.
The paper is organized as follows. In Section 2 we recall some facts
about real interpolation and L´evy processes. In Section 3 we investi-
gate the discrete time approximation. The basic statement is Theorem
3.3 that reduces the stochastic approximation problem to a determin-
istic one in case of the Riemann-approximation (2) (which we call sim-
ple approximation in the sequel). The difference between the simple
and optimal approximation (3) is shown in Theorem 3.4 to be suffi-
ciently small. Theorem 3.5 provides a lower bound for the optimal
L -approximation. Finally, Theorems 3.6 and 3.8 give the connection
2
to the Besov spaces defined by real interpolation. We conclude with
Section 4 where we use the example f(x) = 1 (x) to demonstrate
(K,∞)
howthefractionalsmoothness dependsontheunderlying L´evyprocess.
2. Preliminaries
2.1. Notation. Throughout this paper we will use for A,B,C 0
≥
and c 1 the notation A B for 1B A cB and A = B C for
≥ ∼c c ≤ ≤ ±
B C A B+C.The phrasec`adl`agstandsfor apathwhich isright-
− ≤ ≤
continuous and has left limits. Given q [1, ], the sequence space ℓ
q
∈ ∞
consists of all α = (α ) R such that α := ( ∞ α q)1/q <
N N≥1 ⊆ k kℓq N=1| N|
for q < and α := sup α < , respectively.
∞ ∞ k kℓ∞ N≥1| N| ∞ P
2.2. Real interpolation. First we recall some facts about the real
interpolation method.
4 CHRISTEL GEISS,STEFAN GEISS, ANDEIJA LAUKKARINEN
Definition 2.1. ForBanachspaces X X , where X iscontinuously
1 0 1
⊆
embedded into X , we define for u > 0 the K-functional
0
K(u,x;X ,X ) := inf x +u x .
0 1 x=x0+x1{k 0kX0 k 1kX1}
For θ (0,1) and q [1, ] the real interpolation space (X ,X )
0 1 θ,q
∈ ∈ ∞
consists of all elements x X such that x < where
∈ 0 k k(X0,X1)θ,q ∞
∞[u−θK(u,x;X ,X )]qdu 1q, q [1, )
0 0 1 u ∈ ∞
x :=
k k(X0,X1)θ,q  (cid:2)R (cid:3)
 supu>0u−θK(u,x;X0,X1), q = ∞.

The spaces (X ,X ) equipped with become Banach
0 1 θ,q k · k(X0,X1)θ,q
spaces and form a lexicographical scale, i.e. for any 0 < θ < θ < 1
1 2
and q ,q [1, ] it holds that
1 2
∈ ∞
X (X ,X ) (X ,X ) (X ,X ) X .
0 ⊇ 0 1 θ1,q1 ⊇ 0 1 θ2,q2 ⊇ 0 1 θ2,min{q1,q2} ⊇ 1
For more information the reader is referred to [3, 4].
2.3. The spaces Bθ (E).
2,q
Definition 2.2. For a sequence of Banach spaces E = (E )∞ with
n n=0
E = 0 we let ℓ (E) and d (E) be the Banach spaces of all a =
n 2 1,2
6 { }
(a )∞ E such that
n n=0 ∈
1 1
∞ 2 ∞ 2
a := a 2 and a := (n+1) a 2 ,
k kℓ2(E) k nkEn k kd1,2(E) k nkEn
! !
n=0 n=0
X X
respectively, are finite. Moreover, for θ (0,1) and q [1, ] we let
∈ ∈ ∞
(ℓ (E),d (E)) : θ (0,1),q [1, ]
Bθ (E) := 2 1,2 θ,q ∈ ∈ ∞ .
2,q d (E) : θ = 1,q = 2
1,2
(cid:26)
It can be shown that (cf. [9, Remark A.1])
∞
a 2 (n+1)θ a 2 .
k kBθ2,2(E) ∼c2θ k nkEn
n=0
X
To describe the interpolation spaces Bθ (E) we use two types of func-
2,q
tions. The first one is a generating function for ( a 2 )∞ , i.e. for
k nkEn n=0
a = (a )∞ ℓ (E) we let
n n=0 ∈ 2
∞
T (t) := a 2 tn.
a k nkEn
n=0
X
FRACTIONAL SMOOTHNESS, LE´VY PROCESSES AND APPROXIMATION 5
The second function will be used to describe our stochastic approxima-
tion in a deterministic way: For a ℓ (E) and a deterministic time-net
2
∈
τ = (t )N with 0 = t t = 1 we let
k k=0 0 ≤ ··· ≤ N
N tk 12
A(a,τ) := (t t)(T )′′(t)dt .
k a
−
(cid:18)k=1Ztk−1 (cid:19)
X
For the formulation of the next two theorems which will connect ap-
proximation properties with fractional smoothness special time nets
are needed. Given θ (0,1] and N 1, we let τθ be the time-net
∈ ≥ N
1
k θ
(5) tN,θ := 1 1 for k = 0,1,...,N
k − − N
(cid:18) (cid:19)
for which one has (see [10, relation (4)])
tN,θ t tN,θ tN,θ 1
(6) | k − | | k − k−1| for k = 1,...,N
(1 t)1−θ ≤ (1 tN,θ)1−θ ≤ θN
− − k−1
and t [tN,θ,tN,θ). For θ = 1 we obtain equidistant time-nets. The
∈ k−1 k
following two theorems are taken from [9]. For the convenience of the
reader we comment about the proofs in Remark 2.5 below.
Theorem 2.3 ([9]). For θ (0,1), q [1, ] and a = (a )∞ ℓ (E)
∈ ∈ ∞ n n=0 ∈ 2
one has
kakBθ2,q(E) ∼c kakℓ2(E) + N2θ−1qA(a,τN1) ∞N=1 ℓq
(cid:13)(cid:16) (cid:17) (cid:13)
where c [1, ) depends at most o(cid:13)n (θ,q) and the expre(cid:13)ssions may be
∈ ∞ (cid:13) (cid:13)
infinite.
Theorem 2.4 ([9]). For θ (0,1] and a = (a )∞ ℓ (E) the follow-
∈ n n=0 ∈ 2
ing assertions are equivalent:
(i) a Bθ (E).
∈ 2,2
(ii) 1(1 t)1−θT′′(t)dt < .
0 − F ∞
(iii) There exists a constant c > 0 such that
R
c
A(a,τθ) for N = 1,2,...
N ≤ √N
Remark 2.5. We fix a = (a )∞ ℓ (E) and (θ,q) according to
n n=0 ∈ 2
Theorems 2.3 and 2.4. Then we let β := a and define f =
∞ β h L (R,γ), where γ is the stanndardkGnkaEunssian measure and
n=0 n n ∈ 2
(h )∞ the orthonormal basis of Hermite polynomials. As before, let
n n=0
P
N tk 12 ∞
A(β,τ) := (t t)(T )′′(t)dt with T (t) := β2tn.
k − β β n
(cid:18)k=1Ztk−1 (cid:19) n=0
X X
6 CHRISTEL GEISS,STEFAN GEISS, ANDEIJA LAUKKARINEN
Omitting the notation (E) in the case E = (R,R,...), we have
a = β and a = β . Moreover, [9, Theorem 2.2]
k kℓ2(E) k kℓ2 k kd1,2(E) k kd1,2
gives that a β for θ (0,1) and q [1, ] because
k kBθ2,q(E) ∼c(θ,q) k kBθ2,q ∈ ∈ ∞
of T = T . Hence [9, Lemmas 3.9 and 3.10, Theorem 3.5 (X=W)]
a β
imply Theorem 2.3 of this paper. The equivalence of (i) and (iii) of
Theorem 2.4 follows in the same way by [9, Lemmas 3.9 and 3.10, The-
orem 3.2 (X=W)]. Finally, the equivalence of (i) and (ii) of Theorem
2.4 is a consequence of the proof of [9, Theorem 3.2 (X=W)].
2.4. L´evy processes. We follow the setting and presentation of [17,
Section 1.1] and assume a square integrable mean zero L´evy process
X = (X ) on a stochastic basis (Ω, ,P,( ) ) satisfying
t t∈[0,1] t t∈[0,1]
the usual assumptions, i.e. (Ω, ,P) is comFplete Fwhere the filtration
F
( ) is the augmented natural filtration of X and therefore right-
t t∈[0,1]
F
continuous and := is assumed without loss of generality. The
1
F F
L´evy measure ν with ν( 0 ) = 0 satisfies
{ }
x2ν(dx) <
∞
R
Z
by the square integrability of X (see [16, Theorem 25.3]). Let N be the
associated Poisson random measure and dN˜(t,x) = dN(t,x) dtdν(x)
−
be the compensated Poisson random measure. The L´evy-Itoˆ decompo-
sition (see [16, Theorem 19.2]) can be written under our assumptions
as
X = σW + xN˜(ds,dx).
t t
Z(0,t]×R\{0}
We introduce the finite measures µ on (R) and on ([0,1] R) by
B m B ×
µ(dx) := σ2δ (dx)+x2ν(dx),
0
(dt,dx) := dtµ(dx),
m
whereweagreeaboutµ(R) > 0toavoidpathologies. ForB ((0,1]
R ∈ B ×
) we define the random measure
M(B) := σ dW + xN˜(dt,dx)
t
Z{t∈(0,1]:(t,0)∈B} ZB∩((0,1]×(R\{0}))
and let
Ln := L (([0,1] R)n, (([0,1] R)n), ⊗n) for n 1.
2 2 × B × m ≥
By [12, Theorem 2] there is the chaos decomposition
∞
L := L (Ω, ,P) = I (Ln),
2 2 F n 2
n=0
M
where I (L0) is the space of the a.s. constant random variables and
0 2
I (Ln) := I (f ) : f Ln for n = 1,2,... and I (f ) denotes
n 2 { n n n ∈ 2} n n
FRACTIONAL SMOOTHNESS, LE´VY PROCESSES AND APPROXIMATION 7
the multiple integral w.r.t. the random measure M. For properties of
the multiple integral see [12, Theorem 1]. Especially, I (f ) 2 =
k n n kL2
n! f˜ 2 and
k nkLn2
∞
F 2 = n! f˜ 2
k kL2 k nkLn2
n=0
X
˜
with f being the symmetrization of f , i.e.
n n
1
˜
f (z ,...,z ) = f (z ,...,z )
n 1 n n π(1) π(n)
n!
for all z = (t ,x ) [0,1] R, wXhere the sum is taken over all permu-
i i i
∈ ×
tations π of 1,...,n . For F L the L -representation
2 2
{ } ∈
∞
˜
F = I (f ),
n n
n=0
X
with I (f ) = EF a.s. is unique (note that I (f ) = I (f˜ ) a.s.).
0 0 n n n n
2.5. Besov spaces. Here we recall the construction of Besov spaces
(or spaces of random variables of fractional smoothness) based on the
above chaos expansion.
Definition 2.6. Let D be the space of all F = ∞ I (f ) L
1,2 n=0 n n ∈ 2
such that
∞ P
F 2 := (n+1) I (f ) 2 < .
k kD1,2 k n n kL2 ∞
n=0
X
Moreover,
(L ,D ) : θ (0,1),q [1, ]
Bθ2,q := 2 1,2Dθ,q : θ =∈ 1,q = 2∈ ∞ .
1,2
(cid:26)
2.6. The space of the random variables to approximate. We
M
will approximate random variables from the following space :
Definition 2.7. The closed subspace M L consists of all mean zero
2
⊆
F L such that there exists a representation
2
∈
∞
F = I (f )
n n
n=1
X
with symmetric f such that there are h R and symmetric h
n 0 n
∈ ∈
L (µ⊗n) for n 1 with
2
≥
f ((t ,x ),...,(t ,x )) = h (x ,...,x ) for 0 < t < < t < 1.
n 1 1 n n n−1 1 n−1 1 n
···
The orthogonal projection onto M is denoted by Π : L M L .
2 2
→ ⊆
8 CHRISTEL GEISS,STEFAN GEISS, ANDEIJA LAUKKARINEN
M
Let us summarize some facts about the space :
(a) Representation of Π. For
∞
G = I (α ) L
n n 2
∈
n=0
X
with symmetric α Ln one computes the functions h of the projec-
n ∈ 2 n
tion F = Π(G) by
h (x ,...,x )
n−1 1 n−1
1 tn−1 t2
= n! ... α ((t ,x ),...,(t ,x ),(t ,x ))
n 1 1 n−1 n−1 n n
Z0 Z0 Z0 ZR
µ(dx )
n
(7) dt dt for n 1.
× µ(R) 1··· n ≥
(b) Integral representation of the elements of M. Given F M
∈
with a representation like in Definition 2.7 (the functions h are unique
n
as elements of L (µ⊗n)), we define the martingale ϕ = (ϕ ) by the
2 t t∈[0,1)
L -sum
2
∞
(8) ϕ := h + (n+1)I h 1⊗n ,
t 0 n n (0,t]
Xn=1 (cid:16) (cid:17)
which we will assume to be path-wise c`adl`ag. It follows that
∞
ϕ 2 = h2 + (n+1)2n!tn h 2
k tkL2 0 k nkL2(µ⊗n)
n=1
X
∞
1
= h2 + (n+1)2n!tn f 2
0 µ(R) k n+1kLn2+1
n=1
X
∞
1
= h2 + tn(n+1) I (f ) 2
0 µ(R) k n+1 n+1 kL2
n=1
X
so that
∞
(9) µ(R) sup ϕ 2 + F 2 = (n+1) I (f ) 2 .
k tkL2 k kL2 k n n kL2
t∈[0,1)
n=0
X
Moreover, for t [0,1] we get that, a.s.,
∈
F := E(F ) = ϕ dX .
t t s− s
|F
Z(0,t]
This is analog to the Brownian motion case considered in [7] and [9],
where the representation F = EF + ϕ dB was used together with
(0,1] s s
R
FRACTIONAL SMOOTHNESS, LE´VY PROCESSES AND APPROXIMATION 9
the regularity assumption that (ϕ ) is a martingale or close to a
s s∈[0,1)
martingale in some sense.
(c) Basic examples for elements for M are taken from Lemma
4.2 below: Let Π : L I(X) L be the orthogonal projection
X 2 2
onto I(X) defined in (4)→and let f⊆: R R be a Borel function with
→
f(X ) L , then
1 2
∈
Π (f(X )) = Π(f(X )).
X 1 1
M
This means the elements of occur naturally when applying the
Galtchouk-Kunita-Watanabe projection. It should be noted, that in
the case that σ = 0 and ν = αδ with α > 0 and x R 0 we have
a chaos decomposition of the forxm0 f(X ) = Ef(X )0+∈ ∞\{β}I (1⊗n )
1 1 n=1 n n (0,1]
with β R, so that already f(X ) M.
n ∈ 1 ∈ P
2.7. Dol´eans-Dade stochastic exponential.
Definition 2.8. For 0 a t 1 we let
≤ ≤ ≤
∞ I (1⊗n )
Sa := 1+ n (a,t] ,
t n!
n=1
X
where we can assume that all paths of (Sa) are c`adla`g for any
t t∈[a,1]
fixed a [0,1]. In particular, we let S = (S ) := (S0) .
∈ t t∈[0,1] t t∈[0,1]
The following lemma is standard and we omit its proof.
Lemma 2.9. For 0 a t 1 one has that
≤ ≤ ≤
(i) Sa = 1+ Sa dX a.s.,
t (a,t] u− u
(ii) S = SaS a.s.,
t t aR
(iii) Sa is independent from and E(Sa)2 = eµ(R)(t−a).
t Fa t
3. Approximation of stochastic integrals
In the sequel we will use
∞
:= τ = (t )N : 0 = t < < t = 1 and :=
TN { k k=0 0 ··· N } T TN
N=1
[
assets ofdeterministic time-nets anddefine τ := max t t .
1≤k≤N k k−1
| | | − |
We will consider the following approximations of a random variable
F M with respect to the processes X and S:
∈
10 CHRISTEL GEISS,STEFAN GEISS, ANDEIJA LAUKKARINEN
Definition 3.1. For N 1, Y X,S , F = ϕ dX M,
≥ ∈ { } (0,1] s− s ∈
A = (A )N and τ we let
k k=1 ⊆ F ∈ TN R
(i) asim(F;τ,A) := F N ϕ 1 (Stk−1 1) ,
S − k=1 tk−1 Ak tk − L2
(ii) aopt(F;τ) := inf(cid:13)(cid:13) F P N v (Y Y ) (cid:13)(cid:13), where the infi-
mYum is taken ov(cid:13)e(cid:13)r a−ll k=1-mke−a1surtkab−le vtk−1 (cid:13):L(cid:13)Ω2 R such that
(cid:13) PFtk−1 k−1(cid:13) →
E v (Y Y (cid:13)) 2 < . (cid:13)
| k−1 tk − tk−1 | ∞
Remark 3.2. (i) The definition of asim takes into account the addi-
S
tional sets (A )N to avoid problems with the case that S van-
k k=1
ishes. These extra sets A in asim(F;τ,A) play different roles in
S
Theorem 3.3, Theorem 3.4, and in Theorems 3.5, 3.6 and 3.8. To
recover a more standard form of asim assume that (S ) and
S t t∈[0,1]
(S ) are positive so that we can write
t− t∈[0,1]
ϕ
u
F = ψ (S dX ) with ψ :=
u− u− u u
S
Z(0,1] u
and obtain that
N N
F ϕ (Stk−1 1) = F ψ S (Stk−1 1)
− tk−1 tk − − tk−1 tk−1 tk −
k=1 k=1
X X
N
= F ψ (S S )
− tk−1 tk − tk−1
k=1
X
which is what one expects.
(ii) In the sequel the crucial assumption will be
Ω = S = 0 for all t [0,1].
t
{ 6 } ∈
This can be achieved by the condition ν(( , 1]) = 0 which
−∞ −
implies the almost sure positivity of S and we can adjust S on
a set of measure zero; see [13, Theorem I.4.61] and [16, Theorem
19.2].
Because of the martingale property of (ϕ ) it is easy to check that
t t∈[0,1)
N
aopt(F;τ) = F ϕ (X X )
X − tk−1 tk − tk−1
(cid:13) (cid:13)
(cid:13) Xk=1 (cid:13)L2
(cid:13) (cid:13)
so that for Y = X the s(cid:13)imple and optimal approxim(cid:13)ation coincide.
(cid:13) (cid:13)
The theorem below gives a description of the simple approximation by
a function H (t) that describes, in some sense, the curvature of F M
Y
∈
with respect to Y.