Numerical error for SDE: asymptotic expansion and hyperdistributions

Probability Theory

Numerical error for SDE:

Asymptotic expansion and hyperdistributions

Paul Malliavin

, Anton Thalmaier

a10, rue Saint Louis en l’Isle, 75004 Paris, France

bUniversité d’Évry, laboratoire d’analyse et probabilité, bd François Mitterrand, 91025 Evry cedex, France

Received 13 March 2003; accepted 1 April 2003 Presented by Paul Malliavin

Abstract

The principal part of the error in the Euler scheme for an SDE with smooth coefficients can be expressed as a generalized Watanabe distribution on Wiener space. To cite this article: P. Malliavin, A. Thalmaier, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

2003 Académie des sciences. Published by Éditions scientifiques et médicales Elsevier SAS. All rights reserved.

Résumé

Erreur du schéma d’Euler pour une EDS et hyperdistribution. La partie principale de l’erreur dans l’intégration par le schéma d’Euler d’une EDS avec des coefficients réguliers est une distribution de Watanabe généralisée. Pour citer cet article : P. Malliavin, A. Thalmaier, C. R. Acad. Sci. Paris, Ser. I 336 (2003).

2003 Académie des sciences. Published by Éditions scientifiques et médicales Elsevier SAS. All rights reserved.

1. Algebraic computation of the error

Asymptotic expansion depends upon the choice of a topology for measuring the size of the remainder term;

either a norm topology, or on the opposite side, some weak topology may be chosen. The “weakest” among weak topologies are topologies on spaces of distributions: this will be the point of view of this Note. Our main concern is to establish asymptotic expansions “upstairs”, that is on the probability space itself. We shall deal with the R^d-valued SDE

dξ_W(t)= n

k=1

A_k ξ_W(t)

dW^k+A₀ ξ_W(t)

dt; ξ_W t⁰

=ξ₀, (1)

whereW denotesn-dimensional Brownian motion on Wiener spaceWⁿandA₀, A₁, . . . , Anare smooth bounded vector fields onR^d with bounded derivatives of all orders.

E-mail addresses: sli@ccr.jussieu.fr (P. Malliavin), anton.thalmaier@maths.univ-evry.fr (A. Thalmaier).

1631-073X/03/$ – see front matter 2003 Académie des sciences. Published by Éditions scientifiques et médicales Elsevier SAS. All rights reserved.

doi:10.1016/S1631-073X(03)00189-4

Givenε >0, the Euler scheme of meshεis defined by the following recursion formula:

ξ_Wε(qε)−ξ_Wε

(q−1)ε

= n

k=1

A_k ξ_Wε

(q−1)ε

W^k(qε)−W^k

(q−1)ε +A₀

ξ_Wε

(q−1)ε

ε, ξ_Wε(t)=ξ₀ fort∈ t_ε⁰, t⁰

,

where fort0 we use the notationt_ε= [t/ε]εwith[a]the largest integera. The Euler scheme for all times is then defined as solution of the delayed SDE

dξ_Wε(t)= n

k=1

A_k ξ_Wε(t_ε)

dW^k(t)+A₀ ξ_Wε(t_ε)

dt fortt⁰, ξ_Wε| t_ε⁰, t⁰

=ξ₀. (2)

The remainder termθε(t):=ξW_ε(t)−ξW(t)satisfies the SDE dθ_ε=

n

k=1

A_k

ξ_W(t)+θ_ε(t)

−A_k ξ_W(t)

dW^k(t)+ A₀

ξ_W(t)+θ_ε(t)

−A₀ ξ_W(t)

dt+dχ (t), (3) whereθ_ε(t⁰)=0 and

dχ (t):=

n

k=1

εRk(t)dW^k(t)+^εR0(t)dt, ^εRk(t):=A_k ξ_Wε(t_ε)

−A_k ξ_Wε(t)

.

Let Ak be thed×d matrix defined by differentiating the components of the vector fieldA_k with respect to the coordinate vector fields; then almost surely the derivative of the solutionξ_Wε(t)of (2) with respect to the initial dataξ₀defines a random flow of diffeomorphisms; its Jacobian is given by the matrix-valued delayed SDE

dJ^Wε

t←t⁰=J^Wε

t←t⁰

_n

k=1

A_k ξ_W(t_ε)

dW^k+A₀ ξ_W(t_ε)

dt

, tt⁰,

whereJ^Wε

t←t⁰=id fort∈ [t_ε⁰, t⁰]. The derivatives of^εRk(t)may be computed in terms of the Jacobian matrix:

D_τ,^εRk(t)=1_{τt} Ak

ξ_Wε(t_ε)

J_t^W_←^ε_τ(A)−Ak

ξ_Wε(t)

J_t^W_←^ε_τ(A)

; (4)

the derivativesDτ,θε(t)=:u(t)are computed by differentiating (3). We get du−

n

k=1

Ak(ξW+θε)udW^k−A0(ξW+θε)udt=:dΓ ≡ n

k=1

ΓkdW^k+Γ0dt, (5)

whereΓ0, Γ1, . . . , Γn may be computed using (4) and standard computations of derivatives along the stochastic flow to SDE (1). Then, by Itô’s formula, the following version of the Lagrange formula (variation of constants) is established: in terms of the compensation vector fieldZ:=n

k=1AkΓk, we have u(t)=J_t←t0

t

t⁰

J_t0←τ

dΓ (τ )−Z(τ )dτ

, (6)

where the Itô stochastic integral inside the brackets has to be computed first. We introduce a parameterλ∈ [0,1] and define^λθas the solution of the SDE

d^λθ (t) = n

k=1

A_k

ξ_W(t)+^λθ (t)

−A_k ξ_W(t)

dW^k(t) +

A0

ξW(t)+^λθ (t)

−A0

ξW(t)

dt+λdχ (t), ^λθ t⁰

=0; (7)

as⁰θ (t)=0 for allt, denoting _dλ^dλθ=^λu, we haveθε=₁

0

λudλ. By differentiating (7) with respect toλ, we get the following linear SDE for^λu:

d^λu=d^εQ·^λu+dχ , (8)

where d^εQ=n

k=1Ak(ξW+^λθ )dW^k+A0(ξW+^λθ )dt.

2. Diagonal-regular functions and diagonal-distributions on Wiener space

We consider the diagonal ofR^rwhich is the 1-dimensional subspaceγ= {τ1= · · · =τr}; letΦ^rbe the quotient mapR^r→R^r/γ. A functionU∈L²(R^r)is said to be diagonal-continuous ifE^Φr[U²]is a continuous function, whereE^Φr denotes the conditional expectation underΦ^r; the space^γD^p_r is the subspace ofD_r^psuch that derivatives up to the orderrare diagonal-continuous. We define a diagonal Sobolev norm

f^pγ_D^p

r =E |f|^p+ r

j=1

E^Φj|Dτ₁,...,τ_jf|^2p/2

L^∞(R^j/γ )

.

We remark that solutions of SDE with infinitely bounded differentiable coefficients give rise to functionals belonging to^γD^p_r. A diagonal-distribution (see [8]) is a linear form on^γD_r^psuch that|f, S|cf^γ_D^p_r.

3. Estimation of the error in Sobolev norm of positive order

We get the following estimates of the remainder term in theL^p-norm, resp.^γD^p_r-Sobolev norm; see [2,6,7] for related results.

Proposition 3.1. For anyp∈ ]1,∞[and any integerr >0, we have E

sup

t∈[t⁰,T]

θε(t)^p

R^d

1/p

=O√ ε

; sup

t∈[t⁰,T]

θε(t)γ

D_r^p=O√ ε

. (9)

Proof. Denote by^εJ^W_t←τthe solution to the linear equation (8) with initial condition^εJ^W_τ←τ=id; as its coefficients Ak are bounded, we have uniformly with respect toε:

E sup

t,τ∈[t⁰,T]

^εJ^W_t←τ^2p

cp<∞;

by the same reason the compensation vector fieldZis bounded inL^p; we getE[|^εRk|^2p_Rd] =O(ε^p). Using (6), we have

λu(t)=^εJ^W_t←t0 t

t⁰

n

k=1

εJ^W_t0←τε Rk(τ )

dW^k(τ )+^εJ^W_t0←τε

R0(τ )−Z(τ ) dτ

.

ConsequentlyDτ,kθsatisfies an SDE of the same nature as (3); the second part of (9) is verified along the same lines. ✷

Corollary 3.1 (cf. [4]). The Euler scheme converges quasi-surely.

4. Asymptotic expansion in terms of a diagonal-distribution

We localize the method of Section 3 by using a Taylor expansion atλ=0; denoting the second derivatives by

λv:=(d/dλ)^λuwe have by means of (9) θ=^◦u+1

2

◦v+o(ε).

The question of asymptotic expansion ofθas o(ε)is therefore reduced to the asymptotic expansion of^◦u,^◦vwhich we abbreviate asu,v. We computeufrom (8) forλ=0 which realizes a computation along the path of the original diffusion. In the same way differentiating equation (8) with respect toλand lettingλ=0, the terms

R¹_k(t):=

d

,p=1

∂²Ak

∂ξ∂ξ^p ξ_W(t)

u(t)u^p(t), Z¹(t):=

n

k=1

A_k·R¹_k ξ_W(t)

(10) appear, and we get

v(t)=J^W

t←t⁰

t

t⁰

dτ n

k=1

J^W

t⁰←τ

R¹_k(τ )

dW^k(τ )+J^W

t⁰←τ

R¹₀(τ )−Z¹(τ )

. (11)

Theorem 4.1 (Rate of weak convergence). There existR^d-valued functionsa_k,b_k,conR^d, computable in terms of the coefficients of (1) and its first fourth derivatives, such that for anyf ∈^γD₃^∞−0,

lim

ε→0

1 εE

θ_ε(t) f

= t

t⁰

E n

k,

a_k ξ_W(τ )

D_τ,k² _;τ,f + n

k=1

b_k ξ_W(τ )

D_τ,kf+c ξ_W(τ )

f

dτ. (12)

Remark 1. This result should be compared with [5,1,3].

Proof. Assuming that diagonal distributionsS¹, . . . ,S^dof third order are given, we consider the formal expression S^q =

k(J^W

t←t⁰)^q_kS^k. As the coefficients of the matrixJ^W

t⁰←t belong to^γD₃^∞−0 and as the space of third order diagonal distributions is a module over the algebra^γD₃^∞−0, we deduce that S is a diagonal distribution of third order. Definingu(t)˜ :=J^W

t⁰←tu(t)andv(t)˜ :=J^W

t⁰←tv(t), we are reduced to find a diagonal distributionSsuch that

εlim→0

1 εE

˜ u(t)f

= f,S

, lim

ε→0

1 εE

˜ v(t)f

=0.

To short-hand the notation, we write^W(·):=J^W

t⁰←τ(·); thenu(t)˜ = ˜u₁(t)+ ˜u₂(t)where

˜ u₁(t):=

t

t⁰

n

k=1 Wε

Rk(τ )

dW^k(τ ), u˜₂(t):=

t

t⁰ W_ε

R0(τ )−Z(τ ) dτ.

Using the fact that an Itô integral is a divergence we get E

fu˜₁(t)

= t

t⁰

n

k=1

(D_τ,kf )^W_ε

Rk(τ ) dτ;

by Itô formula applied to the delayed SDE (2) we have

εRk(τ )= − τ

τε

s

Ak

ξW_ε(λ)

·As

ξW_ε(τε)

dW^s(λ)− τ

τε

ε

LAk

ξW_ε(λ)

dλ, (13)

where _ε

LA_k ξ_Wε(λ)

:=1 2

s,,p

A_s

ξ_Wε(τ_ε) A^p_s

ξ_Wε(τ_ε) ∂²A_k

∂ξ∂ξp

ξ_Wε(λ) +

A₀

ξ_Wε(τ_ε)∂A_k

∂ξ

ξ_Wε(λ) . We want to eliminate the first term in (13). To this end we remark

E

(D_τ,rf )^W_ε

Rr(τ )

=E

E^Nτ(D_τ,rf )^W_ε

Rr(τ )

; E E^Nτε(D_τ,rf )

τ

τε

s

Ak

ξ_Wε(λ)

·A_s

ξ_Wε(τ_ε) dW^s(λ)

=0.

By the Ocone–Karatzas formula, we get E^Nτ[D_τ,rf] −E^Nτε[D_τ,rf] =

n

k=1

τ

τε

E^Nλ

D_τ,r² _;λ,kf dW^k(λ) and with^W(·)analogously as above,

E fu˜₁(t)

=E t

t⁰

dτ τ

τε

dλ _n

r,k=1

D_τ,r² _;λ,kf_W Ar

ξW_ε(λ)

·Ak

ξW_ε(τε) +

n

k=1

(Dτ,kf )^W_ε LAk

ξW_ε(λ)

which asε→0 gives rise to the equivalence E

fu˜₁(t) ε

2E t

t⁰

dτ _n

r,k=1

D_τ,r² _;τ,kf_W

(A_r·A_k) ξ_W(τ )

+ n

k=1

(D_τ,kf )^W₀ LA_k

ξ_W(τ ) .

Finally E

fu˜2(t)

=E t

t⁰

f ·_W

R0−^WZ dτ

may be computed as before. There appear integrals along the paths ofD_τ,kf,f, and we get existence ofaˆ,b,ˆ cˆ such that

lim

ε→0

1 εE

˜ u(t)f

= t

t⁰

E n

k,

ˆ a_k

ξ_W(τ )

D²_τ,k;τ,f + n

k=1

bˆ_k ξ_W(τ )

D_τ,kf + ˆc ξ_W(τ )

f

dτ. (14)

It remains to take care ofv: combining (9) and (11) gives immediately˜ v˜=O(ε). To get the sharper estimate o(ε) we use that (10) givesR¹_kas a bilinear expression inu. A typical term is

E h t

t⁰

W∂²Ak

∂ξ∂ξ^puu^pdW^k

= t

t⁰

E

B(τ ) u^p(τ )

dτ, B:=(D_τ,kh)

W∂²Ak

∂ξ∂ξ^pu;

applying formula (14) and takingf =B, we get

E h t

t⁰

W∂²Ak

∂ξ∂ξ^puu^pdW^k

cεB^γ_D^s

2cεh^γ_D^2s

3 u^γ_D^2s

2 ,

and using (9), withθε(·)replaced byu, we obtain the wanted order O(ε). ✷

References

[1] V. Bally, D. Talay, The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function, Probab.

Theory Related Fields 104 (1996) 43–60.

[2] E. Gobet, Weak approximation of killed diffusion using Euler schemes, Stochastic Process. Appl. 87 (2000) 167–197.

[3] A. Kohatsu-Higa, S. Ogawa, Weak rate of convergence for an Euler scheme of nonlinear SDE’s, Monte Carlo Methods Appl. 3 (1997) 327–345.

[4] D. Laurent, Analyse quasi-sure du schéma d’Euler, C. R. Acad. Sci. Paris 315 (1992) 599–602.

[5] D. Talay, L. Tubaro, Expansion of the global error for numerical schemes solving stochastic differential equations, Stochastic Anal. Appl. 8 (1990) 483–509.

[6] D. Talay, Z. Zheng, Quantiles of the Euler scheme for diffusion processes and financial applications, Math. Finance 13 (2003) 187–199.

[7] E. Teman, Analysis of error with Malliavin calculus: application to hedging, Math. Finance 13 (2003) 201–214.

[8] S. Watanabe, Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels, Ann. Probab. 15 (1987) 1–39.