Geometrization of Monte-Carlo numerical analysis of an elliptic operator: strong approximation

Probability Theory

Geometrization of Monte-Carlo numerical analysis of an elliptic operator: strong approximation

Ana Bela Cruzeiro

, Paul Malliavin

, Anton Thalmaier

aGrupo de Física-Matemática UL and Dep. Matemática IST, Av. Rovisco Pais, 1049-001 Lisboa, Portugal b10, rue Saint Louis en l’Isle, 75004 Paris, France

cDépartement de mathématiques, Université de Poitiers, téléport 2, BP 30179, 86962 Futuroscope Chasseneuil, France Received 16 December 2003; accepted 12 January 2004

Presented by Paul Malliavin

Abstract

A one-step scheme is constructed, which, as the Milstein scheme, has the strong approximation property of order 1; in contrast to the Milstein scheme, our scheme does not involve the simulation of iterated Itô integrals of second order. To cite this article:

A.B. Cruzeiro et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).

2004 Published by Elsevier SAS on behalf of Académie des sciences.

Résumé

Monte-Carlo géométrique pour un opérateur elliptique : approximation numérique forte. On propose un schéma à un pas, qui, comme le schéma de Milstein, possède la propriété d’approximation forte à l’ordre 1 ; contrairement au schéma de Milstein, notre schéma ne nécessite pas la simulation d’intégrales itérées de Itô du second degré. Pour citer cet article : A.B. Cruzeiro et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).

2004 Published by Elsevier SAS on behalf of Académie des sciences.

1. Introduction

Numerical integration of SDE consists, an SDE being fixed, in producing a discretization scheme leading to a pathwise approximation of its solution. In this work we discuss the related problem of how to establish a Monte-Carlo simulation to a given real strictly elliptic second order operatorL, which leads to good numerical approximations of the fundamental solution to the corresponding heat operator.

Of course many SDE can be associated toL, each choice corresponding to a parametrization by the Wiener space of the Stroock–Varadhan solution of the martingale problem associated toL. For applications to finance all these parametrizations are equivalent. We shall prove that there exists an optimal parametrization for which the one-step Milstein scheme does not involve the computation of iterated stochastic integrals of second order. For our

E-mail addresses: abcruz@math.ist.utl.pt (A.B. Cruzeiro), sli@ccr.jussieu.fr (P. Malliavin), thalmaier@math.univ-poitiers.fr (A. Thalmaier).

1631-073X/$ – see front matter 2004 Published by Elsevier SAS on behalf of Académie des sciences.

doi:10.1016/j.crma.2004.01.007

search of an optimal parametrization we have to describe the possible parametrizations of the diffusion associated toL; it is sufficient to solve this problem for the Euclidean Laplacian onR^d; in this case it is equivalent to replace the standard Brownian motionWonR^dby an orthogonal transformWwith an Itô differential of the type

dW_k= d

j=1

Ω_W(t)k

jdW_j(t), (1)

wheret→ΩW(t)is an adapted process taking values in the group O(d)of orthogonal matrices. We denote by O(W) the family of all such orthogonal transforms which is isomorphic toP(O(d)), the path space on O(d).

Pointwise multiplication defines a group structure onP(O(d))O(W).

Given onR^dthe data ofd+1 smooth vector fieldsA₀, A₁, . . . , A_d, we consider the Itô SDE dξW(t)=A0

ξW(t) dt+

d

k=1

Ak

ξW(t)

dWk(t), ξW(0)=ξ0. (2)

Throughout this Note we assume ellipticity, that is, for anyξ ∈R^d the vectors A₁(ξ ), . . . , A_d(ξ ) constitute a basis ofR^d; the components of a vector fieldU in this basis are denotedU, A_kξ which gives the decomposition U (ξ )=_d

k=1U, A_kξA_k(ξ ). By change of parametrization we mean the substitution ofWbyW in (2); we then get an Itô process inW. This change of parametrization does not change the infinitesimal generator associated to (2) which has the formL=¹₂

k,α,βA^α_kA^β_kDαDβ+

αA^α₀DαwhereDα=∂/∂ξ^α. The group O(W)operates on the set of elliptic SDE onR^dand the orbits of this action are classified by the corresponding elliptic operatorsL.

2. Definition of the schemeS

Denote bytε:=ε×integer part oft/ε; we define our scheme by Z_W^ε(t)−Z_W^ε(t_ε)=A₀

Z_W^ε(t_ε)

(t−t_ε)+

k

A_k

Z_W^ε(t_ε)

W_k(t)−W_k(t_ε)

+1 2

k,s

(∂_AkA_s)

Z_W^ε(t_ε)

W_k(t)−W_k(t_ε)

W_s(t)−W_s(t_ε)

−εη_k^s +1

2

k,s,i

A_i

Z_W^ε(t_ε)

[A_s, A_i], A_k

Z_{W ε}(tε)

W_k(t)−W_k(t_ε)

W_s(t)−W_s(t_ε)

−εη_k^s , (3) whereW is standard Brownian motion onR^d, and η^s_k the Kronecker symbol defined by η_k^s =1 if k=s and zero otherwise. Denote byP(R^d)the path space onR^d, that is the Banach space of continuous maps from[0, T] intoR^d, endowed with the sup norm:p1−p2∞=sup_t∈[0,T]|p1(t)−p2(t)|_R^d. Fixingξ0∈R^d, letPξ₀(R^d)be the subspace of paths starting fromξ0. Given Borel measuresρ1, ρ2onP(R^d), denote by M(ρ1, ρ2)the set of measurable mapsΨ:P(R^d)→P(R^d)such thatΨ_∗ρ₁=ρ₂; the Monge transport norm (see [8,5]) is defined as

d_M(ρ₁, ρ₂):=

inf

Ψ∈M(ρ1,ρ₂)

Ψ (p)²_∞ρ₁(dp) 1/2

.

Theorem 2.1. Assume ellipticity and assume the vector fieldsA_k along with their first three derivatives to be bounded; fixξ₀∈R^d and let ρ_L be the measure on P_ξ0(R^d) defined by the solution of the Stroock–Varadhan martingale problem [7] for the elliptic operatorL; letρ_S be the measure obtained by the schemeSwith initial valueZ_W^ε(0)=ξ₀. Then

lim sup

ε→0

1

εd_M(ρ_L, ρ_S)=c <∞. (4)

Remark. The proof of Theorem 2.1 will provide an explicit transport functionalΨ₀which puts the statement in a constructive setting; the constantcis effective.

3. The Milstein scheme

The Milstein scheme for SDE (2) (cf., for instance, [6], formula(0.23)or [2], p. 345; see also [3]) is based on the following stochastic Taylor expansion ofA_k along the diffusion trajectory: A_k(ξ_W(t))=A_k(ξ_W(t_ε))+

j(∂_AjA_k)(ξ_W(t_ε))(W_j(t)−W_j(t_ε))+O(ε), which leads to ξW^ε(t)−ξW^ε(tε)=

k

Ak

ξW^ε(tε)

Wk(t)−Wk(tε)

+(t−tε)A0

ξW^ε(tε)

+

i,k

(∂_AiA_k)

ξ_W^ε(t_ε)^t

tε

W_i(s)−W_i(t_ε)

dW_k(s); the computation of_t

t_ε(Wi(s)−Wi(tε))dWk(s)gives the Milstein scheme ξW^ε(t)−ξW^ε(tε)=

k

Ak

ξW^ε(tε)

Wk(t)−Wk(tε)

+(t−tε)A0

ξW^ε(tε) +1

2

i,k

(∂_AiA_k)

ξ_W^ε(t_ε)

W_i(t)−W_i(t_ε)

W_k(t)−W_k(t_ε)

−εη_kⁱ +R, whereR=

i<k[Ai, Ak](ξW^ε(tε))_t

t_ε(Wi(s)−Wi(tε))dWk(s)−(Wk(s)−Wk(tε))dWi(s).

It is well known that the Milstein scheme has the following strong approximation property:

E sup

t∈[0,1]

ξ_W(t)−ξ_W^ε(t)²

=O ε²

. (5)

The numerical difficulty related to the Milstein scheme is how to achieve a fast simulation ofR. The purpose of this work is to show that by a change of parametrization this simulation can be avoided.

4. Horizontal parametrization

Givendindependent vector fieldsA1, . . . , AdonR^d, we take the vectorsA1(ξ ), . . . , Ad(ξ )as basis at the point ξ; the functionsβ_k,sⁱ , called structural functions, are defined (see [1]) by:

β_k, ⁱ (ξ )=

[Ak, A ], Ai

ξ, [Ak, A ](ξ )=

i

β_k, ⁱ (ξ )Ai(ξ ).

The structural functions are antisymmetric with respect to the two lower indices. Consider the connection functions, defined from the structural functions by

Γ_k,sⁱ =1 2

β_k,sⁱ −β_s,i^k +β_i,k^s

. (6)

LetΓkbe thed×d matrix obtained by fixing the indexkin the three indices functionsΓ_k,^∗_∗. Then, by means of the antisymmetry ofβ_∗^∗_,∗in the two lower indices,Γkis an antisymmetric matrix:

2

Γ_k,sⁱ +Γ_k,i^s

=β_k,sⁱ −β_s,i^k +β_i,k^s +β_k,i^s −β_i,s^k +β_s,kⁱ =0.

The matrixΓ_koperates on the coordinate vectors of the basisA_s(ξ )viaΓ_k(A_s)=

iΓ_k,sⁱ A_i. This givesΓ_k(A_s)− Γs(Ak)= [Ak, As]: theith component of the l.h.s. is ¹₂(β_k,sⁱ −β_s,i^k +β_i,k^s −β_s,kⁱ +β_k,i^s −β_i,s^k )=β_k,sⁱ . LetM= R^d×EdwhereEdis the vector space ofd×dmatrices. Define onMvector fieldsA˜k, k=1, . . . , d, as follows:

A˜k(ξ, e)=

e_kA (ξ ),Nk(ξ, e)

, [Nk]^sr(ξ, e)= −

,

e_ke_rΓ _, ^s (ξ ), ξ ∈R^d, e∈Ed. (7)

Denoting for a vectorZonMbyZ^H its projection onR^d, we have:

Proposition 4.1. The vector fieldsA˜_ksatisfy the relation[ ˜A_k,A˜_s]^H=0.

Proof. The horizontal component[∂_A˜

kA˜_s]^His given by [∂_A˜

kA˜s]^H=

i

A˜ⁱ_k∂iA˜^H_s +

q

α,β

[Nk]^αβ∂(^α_β)e^qs

Aq

=

i

e_kAⁱ

e_s∂iA

−

α,β

,

e_ke_βΓ _, ^α

q

η_α^qη^s_βAq

=

i

e_kAⁱ

e_s∂iA

−

α, ,

e_ke_sΓ _, ^α Aα,

using the fact that∂(^α_β)e^sq=η^q_αη_β^s. We finally get[∂_A˜

kA˜_s]^H=

, e_ke_s[∂_A A −

αΓ _, ^αA_α]. Therefore the horizontal component of the commutator is

[ ˜A_k,A˜_s]^H=

,

e_ke_s[A , A] −

α, ,

e_ke_s

Γ _, ^α −Γ^α_, A_α

which vanishes sinceΓ (A )−Γ(A )= [A , A]. ✷

Denote bye^Tthe transposed of the matrixeand letA˜₀(ξ, e)=(A₀(ξ ),−¹₂eJ ),J :=_d

k=1N_k^TN_k. Consider the following Itô SDE on the vector spaceM:

dmW=

k

A˜k(mW)dWk+ ˜A0(mW)dt, mW(0)=(ξ0,Id). (8)

Proposition 4.2. DenotemW(t)=(˜ξW(t), eW(t)), theneW(t)is an orthogonal matrix fort0, and fξ˜_W(t)

− t

0

(Lf )ξ˜_W(s)

dsis a local martingale, for anyf ∈C² R^d

. (9)

Proof. We compute the stochastic differential ofe^Te:

d e^Te

=

k

d e_ke_k

= −

m,k,p

u

e_ke^p_me^u_kΓ_p,u +

v

e_ke_m^pe^v_kΓ_p,v

dW_m +

k

e_k(A˜₀)_k+e_k(A˜₀)_k+

m,p,q,p,q

e^p_me^p_kΓ_p,p e^q_me^q_kΓ_q,q

dt,

where the last term of the drift comes from the Itô contraction

k de_k∗ de_k=

k[N_k^TNk] dt=J dt. The first two terms of the drift are computed by using the definition ofA˜0:

k[e_k(A˜0)_k+e_k(A˜0)_k] = −¹₂[(e^TeJ ) + (e^TeJ ) ]. Writee^Te=Id+σ, then the drift takes the form−(σ J+J σ )/2. We compute the coefficient of dWm:

−

k,p

u

e_ke_m^pe^u_kΓ_p,u +

v

e_ke^p_me_k^vΓ_p,v

= −

p

e^p_m

u

e^Teu

Γ_p,u +

v

e^Tev

Γ_p,v

.

Using the antisymmetryΓ_p, = −Γ_p, we obtain dσ = −

m

dW_m

p

e^p_m

u

σ^uΓ_p,u +

v

σ^vΓ_p,v

−1

2[σ J+J σ] dt. (10)

Eq. (10), together with Eq. (8), gives an SDE with local Lipschitz coefficients for the triple(˜ξ , e, σ ); by uniqueness of the solution, asσ (0)=0, we deduceσ (t)=0 for allt0. ✷

In terms of the newR^d-valued Brownian motionWdefined by dWk(t):=

[eW(t)]^kdW , we have dξ˜_W=

k

A_kξ˜_W(t)

dW_k(t)+A₀ξ˜_W(t)

dt. (11)

5. Reconstruction of the schemeS

We want to prove that our schemeS is essentially the projection of the Milstein scheme (˜ξ_W^ε, e_W^ε)for the solution m_W =(˜ξ_W, e_W) of the SDE (8). In order to write the first componentξ˜_W^ε we have to compute the horizontal part of∂_A˜

kA˜_j, which has been done in the proof to Proposition 4.1: we get ξ˜_W^ε(t)− ˜ξ_W^ε(t_ε)

=A0

ξ˜W^ε(tε)

(t−tε)+

k,

eW^ε(tε)

kA ξ˜W^ε(tε) -(Wk)

+1 2

k,j

,

e_W^ε(t_ε)

k

e_W^ε(t_ε)

j

∂_A A −

i

Γ _, ⁱ A_iξ˜_W^ε(t_ε)

-(W_k)-(W_j)−εη^j_k ,

where-(W_k)=W_k(t)−W_k(t_ε). By (5) E

sup

t∈[0,1]

e_W(t)−e_Wε(t)²

cε², E

sup

t∈[0,1]

ξ˜_W(t)− ˜ξ_Wε(t)²

c ε². (12)

Consider the new processξ_W^. defined by ξ_W^.(t)−ξ_W^.(t_ε)=A₀

ξ_W^.(t_ε)

(t−t_ε)+

k,

e_W(t_ε)

kA ξ_W^. (t_ε)

-(W_k)

+1 2

k,j

,

eW(tε)

k

eW(tε)

j

∂A A−

i

Γ _, ⁱ Ai

ξ_W^.ε(tε)

-(Wk)-(Wj)−εη_k^j .

Lemma 5.1. The processξ_W^. has the same law as the processZ_W^εdefined in (3).

Proof. By Proposition 4.2,W (t)−W (t_ε):=

k[e_W(t_ε)]_k(W_k(t)−W_k(t_ε))are the increments of anR^d-valued Brownian motionW; we get

ξ_W^.(t)−ξ_W^.(t_ε)=A₀ ξ_W^.(t_ε)

(t−t_ε)+

k

A_k

ξ_W^. (t_ε) W_k(t)−W_k(t_ε)

+1 2

k,s

∂_AkA_s−

i

Γ_k,sⁱ A_i

ξ_W^. (t_ε) W_k(t)−W_k(t_ε) W_s(t)−W_s(t_ε)

−εη_k^s .

By Eq. (6), we have 2

iΓ_k,sⁱ Ai = [Ak, As] +

i([Ai, As], AkAi + [Ai, Ak], AsAi), where the first term is antisymmetric in k, s and does not contribute; the remaining sum is symmetric in k, s. Thus we get

−

i,k,sΓ_k,sⁱ Ai-(Wk)-(Ws)=

i,k,sAi[As, Ai], Ak-(Wk)-(Ws)which proves Lemma 5.1. ✷ Lemma 5.2. We haveE[sup_t∈[0,1]ξ_W^.(t)− ˜ξ_W^ε(t)²_Rd]cε².

Proof. The following method of introducing a parameterλand differentiating with respect toλ, is continuously used in [4]. Forλ∈ [0,1], lete^λ:=λeW+(1−λ)eW^ε and define the processξ_W^λ by

ξ_W^λ(t)−ξ_W^λ(tε)=A0

ξ_W^λ(tε)

(t−tε)+

k,

e^λ_W(tε)

kA ξ_W^λ(tε)

-(Wk)

+1 2

k,j

,

e_W^λ(t_ε)

k

e^λ_W(t_ε)

j

∂_A A−

i

Γ _, ⁱ A_i

ξ_W^λε(t_ε)

-(W_k)-(W_j)−(t−t_ε)η^j_k .

Letu^λ_W:=∂ξ /∂λ; thenξ_W^.(t)− ˜ξ_W^ε(t)=₁

0u^λ_W(t)dλ. Denote byA_k, (∂_A A),(Γ _, ⁱ A_i)the matrices obtained by differentiatingA_k, ∂_A A , Γ _, ⁱ A_i with respect toξ, and consider the delayed matrix SDE

dJ_t←t0=

A₀ ξ_W^λ(t_ε)

dt+

k,

e_W^λ(t_ε)

kA ξ_W^λ(t_ε)

dW_k+1 2

k,j ,

e^λ_W(t_ε)

k

e^λ_W(t_ε)

j

×

(∂A A)−

i

Γ _, ⁱ Ai

ξ_W^λε(tε)

-(Wk)dWj(t)+-(Wj)dWk(t)−η^j_kdt Jt←t₀

with initial conditionJt₀←t₀=Id. Then

u^λ_W(T )=JT←t₀

T

0

J_t⁻_←¹_t

0

k,

eW(tε)−eW^ε(tε)

kA ξ_W^λ(tε)

dWk(t)+

k,j

,

eW(tε)−eW^ε(tε)

k

×

e^λ_W(t_ε)

j

∂_A A −

i

Γ _, ⁱ A_i

ξ_W^λε(t_ε)

-(W_k)dW_j(t)+-(W_j)dW_k(t)−η_k^jdt which along with (12) proves Lemma 5.2. ✷

References

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[3] P. Malliavin, Paramétrix trajectorielle pour un opérateur hypoelliptique et repère mobile stochastique, C. R. Acad. Sci. Paris, Sér. A-B 281 (1975) A241–A244.

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Translated and revised from the 1988 Russian original.

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