Wavelet compression of anisotropic integrodifferential operators on sparse tensor product spaces

DOI:10.1051/m2an/2009039 www.esaim-m2an.org

WAVELET COMPRESSION OF ANISOTROPIC INTEGRODIFFERENTIAL OPERATORS ON SPARSE TENSOR PRODUCT SPACES

^∗

Nils Reich

¹

Abstract. For a class of anisotropic integrodifferential operatorsBarising as semigroup generators of Markov processes, we present a sparse tensor product wavelet compression scheme for the Galerkin finite element discretization of the corresponding integrodifferential equationsBu=fon [0,1]ⁿwith possibly largen. Under certain conditions onB, the scheme is of essentially optimal and dimension independent complexityO(h⁻¹|logh|²⁽ⁿ⁻¹⁾) without corrupting the convergence or smoothness requirements of the original sparse tensor finite element scheme. If the conditions onB are not satisfied, the complexity can be bounded byO(h^−(1+ε)), where ε 1 tends to zero with increasing number of the wavelets’

vanishing moments. Herehdenotes the width of the corresponding finite element mesh. The operators under consideration are assumed to be of non-negative (anisotropic) order and admit a non-standard kernelκ(·,·) that can be singular on all secondary diagonals. Practical examples of such operators from Mathematical Finance are given and some numerical results are presented.

Mathematics Subject Classification. 47A20, 65F50, 65N12, 65Y20, 68Q25, 45K05, 65N30.

Received August 6, 2008.

Published online October 9, 2009.

1. Introduction

Sparse tensor product-based wavelet compression for integrodifferential equations (PIDEs) of the form

Bu=f on [0,1]ⁿ, (1.1)

was introduced in [39,46,47] for isotropic integrodifferential operatorsB with distributional Schwartz kernels κ(x, y) that are singular only on the diagonal in [0,1]ⁿ×[0,1]ⁿ. For such operators essentially dimension- independent and asymptotically optimal complexity results have been shown in [46,47].

The assumption thatBadmits a standard kernel however is not always satisfied in practice. For instance in Mathematical Finance the pricing of contracts on baskets of assets where the underlying is modeled by jump processes leads to equations of the form (1.1), where the integrodifferential operator B only admits a kernel that can be singular on all secondary diagonals of [0,1]ⁿ×[0,1]ⁿ and all singularities are of different order.

In this work we construct a sparse tensor product-based wavelet compression scheme for a wide class of such anisotropic operators.

Keywords and phrases. Wavelet compression, sparse grids, anisotropic integrodifferential operators, norm equivalences.

∗Supported in part by scholarships of the Zurich Graduate School in Mathematics (ZGSM) and the Huber Kudlich Foundation at ETH Zurich.

1 ETH Zurich, Seminar for Applied Mathematics, 8092 Zurich, Switzerland. reich@math.ethz.ch

Article published by EDP Sciences c EDP Sciences, SMAI 2009

Even though, due to the different singularity structures, the numerical analysis differs significantly (see [47], Sects. 3 and5below), from a numerical point of view the challenges arising for the discretization of anisotropic operators are essentially the same as for isotropic ones (cf.[47], Sect. 1): Galerkin discretization of integrodiffer- ential equations in general leads to linear systems with densely populated matrices of substantial size. Even on tensor product domains, the straightforward application of standard numerical schemes fails due to the “curse of dimension”: the number of degrees of freedom on a tensor product Finite Element (FE) mesh of widthhin dimensionngrows likeO(h⁻ⁿ) ash→0. The non-locality of the underlying operator thus implies that the FE stiffness matrix consists ofO(h⁻²ⁿ) non-zero entries.

Based on tensor products of univariate wavelet basis functions, in this paper we prove that the complexity of the stiffness matrix can however be reduced toO(h^−(1+ε)) with (for fixed dimension) arbitrarily small 0< ε1 and, under certain conditions, even to O(h⁻¹|logh|²⁽ⁿ⁻¹⁾) without corrupting the convergence of the original FE scheme. Our results are applicable not only to classical pseudodifferential operators but to a wide class of anisotropic integrodifferential operators. To this end, suitable symbol classes are introduced.

The anisotropic discretization technique presented in this work relies on the following two approaches (cf.[47]):

Sparse tensor product spaces as introduced in [7,27,28,44] are used to overcome the “curse of dimension”.

This approach yields essentially dimension independentO(h⁻¹|logh|ⁿ⁻¹) degrees of freedom ash→0 while at the same time (essentially) preserving the approximation rate. Discretizing integrodifferential equations on a sparse tensor product space thus yields matrices containing O(h⁻²|logh|²⁽ⁿ⁻¹⁾) entries. As shown in e.g.[28], these results require greater smoothness of the function to be approximated than the original discretization and this extra regularity increases with the dimensionn.

The non-locality of integral operators can be treated by so-calledwavelet compression. This methodology was introduced by [5] in the very different setting ofisotropic (orstandard) wavelet representation,i.e. the FE basis functions consist of tensor products of scaling functions and wavelets only on thesame level. It was shown that wavelet representation yields an almost sparse representation of certain operators. In [14,15,58,61] this approach was advanced further (on not necessarily tensor product domains) and given a rigorous mathematical foundation based on the requisite that the compressed system has to preserve the stability and convergence properties of the unperturbed discretization. In [51] it was shown that wavelet compression techniques may yield asymptotically optimal complexity (on not necessarily tensor product domains) in the sense that the number of non-zero entries in the resulting matrices grows linearly with the number of degrees of freedom. In contrast to sparse tensor product approximation, this methodology does not require additional smoothness of the approximated function. But, since the number of non-zero matrix entries grows linearly with the degrees of freedom, there still is exponential growth of the number of non-trivial matrix entries as the dimensionntends to infinity. The results on isotropic wavelet compression have been unified in a sophisticated way in [17]. Since it somewhat presents a finalization of the isotropic wavelet compression, we refer to [17] for further details.

Note that, with a slightly different approach but based on analogous principles similar complexity results for the isotropic setting have been presented in [55]. In summary, substitutingh= 2^−J, one finds

• Discretization by sparse tensor product spaces yields O(2^2JJ²⁽ⁿ⁻¹⁾) non-zero entries in the system matrix.

• Wavelet compression of generalfull tensor product spaces yieldsO(2^nJ) non-zero matrix entries.

The following diagram illustrates the connection between the two approaches and our new results:

O(2^2nJ)

[17]et al. isotropic operators

[7]et al.

isotropic operators

//O(2^2JJ²⁽ⁿ⁻¹⁾)

an-/isotropic operators

O(2^nJ)

an-/isotropic operators

//

_ _ _ _ _ _ _

_ O(2^JJ²⁽ⁿ⁻¹⁾)

We shall use the notion of computational “complexity” exclusively to indicate the number of non-zero entries in a given system matrix. With an efficient implementation and quadrature as in e.g. [30] it can be shown that the overall cost of computing and assembling the system matrix is essentially of the same magnitude as its complexity.

As in [47], the complexity results of this work also imply that, under certain conditions, the stiffness matrices of the anisotropic non-local operators under consideration are s^∗-compressible in the sense of [10,25,52]. This shows that, in order to solve the corresponding integrodifferential equations one may employ adaptive wavelet algorithms as in [9,10,26] that converge with the rate of best approximation by an arbitrary linear combination ofN wavelets (so-called bestN-term approximation,cf.[22]).

The outline of this work is as follows:

In Section2 the abstract set-up is presented and notation is fixed.

Section3 provides the main motivation. We briefly introduce finite element asset pricing methods that lead to the abstract anisotropic integrodifferential equations under consideration.

Based on the considerations of Section3, in Section4a new class of anisotropic operator symbols is defined and examples are provided. For the corresponding class of anisotropic operators we then construct the sparse tensor wavelet compression scheme as follows:

In Section 5 fundamental estimates for the entries in the sparse tensor product-based stiffness matrix are derived.

Section6provides the consistency requirements that need to be satisfied by the compression scheme in order to preserve the stability and convergence properties of the sparse tensor product setting without compression.

Based on the two previous sections, in Section 7 the actual compression schemes are defined. Consistency with the sparse tensor product setting without compression is proved.

In Section 8 we provide complexity results for the constructed compression schemes. Based on [46], we show that under certain conditions the complexity of the sparse tensor product setting can be reduced to O

2^JJ²⁽ⁿ⁻¹⁾

non-zero matrix entries provided that the number of vanishing moments is sufficiently large. If these conditions are not satisfied, the complexity can be bounded by O

2^(1+ε)J

, whereε < 1 tends to zero with increasing number of vanishing moments.

Some numerical experiments are presented in Section 9. They coincide with the analytic predictions.

Finally, in Section 10we briefly summarize the main results of this work in order to provide a clear under- standing of the uses and limitations of the presented methods.

2. Galerkin discretization of multidimensional PIDEs

The compression scheme and numerical analysis we present in this work is based on the following generic set-up (cf.[46,47]): On [0,1]ⁿ=:, we consider an integrodifferential equation

Bu=f, (2.1)

with an integrodifferential operator

B=AD+A, (2.2)

whereAD denotes a (possibly vanishing) differential operator ADu=−1

2 n i,j=1

Qij

∂²u

∂xi∂xj

, Qij ∈R, i, j= 1, . . . , n, (2.3)

andAis an integral operator of possibly anisotropic orderα∈Rⁿ,i.e. A:H^α/2()→H^−α/2() continuously.

Here, the Sobolev spacesH^α/2() and its dualH^−α/2() are defined as follows: for u∈C₀^∞(), define ¯uto be the zero extension ofuto all of Rⁿ. Then, fors∈Rⁿ, the spaceH^s() is given by

H^s() :={¯u|u∈C₀^∞()}, (2.4)

where the closure is taken with respect to the norm of the anisotropic Sobolev space H^s(Rⁿ) :=

f ∈ S(Rⁿ) : ⁿ

i=1

(1 +ξ²_i)^si/2f L2(Rⁿ)

<∞

,

where fdenotes the Fourier transform of f ∈ S(Rⁿ). We assume that the operator A admits a kernel representation,

Au(x) =

κ(x, y)u(y)dy, (2.5)

with a distributional kernel functionκ(·,·) that is smooth outside the secondary diagonalsS⊂[0,1]ⁿ×[0,1]ⁿ, i.e.

S ={(x, y)∈[0,1]ⁿ×[0,1]ⁿ : xi=yi, for some i∈ {1, . . . , n}} · The best known example of such integral operators are:

Example 2.1(isotropic operators). Any classical pseudodifferential operatorA:H^q(Rⁿ)→H^−q(Rⁿ) of order 2q∈Rwith symbol in the H¨ormander classS₁²_,^q₀ in the sense of [35,56] admits a distributional kernel function κ(·,·) as in (2.5).

In this case, by the Schwartz kernel theorem (cf. e.g.[54], Sect. VI.7), the function κ(·,·) is singular only on the diagonal in [0,1]ⁿ×[0,1]ⁿand for any σ, σ∈ {0,1,2, . . .}ⁿ there holds

∂x^σ∂y^σκ(x, y)≤cσ,σ|x−y|^{−(n+2q+|σ|+|σ|)}, for allx, y ∈[0,1]ⁿ, (2.6) with some constant cσ,σ independent of x, y ∈ [0,1]ⁿ. A sparse tensor product-based compression scheme especially for such, so-called isotropic, operators has been constructed in [47] and [46], Chapter 2. The more general compression scheme of this work is, of course, also applicable. Throughout, we refer to estimates of the form (2.6) as Calder´on-Zygmund (CZ) estimates.

Denoting byQ= (Qij)_1≤i,j≤n the coefficient matrix of the differential operatorADin (2.3) we shall assume that either Q= 0 or Q>0. The order multiindex α ∈Rⁿ of the integrodifferential operator B=AD+Ais then given by

α=

(2, . . . ,2), ifQ>0 and max{α1, . . . , αn} ≤2,

α, otherwise. (2.7)

For the numerical solution of (2.1), we employ the Galerkin method with respect to a hierarchy of conforming trial spacesVJ ⊂VJ+1⊂H^{α/ 2}(). The variational problem of interest reads: find uJ ∈VJ such that,

BuJ, vJ =f, vJ for all vJ∈VJ. (2.8)

The indexJ represents the meshwidth of order 2^−J. We shall make the following assumptions on the operatorB to ensure that the variational problem (2.8) is well posed – for details we refer toe.g.[53], Proposition III.2.3.

(1) Bsatisfies a G˚arding inequality,i.e. there exist constantsγ >0,C≥0 such that

Bu, u ≥γu²Hα/2 ()−Cu²L²(), for allu∈H^{α/ 2}(). (2.9) (2) B : H^{α/ 2}() → H^−α/2() is continuous, i.e. there exists a constant C > 0 such that for all u, v ∈

H^α/2() there holds

|Bu, v | ≤CuHα/2 ()vHα/2 (). (2.10)

The nested trial spacesVJ⊂VJ+1 we employ in (2.8) shall be sparse tensor product spaces based on a wavelet multiresolution analysis described in the next sections.

To simplify notation, we denote

α:=|α|∞= max{α1, . . . , αn}, for anyα∈Rⁿ.

We shall frequently writeabto express thatais bounded by a constant multiple ofb, uniformly with respect to all parameters on which aand b may depend. Then a∼b means ab and b a. Also, we shall denote N0:=N∪ {0}={0,1,2, . . .}.

2.1. Wavelets on the interval

On [0,1] we shall use the same scaling functions and wavelets as described in [17] based on the construction of [8,16,43] and the references therein.

The trial spacesVj are spanned by single-scale bases Φj ={φj,k:k∈Δj}, where Δj denote suitable index sets. The approximation order of the trial spaces we denote by d,i.e.

d= sup

s∈R: sup

j≥0

infv_j∈Vjv−vj₀ 2^−jsvs

<∞, ∀v∈H^s([0,1])

· (2.11)

Using the single-scale bases constructed in [8] based on B-splines adapted to the interval [0,1] as described in [16], we assume that for each j ≥0, the basis functions φj,k ∈ Φj have compact supports and admit two important properties: φj,k_L2([0,1])= 1 and|suppφj,k| ∼2^−j.

Associated to these primal bases aredual bases Φj ={φj,k :k∈ Δj}, i.e. there holdsφj,k,φj,k =δk,k. Bydwe denote the order ofΦj and assumed≤dfor the remainder of this work. In particular, for B-splines of orderdand duals of orderd≥dsuch thatd+dis even the bases Φj,Φj as in [16] have approximation ordersd andd.

To these single-scale bases there exist biorthogonal complement or wavelet bases Ψj = {ψj,k : k ∈ ∇j}, Ψj ={ψj,k :k∈ ∇j}, where∇j := Δj+1\Δj. Inherited fromφj,k, theψj,k have compact supports and there holds

|suppψj,k| ∼2^−j. (2.12)

The dual pair of wavelet bases Ψ,Ψ is defined by Ψ =

j≥0Ψj,Ψ =

j≥0Ψj, with Ψ₀:= Φ₁,Ψ₀:=Φ₁. There holds

ψj,kL²([0,1])∼1, for allψj,k∈Ψ.

From the biorthogonality of Ψ and Ψ one infers the so-called cancellation property of Ψ (seee.g. [6]),i.e.

|ψj,k, f |2^−j(d+1/2)|f|_Wd,∞ (suppψ_j,k), for eachψj,k∈Ψ. (2.13)

Here|f|_Wd,∞ (Ω):= sup_x∈Ω|∂^df(x)|. Themother wavelet of Ψ we denote byψ,i.e. for any j andk∈ ∇j, ψj,k(x) = 2^j/2ψ(2^jx−k), x∈[0,1]. (2.14) Denoting by Wj,Wj the span of Ψj,Ψj, there holds

Vj+1=Wj+1⊕ Vj, and Vj+1=Wj+1⊕Vj, for allj ≥0, (2.15) and,

Vj=W0⊕. . .⊕ Wj, for allj≥0. (2.16)

Crucial for the consistency of our compression scheme is the fact that the wavelets on [0,1] satisfy the following norm estimates (cf. e.g.[13,16], for the one-sided estimates we refer to [61]):

For an arbitraryu∈H^t([0,1]), 0≤t≤d, with wavelet decomposition u=

∞ j=0

k∈∇j

uj,kψj,k= ∞ j=0

k∈∇j

u,ψj,k ψj,k,

there holds the norm equivalence,

(j,k)

2^2tj|uj,k|²∼ u²H^t([0,1]), if 0≤t < d−1/2, (2.17)

or the one-sided estimate,

(j,k)

2^2tj|uj,k|²u²H^t([0,1]), ifd−1/2≤t < d. (2.18) In caset=dthere only holds,

(j,k) j≤J

2^2tj|uj,k|²Ju²H^t([0,1]), ift=d. (2.19)

We conclude this section by an explicit example of wavelets on [0,1] with approximation order d= 2:

Example 2.2. The wavelets comprise of piecewise linear continuous functions on [0,1] vanishing at the end- points. The mesh for levelj ≥0 is defined by the nodesxj,k :=k2^−(j+1) withk∈ ∇j :={0, . . . ,2^j+1}. There holdsNj := dimVj = 2^j+1−1 and thereforeMj:= dimVj−dimVj−1= 2^j.

On level j= 0 we have N₀=M₀= 1 andψ₀,1is defined as the piecewise linear function with valuec₀>0 atx₀,1= ¹₂ and 0 at the endpoints 0, 1.

Forj >0 we firstly definecj:= 2^j/2. Then the waveletψj,1is defined as the piecewise linear function such that ψj,1(xj,1) = 2cj, ψj,1(xj,2) = −cj and ψj,1(xj,s) = 0 for all other s= 1,2. Similarly, the waveletψj,M_j

takes the values ψj,M_j(xj,N_j) = 2cj, ψj,M_j(xj,N_j−1) =−cj and zero at all other nodes. For 1< k < Mj the wavelet ψj,k is defined byψj,k(xj,2k−2) =−cj, ψj,k(xj,2k−1) = 2cj, ψj,k(xj,2k) =−cj andψj,k(xj,s) = 0 for all others= 2k−2,2k−1,2k.

Remark 2.3. Note that there is a strong link between the order of the operator, the approximation order of the multiresolution analysis and the number of vanishing moments of the wavelets which already restricts the possible choice of wavelet bases. In fact, the analysis of the so-called second compression we adapt from [17,51]

refers exclusively to biorthogonal spline wavelets whose singular supports are well defined and not dense in the wavelets’ supports. We refer to [29] for more specific illustrations.

2.2. Sparse tensor product spaces

Forx= (x₁, . . . , xn)∈[0,1]ⁿ, we denote,

ψ_j,k(x) :=ψj₁,k₁⊗. . .⊗ψj_n,k_n(x₁, . . . , xn) =ψj₁,k₁(x₁). . . ψj_n,k_n(xn).

Using Fubini’s theorem one infers that the scaling and cancellation properties (2.12), (2.13) of the univariate wavelets carry forward to their tensor products. In particular,

|suppψ_j,k|= n i=1

|suppψj_i,k_i| ∼2^{−(j1+...+jn)},

and eachψ_j,k hasdvanishing moments which implies the cancellation property

|v, ψj,k |2^−12|j|12^{−dmax{j1,...,jn}}|v|_Wd,∞ (suppψ_j,k). (2.20) On= [0,1]ⁿ, we define the subspaceVJ ⊂H^α/2() as the (full) tensor product of the spaces defined on [0,1]

VJ :=

n i=1

VJ, (2.21)

which can be written using (2.16) as

VJ = span{ψ_j,k : ki∈ ∇j_i,0≤ji≤J, i= 1, . . . , n}

=

J j₁,...,j_n=0

Wj₁⊗. . .⊗ Wj_n.

We define the regularityγ >|α|_∞/2 of the trial spaces by

γ= sup{s∈R : VJ ⊂H^s()} · (2.22)

It is known that based on the spline wavelets constructed in Example2.2the regularity index satisfiesγ=d−1/2.

The sparse tensor product spacesVJ are defined by

VJ := span{ψ_j,k : ki∈ ∇j_i, i= 1, . . . , n; 0≤ |j|₁≤J} =

0≤|j|1≤J

Wj₁ ⊗. . .⊗ Wj_n. (2.23)

One readily infers thatNJ := dim(VJ) =O(2^nJ) whereasNJ := dim(VJ) =O(2^JJⁿ⁻¹) as J tends to infinity.

However, both spaces have similar approximation properties in terms of the Finite Element meshwidthh= 2^−J, provided the function to be approximated is sufficiently smooth. To characterize the necessary extra smoothness we introduce the spacesH^s([0,1]ⁿ),s∈Nⁿ₀, of all measurable functionsu: [0,1]ⁿ→R, such that the norm,

u_H^s₍₎:=

0≤α_i≤s_i, i=1,...,n

∂^α₁¹. . . ∂_n^αnu²L²()

₁/2

,

is finite. That is

H^s([0,1]ⁿ) = n

i=1

H^si([0,1]). (2.24)

For arbitrarys∈Rⁿ_≥0, we defineH^sby interpolation. Because of the underlying tensor product structure (2.24), one infers from (2.17)–(2.19) that for

u=

(j,k)

u_j,kψ_j,k=

(j,k)

u_j,kψj₁k₁⊗. . .⊗ψj_nk_n,

there holds the norm equivalence

(j,k)

2^{2s1j1+...+2snjn}|u_j,k|²∼ u²_Hs, if 0≤si< d−1/2 for alli, (2.25)

and the one-sided bounds

(j,k)

2^{2s1j1+...+2snjn}|u_j,k|² u²_Hs, if 0≤si< dfor alli, (2.26)

k

2^{2s1j1+...+2snjn}|uj,k|² u²_Hs, ifsi=dfor some i. (2.27) Note that a slightly refined version of the one-sided bounds (2.26), (2.27) can be found in [48], proof of Theo- rem. 5.1.

By (2.21), one may decompose anyu∈L²() into

u(x) =

j_i≥0 i=1,...,n

k_i∈∇_ji

u_j,kψ_j,k(x) =

j_i≥0 i=1,...,n

k_i∈∇_ji

u_j,kψj₁,k₁(x₁). . . ψj_n,k_n(xn).

In this style, the sparse grid projectionPJ :L²()→VJ is defined by truncation of the wavelet expansion:

(PJu)(x) :=

0≤|j|1≤J

k∈∇j

u_j,kψ_j,k(x), (2.28)

where∇_j=∇₍j₁,...,j_n):=∇j₁×. . .× ∇j_n.

2.3. Approximation rates for anisotropic operators on sparse tensor product spaces

For the proof of the following convergence and stability results, we refer to [48] and [28,60], respectively.

By [60], Proposition 3.1, and [48], Theorem 5.1, the sparse tensor product projectionPJ in (2.28) satisfies:

Lemma 2.4. Suppose for each i = 1, . . . , n there holds 0 ≤ αi/2 < γ, with γ given by (2.22), then for u∈H^α/2()there holds:

(1) Stability of PJ:

PJuH^α/2()uH^α/2(). (2.29)

(2) Approximation property of PJ: let ^α₂ⁱ ≤ti ≤d, i= 1, . . . , n, with dgiven by (2.11). For u∈ H^t() there holds

u−PJuH^α/2()

⎧⎪

⎨

⎪⎩

2^(α2−t)Ju_H^t₍₎ if

α= 0 or ti=dfor alli, 2^(α2−t)JJⁿ⁻¹² u_H^t₍₎ otherwise,

(2.30)

where we denotet= (t₁, . . . , tn)and(^α₂ −t) = max{^α₂¹ −t₁, . . . ,^α₂ⁿ−tn}. Thus, there holds:

Proposition 2.5. Let α, as defined in (2.7), denote the order of the integrodifferential operator B in (2.2).

Then the sparse tensor product spaces VJ in (2.23)based on the wavelets introduced in Section 2.1satisfy:

(1) The Galerkin discretization of (2.1) based on sparse tensor product spaces VJ as defined in (2.23) is stable,i.e. there existJ₀>0 andc₁>0,c₂≥0 such that for anyJ≥J₀ there holds

| BvJ, vJ | ≥c₁vJ²_Hα/2 ()−c₂vJ²L2(), for allvJ ∈VJ, (2.31)

and there exists somec₃>0such that for all J ≥J₀,

|BvJ, wJ | ≤c₃vJH^α/2 ()wJH^α/2 (), for allvJ, wJ ∈VJ. (2.32) In particular, the variational problem (2.8)admits a unique solution.

(2) Let u and uJ denote the solutions of the original equation (2.1) and the variational problem (2.8), respectively. The best convergence of the sparse tensor product Galerkin scheme is determined by

u−uJH^α/2 ()2^{−(d−|α|∞/2−ν)J}u_H^ρ₍₎, (2.33) provided u∈ H^ρ(). The anisotropic smoothness parameterρ∈Rⁿ>0 is given by

ρi =d− |α|∞

2 −αi

2

, (2.34)

for each i= 1, . . . , n, and furthermore

ν=

⎧⎨

⎩

(n−1)d

nd−1 , ifα= (0, . . . ,0) and henceρ= (d, . . . , d),

0, otherwise.

Proof. The stability estimates (2.31), (2.32) are obtained exactly as in [47], Proposition 2.1. The convergence rate (2.33) in the sparse tensor product setting is given by [48], Proposition 5.2.

Having set up the general numerical basis of our approach, in the next section we briefly describe our main applications.

3. Motivation: Pricing of financial derivatives in models with jumps

Even though the results of the present work can be applied to a wide range of non-local operators, our main motivation arises from Mathematical Finance. In this section we illustrate how high-dimensional equations of the form (2.1) naturally occur in this field.

3.1. Pricing equations

Consider arbitrage-free valuesu(x, T) of contingent claims on baskets ofs∈Nassets. The log-returns of the underlying assets are modeled by a L´evy or, more generally, a Feller processX with state spaceRⁿ,s≤n, and X₀=x. For example, the compression techniques constructed in this work can be applied when X is a L´evy copula process (thenn=s≥2,cf.[24]) or the price process of a Barndorff-Nielsen-Shephard (BNS) stochastic volatility model (then s= 1,n= 2,cf.[3]).

By the fundamental theorem of asset pricing (seee.g.[18]), an arbitrage free priceuof an European contingent claim with payoffg(·) is given by the conditional expectation

u(x, t) =E(g(Xt)|X₀=x),

under ana priori chosen risk-neutral martingale measure equivalent to the historical measure (see e.g.[19,20]

for measure selection criteria).

Deterministic methods to computeu(x, T) are based on the solution of the corresponding backward Kolmogorov equation (for the derivation seee.g.[21], [38], Sect. 7.3, as well as [24,41,49])

ut+Bu= 0, u|t=T =g. (3.1)

Here B denotes the infinitesimal generator of X with domain D(B). For the Galerkin-based Finite Element implementation, equation (3.1) is converted into variational form as illustrated in Section 2. Formally, the resulting problem reads: findusuch that

∂

∂tu, v

+Bu, v

E(u,v)

= 0, for allv∈ D(B). (3.2)

In the classical setting of Black-Scholes, X is a geometric Brownian Motion and B is a diffusion operator so that a closed form solution of (3.1) and (3.2) for plain vanilla contracts is possible in certain cases. For more general L´evy or Feller price processesX,Bis in general a pseudodifferential operator with symbolψ^X, i.e.

(Bu)(x) = (ψ^X(x, D)u)(x) =−

Rⁿe^{i ξ,x}ψ^X(x, ξ)u(ξ)dξ. (3.3) IfX is a pure jump process and henceB=Ais an integral operator, one obtains the kernel representation (2.5) ofA=ψ^X(x, D) by writing for any u∈ S(Rⁿ),

Au(x) = −

Rⁿ Rⁿe^{i x−y,ξ}ψ^X(x, ξ)u(y)dydξ.

Thus, the kernelκ(·,·) in (2.5) can be represented as κ(x, y) :=

Rⁿe^{i x−y,ξ}ψ^X(x, ξ)dξ, (3.4)

the inverse Fourier transform (in the sense of oscillatory integrals, seee.g.[36], Eq. (18.1.7)) ofψ^X atx−y.

Remark 3.1. IfX is a pure jump L´evy process with absolutely continuous L´evy measure then the following relation holds betweenκ(·,·) and the densityk(·) of the L´evy measure ofX:

κ(x, y) =

Rⁿ\{0} Rⁿ

e^{i x−y−z,ξ}−e^{i x−y,ξ}+ iz, ξ

1 +|z|²e^{i x−y,ξ}

k(z)dξdz, (3.5)

in the sense of distributions. By [34], Lemma 2.8, for anyξ∈Rⁿ,z∈Rⁿ there holds e^{−i z,ξ}−1 + iz, ξ

1 +|z|²

≤7· |z|²

1 +|z|²·(1 +|ξ|²).

Hence, the distributional kernelκ(·,·) in (3.5) is indeed well defined, sincek(·) is a L´evy kernel that satisfies

Rⁿ(|z|²∧1)k(z)dz <∞.

For an extensive description of L´evy processes we refer to the monographs [4,50].

For the numerical solution of the variational problem (3.2) we employ variational Galerkin methods developed in [1,2,12,24,31,40–42,46]. We conclude this introductory part by illustrating this approach in case the underlying processX is a L´evy process (for a more general survey we refer to [32,33]).

3.2. The Finite Element Method for option pricing in multidimensional L´ evy models

The considerations of this section are based on [24,49]. Suppose X is a L´evy process with state space Rⁿ and characteristic exponent

ψ^X(ξ) =−iγ, ξ +1

2ξ,Qξ +

Rⁿ\{0}

1−e^{i ξ,y}+ iξ, y 1 +|y|²

ν(dy),

whereγ∈Rⁿ is the drift vector,Q ∈R^n×n is the covariance matrix andν(dy) is the L´evy measure ofX. Assume the risk-neutral dynamics of s=n >1 assets are given by

Stⁱ=S₀ⁱe^rt+Xit, i= 1, . . . , n,

under a risk-neutral measure such that e^Xi is a martingale with respect to the canonical filtration Ft⁰ :=

σ(Xs, s≤t),t≥0, of the multivariate processX.

Consider an European option with maturityT <∞and payoff g(S) which is assumed to be Lipschitz. The valueV(t, St) of this option is given by

V(t, S) =E

e^−r(T−t)g(ST)|St=S

, (3.6)

and, sufficient smoothness provided, it can be computed as the solution of a partial integrodifferential equation.

Theorem 3.2. Assume that V(t, S)in(3.6)satisfies

V(t, S)∈C^1,2((0, T)×Rⁿ>0)∩C⁰

[0, T]×Rⁿ_≥0 . Then V(t, S)is the solution of the following PIDE:

∂V

∂t(t, S) +1 2

n i,j=1

SiSjQij

∂²V

∂Si∂Sj

(t, S) +r n i=1

Si

∂V

∂Si

(t, S)−rV(t, S) (3.7)

+

Rⁿ

V(t, Se^z)−V(t, S)− n i=1

Si(e^zi−1) ∂V

∂Si

(t, S) ν(dz) = 0, in (0, T)×Rⁿ_≥0 where V(t, Se^z) :=V(t, S₁e^z1, . . . , Sne^zn), and the terminal condition is given by

V(T, S) =g(S) ∀S ∈Rⁿ_≥0. (3.8)

Proof. [49], Theorem 4.2.

If the marginal L´evy measuresνi,i= 1, . . . , n, ofν are absolutely continuous and admit densities νi(dz) = ki(z)dz with constantsGi>0,Mi>0,i= 1, . . . , n, such that

ki(z)

e^Giz, for allz <−1,

e^−Miz, for allz >1, (3.9)

the PIDE (3.7) can be transformed into a simpler form.

Corollary 3.3. Suppose the marginal L´evy measures νi, i = 1, . . . , n, satisfy (3.9) with Mi > 1, Gi > 0, i= 1, . . . , n. Furthermore, let

u(τ, x) = e^rτV

T−τ,e^{x1+(γ1−r)τ}, . . . ,e^{xn+(γn−r)τ}

,

where

γi=Qii

2 +

R(e^zi−1−zi)νi(dzi).

Then, usatisfies the PIDE

∂u

∂τ +AD[u] +A[u] = 0, (3.10)

in (0, T)×Rⁿ with initial conditionu(0, x) :=u₀. The differential operator is defined for ϕ∈C₀²(Rⁿ)by AD[ϕ] =−1

2 n i,j=1

Qij

∂²ϕ

∂xi∂xj

, (3.11)

and the integrodifferential operator by

A[ϕ] =−

Rⁿ

ϕ(x+z)−ϕ(x)− n i=1

zi

∂ϕ

∂xi

(x) ν(dz). (3.12)

The initial condition is given by

u₀=g(e^x) :=g(e^x1, . . . ,e^xn). (3.13)

Proof. [49], Corollary 4.3.

For anyu, v∈C₀^∞(Rⁿ) we associate with the diffusion partADthe bilinear form ED(u, v) = 1

2 n i,j=1

Qij Rⁿ

∂u

∂xi

∂v

∂xj

dx.

To the jump partA we associate the so-called canonical bilinear jump form EJ(u, v) =−

Rⁿ Rⁿ

u(x+z)−u(x)− n

i=1

zi

∂u

∂xi

(x) v(x)dx ν(dz),

and set

E(u, v) =ED(u, v) +EJ(u, v). (3.14)

Herewith, we can now formulate the formal parabolic problem (3.2) rigorously:

Find u∈L²((0, T);D(E))∩H¹((0, T);D(E)^∗) such that ∂u

∂τ

v_D(E)∗,D(E)+E(u, v) = 0, τ ∈(0, T), ∀v∈ D(E), (3.15) u(0) =u₀.

HereD(E) denotes the domain of the bilinear formE(·,·) ofX. The well-posedness of (3.15) has been analyzed in [24,49] where it also has been shown thatD(E) can explicitly be characterized in terms of anisotropic Sobolev spaces.

For implementation, the variational problem (3.15) needs to be localized to a bounded domain, discretized in space, and a time stepping scheme has to be applied. More precisely, these three steps are accomplished as follows:

1. Localization. For the localization we find that in Finance truncation of the original x-domain Rⁿ to ΩR:= [−R, R]ⁿ,R >0, corresponds to approximating the solutionuof (3.7) by the priceuRof a barrier option on ΩR. In log-priceuR is given by

uR(t, x) =E

g(e^XT)1_{T <τ_Ω

R,t}|Xt=x

, (3.16)

whereτ_ΩR,t= inf{s≥t|Xs∈/ΩR}denotes the first exit time ofXtfrom ΩR after timet. In case of semiheavy tails (3.9), the solution of the localized problem uR converges pointwise exponentially to the solutionuof the original problem.

Theorem 3.4. Suppose the payoff function g:Rⁿ→Rsatisfies g(S)

n i=1

Si+ 1, ∀S∈Rⁿ_≥0,

and the marginal measures νi satisfy (3.9) with Mi > 1, Gi > 1, i = 1, . . . , n. Then, there exist constants α, β >0such that

|u(t, x)−uR(t, x)|e^{−αR+βx∞}.

Proof. [49], Theorem 4.15.

Remark 3.5. Theorem3.4yields that as long as the underlying L´evy measure admits semiheavy tails and the payoff function g is not growing stronger than linearly, the option priceu as given (3.7) can be approximated by the priceuRof a corresponding down-and-out barrier option with “active domain” ΩR. The definition ofuR

is given in (3.16).

Heuristically speaking, as long as the price of the underlying remains within the active domain ΩR of uR

the values of the two options coincide. Clearly, as one increases the size of ΩR the difference between the two option values has to decrease. Due to the semiheavy tails of the underlying L´evy measure (i.e. the marginal densities decay exponentially), this difference decreases exponentially with increasing size of ΩR.

SinceuR converges exponentially to the desired solutionuof (3.7), from now on we shall assume thatR >0 and hence ΩRis chosen sufficiently large so that it suffices to calculateuRinstead ofu. SinceuR is the price of a down-and-out barrier option one is left with a Dirichlet problem (cf. e.g. [11], Sect. 12.1.2): more precisely, for any functionuwith support in ΩR we denote byuits extension by zero to the whole ofRⁿ and define

ER(u, v) =E(u, v), with

D(ER) ={u|u∈C₀^∞(ΩR)},

where the closure is taken with respect to the natural norm · _E ofD(E),i.e. u²_E =E(u, u) +u²_L2. Thus, we can restate the variational form (3.15) on the bounded domain and the existence and uniqueness results for (3.15) remain valid.

Find uR∈L²((0, T);D(ER))∩H¹((0, T);D(ER)^∗) such that ∂uR

∂τ , v

+ER(uR, v) = 0, ∀τ ∈(0, T),∀v∈ D(ER), (3.17) uR(0) =u₀|ΩR.