Table Of ContentCUBATURE METHODS FOR STOCHASTIC (PARTIAL)
DIFFERENTIAL EQUATIONS IN WEIGHTED SPACES
PHILIPPDO¨RSEK,JOSEFTEICHMANN,ANDDEJANVELUSˇCˇEK
Abstract. The cubature on Wiener space method, a high-order weak ap-
proximationscheme,isestablishedforSPDEsinthecaseofunbounded char-
2
acteristics and unbounded payoffs. We first introduce a recently described
1
flexiblefunctionalanalyticframework,socalledweightedspaces,whereFeller-
0
like properties hold. A refined analysis of vector fields on weighted spaces
2
thenyieldsoptimalconvergenceratesofcubaturemethodsforstochasticpar-
n tialdifferentialequations ofDaPrato-Zabczyktype. Theubiquitousstability
a for the local approximation operator withinthe functional analytic setting is
J proved forSPDEs,however, inthe infinitedimensional case weneed anewly
9 introduced assumption on weak symmetry of the cubature formula. In finite
1 dimensions,weusetheUFGconditiontoobtainoptimalratesofconvergence
onnon-uniformmeshesfornonsmoothpayoffswithexponential growth.
]
R
P
h. 1. Introduction
t
a CubatureonWienerspace,arealizationofthe abstractKLVhighordermethod
m
after Shigeo Kusuoka [21], Terry Lyons and Nicolas Victoir [25], is a weak ap-
[ proximation scheme for stochastic differential equations. Significant advantages
in comparison to other weak approximation schemes such as Taylor methods, see
1
v [19], are that it respects the geometry of the problem, and that at least theoretic-
4 ally, it is possible to reach arbitrarily high rates of convergence without requiring
2 the calculation of higher derivatives, see [25, Theorem 2.4, Proposition 2.5]. The
0 concrete construction of such cubature paths of high order is still quite difficult,
4
see [16] for paths up to order 11 for a single driving Brownian motion. Cubature
.
1 schemes provide a time-discretization approximating the unknown expected value
0
of a functional of the solution process of the SPDE by an expectation of an iter-
2
atively constructed function on a high-dimensional discrete product space. Often
1
: a direct evaluation of the functional on the discrete probability space is too ex-
v
pensive,therefore,severalmethods tospeed-upthe evaluationofcubature schemes
i
X such as recombination [23, 31] or tree-based branching [6] have emerged. Other-
r wise the functionals have to be evaluated with Monte Carlo or Quasi Monte Carlo
a
algorithms on the discrete product probability space.
High-order weak approximation schemes provide interesting lower complexity
alternatives to standard multi-level Monte Carlo schemes if Quasi Monte Carlo
algorithmsordeterministicalgorithmscanbeappliedfortheevaluationofthecon-
structed functionals. Indeed, multi-level Monte Carlo schemes lead to complexity
estimates of order (almost) (ǫ−2), i.e. to reach accuracy ǫ a number of opera-
O
tions of order ǫ−2 is necessary. In contrast, QMC evaluations of weak, high-order
approximation schemes of order k lead to complexity estimates of order (almost)
(ǫ−1−1/k), as long as the QMC integration yields optimal convergence (this in
O
turn also depends on the dimension of integration space, which is moderate for
2000 Mathematics Subject Classification. Primary60H15, 65C35;Secondary46N30.
Key words and phrases. Cubature on Wiener space, stochastic partial differential equations,
highorderweakapproximationscheme.
1
2 PHILIPPDO¨RSEK,JOSEFTEICHMANN,ANDDEJANVELUSˇCˇEK
high order methods). Hence we believe that it is worth analyzing in depth the
functional analytic framework of cubature schemes, i.e. we aim for constructing a
flexible enough pool of Banach spaces of payoffs and Banach spaces of character-
istics, where relevant problems from practice can be embedded.
In this work, we shall relax the regularity assumptions of the cubature method,
similarly as was done in [12, 10, 13] for the splitting approach of Syoiti Ninomiya
and Nicolas Victoir [27]. Consider a stochastic differential equation on Rn in its
Stratonovich form,
d
(1.1) dXx = V (Xx) dBj.
t j t ◦ t
i=0
X
All initial work was based on the fundamental assumption that the vector fields
V : Rn Rn are bounded and C∞-bounded. This is also a typical assumption
j
→
in other approximation methods for stochastic differential equations, e.g., in [33].
Some success in relaxing these assumptions, which are actually rarely satisfied in
practicalproblemswasachieved,atleastforapproximationsofthesplittingtype,in
works by Tanaka and Kohatsu-Higa[34] and Alfonsi [1]. While in the first one the
focus was on extensions to L´evy driving noise and in the second to CIR processes,
in both approaches it was recognized that polynomially bounded payoffs are the
correct context for problems with Lipschitz continuous vector fields.
Another approach was suggested in [12]. There, splitting schemes were ana-
lyzed on general weighted spaces, allowing in particular the approximation of Da
Prato-Zabczykstochasticpartialdifferentialequations wherethe driftpart,the in-
finitesimalgeneratorofastronglycontinuoussemigroupontheinfinitedimensional
state space, is not even continuous.
All these approaches profited from the special structure of splitting schemes, as
there the stability or power boundedness of the discrete approximation operator
can be shown by investigating every part separately. Instead, we follow a similar
idea as was applied to the stochastic Navier-Stokes equations in [11]. We extend
the results of [3] to more general coefficients and payoffs. This allows us to obtain
methodsoforderhigherthan2withouthavingtoresorttoextrapolation,see[4,28].
While the use ofthe weightedspaces from[30, 12] is also mandatoryhere, we shall
provide a refined analysis of the vector fields defined on these spaces. This will
allow us to do a Taylor expansion of the cubature approximations to compute the
local approximation order.
While dealing with the stability, we shall use two different approaches. In the
finite dimensionalcasewithsufficiently smoothvectorfields, the Gronwallinequal-
ity yields the claim in a straightforward manner under a reasonable assumption
of compatibility between the vector fields and the weight function. In the infinite
dimensional case, we apply the method of the moving frame from [35]. This leads
totimedependentvectorfieldsthatarenonsmoothinthetimecomponent. Asthis
makes a Taylor expansion impossible, we introduce a weak symmetry condition on
cubature paths, an assumption usually satisfied by cubature schemes. This allows
ustoobtainstabilitynotonlyforDaPrato-Zabczykequationswithpseudocontract-
ive generator, but also for stochastic differential equations on infinite dimensional
state spaces, where the vector fields depend roughly, i.e., continuously, but not
differentiably, on time.
Finally, we consider the effects of the UFG condition in our setting. Under the
same assumptions on the coefficients as in [5], we are able to prove optimal rates
of convergence on non-uniform meshes for nonsmooth payoffs that are allowed to
grow exponentially.
CUBATURE METHODS FOR SPDES 3
There are many successful discretisation schemes for stochastic partial differ-
ential equations. [17] gives an overview of strong and pathwise schemes. Weak
approximation schemes are more difficult. Recently, it was proved in [9] that an
implicit Euler scheme convergesalmostwith weak rate 1/2for equations drivenby
space-time white noise, doubling the corresponding strong rate of convergence;see
also the references in [9] for more background on weak approximation schemes for
stochastic partial differential equations with space-time white noise. In contrast,
we restrict ourselves to finite-dimensional driving noise, but obtain the same weak
rate of convergence as for finite-dimensional state spaces.
In our proofs, C denotes a generic positive real constant that can change from
line to line.
2. ψ spaces
B
We recall the following definition of spaces of functions with controlled growth,
see also [30, 12, 11, 10, 13]. Notice that we obtain Feller-like properties for SPDEs
in this setting.
Definition 2.1. Let (X, ) be the dual space of a separable Banach space,
X
k·k
and ϕ: X (0, ) be bounded from below by some δ > 0. For a Banach space
→ ∞
(Y, ), we set
Y
k·k
(2.1) Bϕ(X;Y):= f: X Y : supϕ(x)−1 f(x) < ,
Y
→ k k ∞
(cid:26) x∈X (cid:27)
endowed with the ϕ-norm
(2.2) f := supϕ(x)−1 f(x) .
ϕ Y
k k k k
x∈X
Letk 0. Ifϕ=(ϕ ) ,ϕ : X (0, )boundedfrombelowbysomeδ >0,
j j=0,...,k j
≥ → ∞
j =0,...,k, we set
Bϕ(X;Y):= f Ck(X;Y): supϕ (x)−1 Djf(x) <
k ∈ j k kLj(X;Y) ∞
x∈X
(cid:8)
(2.3) for j =0,...,k .
Bϕ(X;Y) is endowed with the norm (cid:9)
k
k
(2.4) f := f + f ,
k kϕ,k k kϕ0 | |ϕj,j
j=1
X
where the seminorms are given by
|·|ϕj,j
(2.5) f := supϕ (x)−1 Djf(x) .
| |ϕj,j j k kLj(X;Y)
x∈X
Here, L (X;Y) denotes the spaceof bounded multilinear forms a: Xj Y, and is
j
→
endowed with the norm
(2.6) a := sup a(h ,...,h ) .
k kLj(X;Y) k 1 j kY
khik≤1,i=1,...,j
For simplicity, we set L (X;Y) := Y; we remark that L (X;Y) is the space of
0 1
bounded linear operators X Y, and in this case, the above norm is the usual
operator norm. If Y =R, we→define Bϕ(X):=Bϕ(X;R) and Bϕ(X):=Bϕ(X;R).
k k
Definition 2.2. Let (X, ) be the dual space of a separable Banach space. A
X
k·k
functionϕ iscalledadmissible weight function ifandonlyif ϕ: X (0, )is such
→ ∞
that K := x X: ϕ(x) R is weak- compact for all R>0.
R
{ ∈ ≤ } ∗
It is called D-admissible weight function if and only if it is an admissible weight
function and for every x X, there exists some R > 0 such that B (x) K for
ε R
∈ ⊂
some ε>0, where B (x):= y X: y x ε is the closed ε-ball around x.
ε X
{ ∈ k − k ≤ }
4 PHILIPPDO¨RSEK,JOSEFTEICHMANN,ANDDEJANVELUSˇCˇEK
It is calledC-admissible weight function if and only if ϕ is bounded from below,
weak- lower semicontinuous, and if for every x X, there exists some ε>0 such
∗ ∈
that ϕ is bounded on B (x).
ε
Remark 2.3. We do not require C-admissible weight functions to be admissible.
However, ϕ is D-admissible if and only if it is admissible and C-admissible.
Theorem 2.4. Let k N, and assume that ϕ = (ϕ ) is a vector of C-
j j=0,...,k
admissible weight functi∈ons. Then, Bϕ(X;Y) is a Banach space.
k
Proof. Let (fn)n∈N be a Cauchy sequence in this space. It is clear that fn admits
a pointwise limit f. Moreover, it follows that for every x X and every closed
∈
ε-ball B (x), f are Cauchy sequences in Ck(B (x);Y). But this entails that
ε n|Bε(x) ε
f Ck(B (x);Y). As differentiability is a local property, we see that f
|Bε(x) ∈ ε ∈
Ck(X;Y). Thenecessaryestimatesforf anditsderivativesarenoweasytosee. (cid:3)
Remark 2.5. A counterexample showing the necessity of C-admissibility in The-
orem 2.4 is given in Appendix A.
Definition 2.6. Let (X, ) be the dual space of a separable Banach space, its
X
k·k
predual being W, X = W∗, and (Y, ) a Banach space. The space of bounded
Y
k·k
smooth cylindrical functions is defined by
(X,Y):= f: X Y : f =g( ,w ,..., ,w )
1 n
A → h· i h· i
for some g C∞(Rn;Y),
(cid:8) ∈ b
(2.7) w W, i=1,...,n, n N .
i
∈ ∈
Here, , denotes the dual pairing of X and W. For Y = R, w(cid:9)e set (X) :=
(X,Rh·).·i A
A
Definition 2.7. Let (X, ) be the dual space of a separable Banach space and
X
k·k
(Y, ) be a Banach space. Let ψ be an admissible weight function on X.
Y
Tkh·ke space ψ(X;Y) is the closure of (X,Y) in Bψ(X;Y). For Y =R, we set
ψ(X):= ψ(BX;R). A
B B
Remark 2.8. [12, Theorem 4.2] shows that our definition of ψ(X) here agrees
B
with our earlier definition from [12, Definition 2.2]. Due to [12, Theorem 2.7], the
functionsin ψ(X)arecharacterizedbythepropertythatbothf C((K ) )
B |KR ∈ R w∗
and
(2.8) lim sup ψ(x)−1 f(x) =0.
R→∞x∈X\KR | |
Definition 2.9. Let (X, ) be the dual space of a separable Banach space and
X
k·k
(Y, )beaBanachspace. Letψ =(ψ ) withψ D-admissibleweightfunc-
Y j j=0,...,k j
k·k
tions for j =0,...,k. The space ψ(X;Y) is the closure of (X,Z) in Bψ(X;Y).
Bk A k
For Y = R, we set ψ(X) := ψ(X;R). In particular, by Theorem 2.4, it follows
Bk Bk
that ψ(X) is a separable Banach space.
Bk
One essential property of ψ(X) spaces is that the dual space of this separable
B
Banachspace is a well understood space of Radon measures, such as in the case of
C (X) for locally compact spaces X.
0
Theorem 2.10 (Riesz representation for ψ(X)). Let ℓ: ψ(X) R be a con-
B B →
tinuous linear functional. Then, there exists a finite signed Radon measure µ on X
such that
(2.9) ℓ(f)= f(x)µ(dx) for all f ψ(X).
∈B
ZX
CUBATURE METHODS FOR SPDES 5
Furthermore,
(2.10) ψ(x)|µ|(dx)=kℓkL(Bψ(X),R),
ZX
where µ denotes the total variation measure of µ.
| |
As every such measure defines a continuous linear functional on ψ(X), this
B
completely characterizes the dual space of ψ(X).
B
This allows for the introduction of the generalized Feller property, such that we
canspeakaboutstronglycontinuoussemigroupsonspacesoffunctionswithgrowth
controlled by ψ, in particular functions which are in general unbounded.
Let (P ) be a family of bounded linear operators P : ψ(X) ψ(X) with
t t≥0 t
B → B
the following properties:
(F1) P =I, the identity on ψ(X),
0
B
(F2) P =P P for all t, s 0,
t+s t s
≥
(F3) for all f ψ(X) and x X, lim P f(x)=f(x),
t→0+ t
(F4) there exi∈stBa constant C∈ R and ε > 0 such that for all t [0,ε],
∈ ∈
P C,
k tkL(Bψ(X)) ≤
(F5) P is positive for all t 0, that is, for f ψ(X), f 0, we have P f 0.
t t
≥ ∈B ≥ ≥
Alluding to [18, Chapter 17],such a family ofoperatorswill be calleda generalized
Feller semigroup.
We shall now prove that semigroups satisfying F1 to F4 are actually strongly
continuous, a direct consequence of Lebesgue’s dominated convergence theorem
with respect to the measure existing due to Riesz representation.
Theorem2.11. Let(P ) satisfyF1toF4. Then, (P ) isstronglycontinuous
t t≥0 t t≥0
on ψ(X), that is,
B
(2.11) lim P f f =0 for all f ψ(X).
t ψ
t→0+k − k ∈B
Proof. By [14, Theorem I.5.8], we only have to prove that t ℓ(P f) is right
t
7→
continuous at zero for every f ψ(X) and every continuous linear functional
ℓ: ψ(X) R. Due to Theorem∈2.B10, we know that there exists a signed measure
B →
ν onX suchthatℓ(g)= gdν foreveryg ψ(X). ByF4,weseethatforevery
X ∈B
t [0,ε],
∈ R
(2.12) P f(x) Cψ(x).
t
| |≤
Due to (2.10), the dominated convergence theorem yields
(2.13) lim P f(x)ν(dx)= f(x)ν(dx),
t
t→0+ZX ZX
andtheclaimfollows. Here,theintegrabilityofψwithrespecttothetotalvariation
measure ν enters in an essential way. (cid:3)
| |
3. Vector fields and directional derivatives
When we ask for convergence rates we have to specify large enough sets of test
functions within the basic ψ(X)-spaces. For this purpose we need to analyze
B
directional derivatives and their functional analytic behavior. This can be done
within the setting of ψ(X;Y) spaces.
Bk
Let (X, ) be the dual space of a separable Banach space. Given (Z, )
X Z
k·k k·k
thedualspaceofanotherseparableBanachspacethatisembeddedinX,wederive
conditions on V : Z X such that the directional derivative g ψˆ (Z), where
→ ∈Bk−1
(3.1) g(z):=Df(z)(V(z)) for z Z
∈
6 PHILIPPDO¨RSEK,JOSEFTEICHMANN,ANDDEJANVELUSˇCˇEK
and f ψ(X). Here, ψ = (ψ ) and ψˆ = (ψˆ ) are vectors of
∈ Bk j j=0,...,k j j=0,...,k−1
D-admissible weight functions on X and Z, respectively.
We shall assume that V Bϕ (Z;X) for some vector ϕ = (ϕ ) of
∈ k−1 j j=0,...,k−1
C-admissible weight functions on Z. Then, V is k 1 times continuously Fr´echet
−
differentiable. As f Ck(X), the Leibniz rule yields
∈
(3.2)
j
1
Djg(z)(h , ,h )= g (z,h , ,h ), j =0,...,k 1.
1 ··· j i!(j i)! j,i σ1 ··· σj −
Xi=0 − σX∈Sj
Here, denotes the symmetric group with j elements, and
j
S
(3.3) g (z,h , ,h ):=Di+1f(z)(h , ,h ,Dj−iV(z)(h , ,h )).
j,i 1 j 1 i i+1 j
··· ··· ···
In particular, if we assume that for some constant C >0,
j
j
(3.4) ψˆ (z) C−1 ψ (z)ϕ (z) for j =0, ,k 1,
j i+1 j−i
≥ i ··· −
i=0(cid:18) (cid:19)
X
it follows that g Bψˆ (Z).
∈ k−1
Itisnotsostraightforwardtoprovethatgcanalsobeapproximatedbyfunctions
in (X), which would imply g ψˆ (Z). In [10], a general theory for multiplic-
A ∈ Bk−1
ation operators on ψ spaces is derived. Here, we take a different route, focusing
B
on the problem at hand. The following definition is essential.
Definition 3.1. Givena Banachspace(X, )andthe dualspace(Z, ) ofa
X Z
separable Banach space. Let V Bϕ(Z;X)kw·kith ϕ a given vector of C-akd·mk issible
weight functions on Z. We say t∈hatkV ϕ(Z;X) if and only if for every y X∗,
∈Ck ∈
there exists a constant C > 0 such that for all R > 0, there exists a sequence
V,y
(vn)n∈N ⊂A(Z) with supn∈Nkvnkϕ,k ≤CV,y such that, with v :=y◦V,
(3.5) lim v v =0.
n→∞k − nkCk(BR(0))
Here, B (0) is the closed unit ball of radius R in Z, and
R
k
(3.6) g := sup Djg(z) .
k kCk(BR(0)) k kLj(Z)
j=0z∈BR(0)
X
Remark 3.2. Itisclearthatvectorfieldssuchasthosefrom[13,Section2.2]satisfy
theaboveassumption. Moregenerally,ifZ isaHilbertspaceandiscompactlyem-
bedded into a larger Hilbert space Y such that y V can be extended to a smooth
mapping Y R lying in Ck(Y;R) for all y X◦∗, then the above assumption is
satisfied, i.e.→, V ϕ(Z;X)b for every vector∈ϕ of C-admissible weight functions
∈ Ck
on Z. Indeed, the extension of y V and its derivatives are continuous on Y,
◦
whence uniformly continous on the compact set B (0). Let us fix a sequence of
R
increasingfinite-dimensional,orthogonalprojectionsπ id convergingstrongly
n Y
→
to the identity: composing the extension of y V with π yields a pointwise con-
n
◦
verging, equicontinous sequence of cylindrical function on B (0), which is – up to
R
a smoothing argument – the desired assertion.
See also [13, Theorem 5] and [10, Theorem 2.39] for comparable arguments. In
particular,this implies that Nemytskiioperatorsare included inour setup if Z is a
Sobolev space of sufficiently smooth functions, see also [10, Example 2.48].
This definition should also be compared to the form of the multiplicative noise
suggestedin[9,Remark2.3]. Itissimilarinspirittothe definitionof ϕ(H;H),as
Ck
there, A is assumed to be a negative self-adjoint operator with a compact inverse.
Hence, if we consider a single component of the noise, x σ˜(( A)−1/4x), with
7→ −
σ˜: H H a C3-function with derivatives bounded up to order 3, it satisfies our
→
CUBATURE METHODS FOR SPDES 7
assumptions givenaboveand hence lies in ϕ(H) with ϕ (x):=(1+ x 2 )1/2 and
C3 0 k kH
ϕ (x):=1, j 1.
j
≥
Theorem 3.3. Fix k 1. Let ψ = (ψ ) be a vector of D-admissible
i i=0,...,k
≥
weight functions on X, and ψˆ = (ψˆ ) a vector of D-admissible and ϕ =
j j=0,...,k−1
(ϕ ) a vector of C-admissible weight functions on Z. Suppose (3.4).
j j=0,...,k−1
Then, the Lie derivative : ϕ (Z;X) ψ(X) ψˆ (Z) defined through
L Ck−1 ×Bk →Bk−1
(3.7) (V,f)(z):= f(z):=Df(z)(V(z))
V
L L
is a bilinear, bounded operator.
Remark 3.4. Clearly,itisnecessarythatV ϕ(Z;X)if f ψˆ(Z)issupposed
∈Ck LV ∈Bk
to holdfor f ψ (X)for a sufficiently largeclassofweightfunctions ψ. Indeed,
∈Bk+1
choose ψ (x) := ρ( x ) with some increasing, left continuous and superlinear
0 X
k k
function ρ, and ψ arbitrary D-admissible weight functions on X. Then, f := y
j
∈
ψ (X) for all y X∗. Hence, f(z) = y(V(z)), and y V ψˆ(Z) implies
Bthka+t1V ϕ(Z;X)∈. LV ◦ ∈ Bk
∈Ck
Proof. The claimed boundedness of was remarked above, and follows straight
L
away from (3.4).
Hence,weonlyneedtoprovethat f ψˆ (Z)forgivenV ϕ (Z;X)and
LV ∈Bk−1 ∈Ck−1
f =g( ,w , , ,w ) (X);theresultthenfollowsfromadensityargument.
1 n
h· i ··· h· i ∈A
Fix ε>0. We shall constructg (Z) such that f g <Cε with some
ε ∈A kLV − εkψˆ,k
constant C >0 independent of ε.
Choose a dual set of vectors (ζ ) Z of (w ) , i.e., ζ ,w = δ .
i i=1,...,n i i=1,...,n i j ij
Let Z := span ζ : i=1, ,n , and defin⊂e π: X Z by πx := h n xi,w ζ .
n { i ··· } → n i=1h ii i
Then, f π =f, and
◦ P
n
(3.8) f(z)= Df(z)(ζ ) V(z),w .
V i i
L h i
i=1
X
Clearly,w X∗,andthusbyDefinition3.1,thereexistsC :=max C >
i ∈ V i=1,...,n V,wi
0 such that for all R>0, we can find vi (Z) with vi C and
R,ε ∈A k R,εkϕ,k−1 ≤ V
(3.9) kwi◦V −vRi,εkCk−1(BR(0)) <ε,
where B (0) denotes the closedunit ball in Z. Setting g := n Df()(ζ )vi
R ε i=1 · i R,ε ∈
(Z), it follows that with a constant C >0 independent of R>0,
f
A P
(3.10) kLVf −gεkCk−1(BR(0)) <Cfε.
ChooseR >0largeenoughsuchthatψ (z)>ε−1 for z >R . Thisis possible
ε j Z ε
k k
as the embedding Z X is continuous. Hence, as f and all its derivatives are
→
bounded,
(3.11) ψˆ (z)−1 Dj f(z) <C ε for z >R , j =0,...,k 1,
j k LV kLj(Z) f k kZ ε −
where C is independent of ε. Furthermore,
f
(3.12) ψˆ (z)−1 Djg (z) C ε for z >R , j =0,...,k 1,
j k ε kLj(Z) ≤ f,V k kZ ε −
where C > 0 depends on f and V, but not on ε or R . Plugging the results
f,V ε
together proves the claim. (cid:3)
Let us consider two special cases.
8 PHILIPPDO¨RSEK,JOSEFTEICHMANN,ANDDEJANVELUSˇCˇEK
Corollary3.5. Let(H, )beaHilbertspace, (Z, )acontinuouslyembedded
H Z
k·k k·k
Hilbert space. Define the D-admissible weight functions ψ (x):=cosh( x ) on H
j H
k k
and ψˆ (x) := cosh( x ) on Z and the C-admissible weight functions ϕ (x) := 1
j Z j
k k
on Z, j 0. Then, for every k 0, the mapping
≥ ≥
(3.13) : ψ(X) ϕ (Z;X) ψˆ (Z), (f,V) f,
L Bk ×Ck−1 →Bk−1 7→LV
given by f(x):=Df(x)V(x), is bounded and bilinear.
V
L
Remark 3.6. If Z = H, this has the simple interpretation that bounded vector
fields map cosh-weighted spaces into themselves.
Proof. This is straightforwardfrom Theorem 3.3, as the ψˆ defined there is only a
j
multiple of ψˆ in this case. (cid:3)
j
The following special case is very useful in the analysis of stochastic partial
differential equations of Da Prato-Zabczyk type.
Corollary3.7. Let(H, )beaHilbertspace, (Z, )acontinuouslyembedded
H Z
Hilbert space. Fix n k·Nk. Define the D-admissibkle·kweight functions ψ (x) :=
j
∈
(1+ x 2 )(n−j)/2 on H and ψˆ (x) := (1+ x 2)(n−j)/2 on Z, j = 0,...,n 1,
k kH j k kZ −
and the C-admissible weight functions ϕ (x) := (1+ x 2)1/2 and ϕ (x) := 1 on
Z, j N. Then, for k n 1, the mapp0ing k kZ j
∈ ≤ −
(3.14) : ψ(X) ϕ (Z;X) ψˆ (Z), (f,V) f,
L Bk ×Ck−1 →Bk−1 7→LV
given by f(x):=Df(x)V(x), is bounded and bilinear.
V
L
Remark 3.8. This means that linearly bounded vector fields Z X with bounded
→
derivatives (hence also Lipschitz continuous) map polynomially bounded functions
topolynomiallyboundedfunctions,withthesameweights. Inparticular,ifA: domA
H H isadenselydefined,closedoperator,thenV ϕ(domA;H)forallk 0, ⊂
→ A ∈Ck ≥
where ϕ is defined as in Corollary 3.7, domA is endowed with the operator norm,
and V (x):=Ax for x domA.
A
∈
Proof. Calculating
j
j
(1+ x 2)(n−1)/2(1+ x 2)1/2+ (1+ x 2)(n−i−1)/2
k kZ k kZ i k kZ
i=0(cid:18) (cid:19)
X
(3.15) Cψˆ (x),
j
≤
the claim again follows from an application of Theorem 3.3. (cid:3)
4. Stability of cubature schemes
We shall now prove stability of cubature on Wiener space in the setting of
weighted spaces. Consider from now on the following setup. Let on [0,1] be
given paths (ω(1)) , ω(1)(s) = (ω(1),j(s)) , ω(1),0(s) = s, and weights
i i=1,...,N i i j=0,...,d i
(λ ) of a cubature on Wiener space of order m 1 for a d-dimensional
i i=1,...,N
≥
Brownian motion, i.e., for all multi-indices (j ,...,j ) with k+# i: j =0 m
1 k i
{ }≤
and a d-dimensional Brownianmotion (Bj) ,
t j=1,...,d,t≥0
(4.1) E dBj1 dBjk
··· ◦ s1···◦ sk
(cid:20)Z Z0≤s1≤···≤sk≤1 (cid:21)
N
= λ dω(1),j1(s ) dω(1),jk(s ).
i ··· i 1 ···◦ i k
i=1 Z Z0≤s1≤···≤sk≤1
X
CUBATURE METHODS FOR SPDES 9
Here, we have set B0 :=t and dB0 :=dt for ease of notation. For a general time
t ◦ t
interval [0,∆t], we set
(4.2) ω(∆t),0(s):=s and ω(∆t),j(s):=√∆tω(1),j(s/∆t), j =1,...,d,
i i i
so that (ω(∆t)) and (λ ) define a cubature formula on Wiener space
i i=1,...,N i i=1,...,N
of order m on [0,∆t]. The approximation of the Markov semigroup (P ) , given
t t≥0
by P f(x) := E[f(Xx)] for a function f: H R, where (Xx) solves the Stra-
t t → t t≥0
tonovich stochastic differential equation
d
(4.3) dXx = V (Xx) dBj, Xx =x,
t j t ◦ t 0
j=0
X
on some state space H, then reads
(4.4) P f(x) Qn f(x),
t ≈ (t/n)
where the one step approximation operator is defined by
N
(4.5) Q f(x):= λ f(Xx (ω(∆t))),
(∆t) i ∆t i
i=1
X
with Xx(ω(∆t)) the solution of the problem
t i
d
(4.6) dXx(ω(∆t))= V (Xx(ω(∆t)))dω(∆t),j(s), Xx(ω(∆t))=x.
s i j s i i 0 i
j=0
X
Under certain smoothness assumptions on the vector fields V , j = 0,...,d, and
j
the payoff f, we expect that
(4.7) P f(x) Qn f(x) Cn−(m−1)/2,
| f − (t/n) |≤
where the constant C > 0 can depend on f, V , j = 0,...,d, and x H. For the
j
∈
case H finite-dimensional and f and V bounded and C∞-bounded, j = 0,...,d,
j
it is known that C depends on the supremum norms of f and its derivatives, but
not on x H, see [25]. For more background on the method, see [25, 5, 3]. An
∈
alternative approach can be found in [21, 22]. Its implementation as a splitting
method is given in [27], see also [26, 1, 34].
Our strategy is as follows. First, we consider the finite dimensional case. Here,
the analysis is straightforward. Afterwards, we turn to the infinite dimensional
setting. Here, our aim is to prove stability for Da Prato-Zabczyk equations with
pseudodissipative generator. We prove first the auxiliary result in Theorem 4.4,
which might be of independent interest. The method of the moving frame then
yields firstTheorem4.7, and the Sz˝okefalvi-Nagytheoremallows us to conclude in
Corollary 4.8.
4.1. Finite dimensional state space. Given a Stratonovich SDE on Rn,
d
(4.8) dXx = V (Xx) dBj, Xx =x,
t j t ◦ t 0
j=0
X
we let the local discretisation of P f(x):=E[f(Xx)] be defined by
t t
N
(4.9) Q f(x):= λ f(Xx (ω(∆t))),
(∆t) i ∆t i
i=1
X
10 PHILIPPDO¨RSEK,JOSEFTEICHMANN,ANDDEJANVELUSˇCˇEK
where Xx(ω(∆t)) is the solution of the problem
t i
d
(4.10) dXx(ω(∆t))= V (Xx(ω(∆t)))dω(∆t),j(s), Xx(ω(∆t))=x.
s i j s i i 0 i
j=0
X
Theorem 4.1. Let ψ be an admissible weight function on Rn, and assume that
(4.11) V V ψ(x) + V ψ(x) Cψ(x) for i=0, ,d and j =1, ,d,
i j i
| | | |≤ ··· ···
where we require that all the necessary derivatives are well-defined.
Then, there exists a constant C˜ >0 independent of ∆t>0 such that
(4.12) Q ψ(x) exp(C˜∆t)ψ(x).
(∆t)
≤
Proof. We define the intermediate operator
N
(4.13) Q f(x):= λ f(Xx(ω(∆t))) for s [0,t]
(∆t,s) i s i ∈
i=1
X
and note that Q =Q . The definition of the iteration step yields
(∆t) (∆t,∆t)
d s
(4.14) ψ(Xx(ω(∆t)))=ψ(x)+ V ψ(Xx(ω(∆t)))dω(∆t),j(r)
s i j r i i
j=0Z0
X
s d
=ψ(x)+ V ψ(Xx(ω(∆t)))dr+ V ψ(x)ω(∆t),j(s)
0 r i j i
Z0 j=1
X
d d s r
+ V V ψ(Xx(ω(∆t)))dω(∆t),k(q)dω(∆t),j(r).
k j q i i i
j=1k=0Z0 Z0
XX
By (4.11),
s s
(4.15) V ψ(Xx(ω(∆t)))dr C ψ(Xx(ω(∆t)))dr.
0 r i ≤ r i
Z0 Z0
Furthermore, as ω(∆t),j(s) C(∆t)1/2 and ∂ ω(∆t),j(s) C(∆t)−1/2, Fubini’s
| i | ≤ |∂s i | ≤
theorem yields
s r
V V ψ(Xx(ω(∆t)))dω(∆t),k(q)dω(∆t),j(r)
k j q i i i
Z0 Z0
s ∂
C ω(∆t),j(s) ω(∆t),j(q)ψ(Xx(ω(∆t))) ω(∆t),j(q) dq
≤ | i − i | q i ∂q i
Z0 (cid:12) (cid:12)
s (cid:12) (cid:12)
(4.16) ≤C ψ(Xqx(ωi(∆t)))dq. (cid:12)(cid:12) (cid:12)(cid:12)
Z0
Thus, we see that
N
(4.17) Q ψ(x)= λ ψ(Xx(ω(∆t)))
(∆t,s) i s i
i=1
X
d N s
ψ(x)+ V ψ(x) λ ω(∆t),j(s)+C Q ψ(x)dr.
≤ j i i (∆t,r)
j=1 i=1 Z0
X X
Definingα (x):= d V ψ(x) N λ ω(∆t),j(s),theGronwallinequalityyields
∆t,s j=1 j i=1 i i
P P s
(4.18) Q ψ(x) ψ(x)+α (x)+ (ψ(x)+α (x))Cexp(C(s r))dr.
(∆t,s) ∆t,s ∆t,r
≤ −
Z0
