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ON ERROR OPERATORS RELATED TO THE ARBITRARY FUNCTIONS PRINCIPLE 3 Nicolas Bouleau ∗ 1 0 2 October 2006 n a J Abstract. The error on a real quantity Y due to the graduation of the measuring 7 2 instrument may be asymptotically represented, when the graduation is regular and fines down, by a Dirichlet form on R whose square field operator does not depend ] R on the probability law of Y as soon as this law possesses a continuous density. This P feature is related to the “arbitrary functions principle” (Poincaré, Hopf). We give h. extensions of this property to Rd and to the Wiener space for some approximations t a of the Brownian motion. This gives new approximations of the Ornstein-Uhlenbeck m gradient. These results apply to the discretization of some stochastic differential [ equations encountered in mechanics. 1 v Key words : arbitrary functions, Dirichlet forms, Euler scheme, Girsanov theorem, 5 8 mechanical system, Rajchman measure, square field operator, stable convergence, 3 stochastic differential equation. 6 . 1 Introduction. 0 3 1 The approximation of a random variable Y by an other one Y yields most often a : n v Dirichlet form. The framework is general, cf. Bouleau [3] whose results are recalled i X I.1 below. r § a Usually, when this Dirichlet form exists and does not vanish, the conditional law ofY givenY = y isnotreduced toaDiracmass, andthevarianceofthisconditional n law yields the square field operator Γ. On the other hand when the approximation is deterministic, i.e. when Y is a function of Y say Y = η (Y), then most often the n n n symmetric bias operator A˜ and the Dirichlet form vanish, cf. Bouleau [3] examples 2.1 to 2.9 and remark 5. Nevertheless, there are cases where the conditional law of Y given Y is a Dirac n mass, i.e. Y is a deterministic function of Y, and where the approximation of Y by n Y yields even so a non zero Dirichlet form on L2(P ). n Y This phenomenon is interesting, insofar as randomness (here the Dirichlet form) isgeneratedbyadeterministic device. Initssimplest form, thephenomenon appears ∗Ecole des Ponts, ParisTech, 28 rue des Saints Pères, 75007 Paris, France; e-mail : [email protected] 1 precisely when a quantity is measured by a graduated instrument to the nearest graduation when looking at the asymptotic limits as the graduation fines down. The first part of this article is devoted to functional analytical tools that we need afterwards. We first recall the properties of the bias operators and the Dirichlet form associated with an approximation. Next we prove a Girsanov-type theorem for Dirichlet forms which has its own interest, i.e. an answer to the question of an absolutely continuous change of measure for Dirichlet forms. At last we recall some simple properties of Rajchman measures. The second part is devoted to the case of a real or finite dimensional quantity measured with equidistant graduations. The mathematical argument here is basi- cally the arbitrary functions methodabout which we give a short historical comment. Several infinite dimensional extensions of the arbitrary functions principle are studied in the third part. The first one is about approximations of continuous mar- tingales whose brackets are Rajchman measures. Then we consider the case of the Wiener space on which the preceding results may be improved and other asymptotic properties are obtained concerning the approximation of the Ornstein-Uhlenbeck gradient. Eventually we apply these results to the approximation of stochastic differential equations encountered in mechanics and solved by the Euler scheme. I. Functional analytical tools. I.1. Approximation, Dirichlet forms and bias operators. Our study uses the theoretical framework concerning the bias operators and the Dirichlet form generated by an approximation proposed in Bouleau [3]. We recall here the definitions and main results for the convenience of the reader. Here, considered Dirichlet forms are always symmetric. Let Y be a random variable defined on (Ω, ,P) with values in a measurable A space (E, ) and let Y be approximations also defined on (Ω, ,P) with values n F A in (E, ). We consider an algebra of bounded functions from E into R or C F D containing the constants and dense in L2(E, ,P ) and a sequence α of positive Y n F numbers. With and (α ) we consider the four following assumptions defining the n D four bias operators ϕ , there exists A[ϕ] L2(E, ,P ) s.t. χ (H1) ∀ ∈ D ∈ F Y ∀ ∈ D lim α E[(ϕ(Y ) ϕ(Y))χ(Y)] = E [A[ϕ]χ]. n n n Y (cid:26) →∞ − ϕ , there exists A[ϕ] L2(E, ,P ) s.t. χ (H2) ∀ ∈ D ∈ F Y ∀ ∈ D lim α E[(ϕ(Y) ϕ(Y ))χ(Y )] = E [A[ϕ]χ]. n n n n Y (cid:26) →∞ − ϕ , there exists A[ϕ] L2(E, ,P ) s.t. χ (H3) ∀ ∈ D ∈ F Y ∀ ∈ D lim α E[(ϕ(Y ) ϕ(Y))(χ(Y ) χ(Y))] = 2E [A[ϕ]χ]. ( n n n n Y →∞ −e − − ϕ , there existsA[ϕ] L2(E, ,P ) s.t. χ (H4) ∀ ∈ D \ ∈ F Y ∀ ∈eD lim α E[(ϕ(Y ) ϕ(Y))(χ(Y )+χ(Y))] = 2E [A[ϕ]χ]. n n n n Y (cid:26) →∞ − \ 2 We first note that as soon as two of hypotheses (H1) (H2) (H3) (H4) are fulfilled (with the same algebra and the same sequence α ), the other two follow thanks n D to the relations A+A A A A = A = − . 2 \ 2 When defined, the operator A which considers the asymptotic error from the point e of view of the limit model, will be called the theoretical bias operator. The operator A which considers the asymptotic error from the point of view of the approximating model will be called the practical bias operator. Because of the property A[ϕ],χ = ϕ,A[χ] h iL2(PY) h iL2(PY) the operator A will be calleed the symmetric biaes operator. The operatorA which is often (see theorem 2 below) a first order operator will \ be called the seingular bias operator. Theorem 1. Under the hypothesis (H3), a) the limit α [ϕ,χ] = lim nE[(ϕ(Y ) ϕ(Y))(χ(Y ) χ(Y)] ϕ,χ (1) n n E n 2 − − ∈ D defines a celosable positive bilinear form whose smallest closed extension is denoted ( ,D). E b) ( ,D) is a Dirichlet form E c) ( ,D) admits a square field operator Γ satisfying ϕ,χ E ∀ ∈ D Γ[ϕ] = A[ϕ2] 2ϕA[ϕ] (2) − E [Γ[ϕ]χ] = limα E[(ϕ(Y ) ϕ(Y))2(χ(Y )+χ(Y))/2] (3) Y n en e n n − d) ( ,D) is local if and only if ϕ E ∀ ∈ D limα E[(ϕ(Y ) ϕ(Y))4] = 0 (4) n n n − this condition is equivalent to λ > 2 lim α E[ ϕ(Y ) ϕ(Y) λ] = 0. n n n ∃ | − | e) If the form ( ,D) is local, then the principle of asymptotic error calculus is E valid on = F(f ,...,f ) : f , F 1(Rp,R) i.e. 1 p i D { ∈ D ∈ C } lim αeE[(F(f (Y ),...,f (Y )) F(f (Y),...,f (Y))2] n n 1 n p n 1 p − = E [ p F (f ,...,f )F (f ,...,f )Γ[f ,f ]]. Y i,j=1 i′ 1 p j′ 1 p i j An operator B from into L2(P ) wPill be said to be a first order operator if it Y D satisfies B[ϕχ] = B[ϕ]χ+ϕB[χ] ϕ,χ ∀ ∈ D 3 Theorem 2. Under (H1) to (H4). If there is a real number p 1 s.t. ≥ limα E[(ϕ(Y ) ϕ(Y))2 ψ(Y ) ψ(Y) p] = 0 ϕ,ψ n n n n − | − | ∀ ∈ D thenA is first order. \ In particular, if the Dirichlet form is local, by the d) of theorem 1, the operator A is first order. \ I.2. Girsanov-type theorem for Dirichlet forms. An error structure is a probability space (Ω, ,P) equipped with a local Dirichlet A form with domain D dense in L2(Ω, ,P) admitting a square field operator Γ, see A Bouleau [2]. We denote A the domain of the associated generator. D Theorem 3. Let (Ω, ,P,D,Γ) be an error structure. Let be f D L such that ∞ A ∈ ∩ f > 0, Ef = 1. We put P = f.P. 1 a) The bilinear form defined on A L by 1 ∞ E D ∩ 1 [u,v] = E fvA[u]+ vΓ[u,f] (5) 1 E − 2 (cid:20) (cid:21) is closable in L2(P ) and satisfies for u,v A L 1 ∞ ∈ D ∩ 1 [u,v] = A u,v = u,A v = E[fΓ[u,v]] (6) 1 1 1 E −h i −h i 2 where A [u] = A[u]+ 1 Γ[u,f]. 1 2f b) Let (D , ) be the smallest closed extension of ( A L , ). Then D D , 1 1 ∞ 1 1 E D ∩ E ⊂ is local and admits a square field operator Γ , and 1 1 E Γ = Γ on D 1 in addition A A and A [u] = A[u]+ 1 Γ[u,f] for all u A. D ⊂ D 1 1 2f ∈ D Proof. 1) First, using that the resolvent operators are bounded operators sending L into A L , we see that A L is dense in D (equipped with the usual ∞ ∞ ∞ D ∩ D ∩ norm ( . 2 + [.])1/2), hence also dense in L2(P ). k kL2 E 1 2) Using that D L is an algebra, for u,v A L we have ∞ ∞ ∩ ∈ D ∩ 1 1 1 [u,v] = E[fvA[u]+ vΓ[u,f]] = E[Γ[fv,u] vΓ[u,f]] = E[fΓ[u,v]]. 1 E − 2 2 − 2 So, defining A as in the statement, we have u,v A L 1 ∞ ∀ ∈ D ∩ [u,v] = E [vA u] = E [uA v]. 1 1 1 1 1 E − − 4 The operator A is therefore symmetric on A L under P . Hence the form 1 ∞ 1 1 D ∩ E defined on A L is closable, (Fukushima et al. [5], condition 1.1.3 p 4). ∞ D ∩ 3) Let (D , ) be the smallest closed extension of ( A L , ). Let be u D 1 1 ∞ 1 E D ∩ E ∈ and u A L , with u u in D. Using [u u ] f [u u ] and n ∞ n 1 n m n m the clos∈edDness∩of we get u→ u in D , hencEe D −D . ≤Nokwkb∞yEusua−l inequali- 1 n 1 1 E → ⊂ ties we see that Γ[u ] is a Cauchy sequence in L1(P ) and that the limit Γ [u] does n 1 1 not depend on the particular sequence (u ) satisfying the above condition. Then n following Bouleau [2], Chap. III 2.5 p.38, the functional calculus extends to D , 1 § the axioms of error structures are fulfilled for (Ω, ,P ,D ,Γ ) and this gives with 1 1 1 A (cid:3) usual arguments the b) of the statement. I.3. Rajchman measures. In the whole paper, if x is a real number, [x] denotes the entire part of x and x = x [x] the fractional part. { } − Definition 1. A measure µ on the torus T1 is said to be Rajchman if µˆ = e2iπnxdµ(x) 0 when n . → | | ↑ ∞ T1 Z The set of Rajchman measures is a band : if µ and if ν µ then ν , R ∈ R ≪ | | ∈ R cf. Rajchman [18] [19], Lyons [15]. Lemma. Let X be a real random variable and let Ψ (u) = EeiuX be its character- X istic function. Then lim Ψ (u) = 0 P . X X u ⇐⇒ { } ∈ R | |→∞ Proof. a) If lim Ψ (u) = 0 then Ψ (2πn) = (P )ˆ(n) 0. u X X X b) Let ρ be a|p|→ro∞bability measure on T1 s.t. ρ {.}From→ ∈ R ∞ ((u [u])2iπx)p e2iπux = e2iπ[u]x − p! p=0 X we have ∞ ((u [u])2iπ)p e2iπuxρ(dx) = − a ([u]) p p! Z p=0 X with a (n) = xpe2iπnxρ(dx) hence a (n) 1 and lim a (n) = 0 since p p n p xpρ(dx) , so | | ≤ | |→∞ ∈ R R lim e2iπuxρ(dx) = 0. u | |→∞Z 5 Now if P , since 1 .P P we have X x [p,p+1[ X X { } ∈ R { ∈ } { } ≪ { } lim E[e2iπuX] = lim E[e2iπuX1 ] X [p,p+1[ u u { ∈ } | |→∞ | |→∞ p X (cid:3) which goes to zero by dominated convergence. A probability measure on R satisfying the conditions of the lemma will be called Rajchman. Examples. Thanks to the Riemann-Lebesgue lemma, absolutely continuous measures are in . It follows from the lemma that if a measure ν satisfies ν⋆ ⋆ν R ··· ∈ R then ν . There are singular Rajchman measures, cf. Kahane and Salem [13]. ∈ R The preceding definitions and properties extend to Td : a measure µ on Td is said to be in if µˆ(k) 0 as k in Zd. The set of measures in is a R → → ∞ R band. If X is Rd-valued, lim Eei u,X = 0 is equivalent to P where u h i X | |→∞ { } ∈ R x = ( x ,..., x ). 1 d { } { } { } II. Finite dimensional cases. d In the whole article = denotes the convergence in law, i.e. the convergence ⇒ of the probability laws on bounded continuous functions. The arbitrary functions principle may be stated as follows: Proposition 1. Let X,Y,Z be random variables with values in R, R, and Rm resp. Then d ( nX +Y ,X,Y,Z) = (U,X,Y,Z) (7) { } ⇒ where U is uniform on the unit interval independent of (X,Y,Z), if and only if P X is Rajchman. Proof. If µ is a probability measure on T1 Rm, let us put × µˆ(k,ζ) = e2iπkx+ ζ,y µ(dx,dy), h i Z then µ =d µ iff µˆ (k,ζ) µˆ(k,ζ) k Z, ζ Rm. n n ⇒ → ∀ ∈ ∀ ∈ a) If P X ∈ R Pˆ (k,ζ ,ζ ,ζ ) = E[exp 2iπk(nX +Y)+iζ X +iζ Y +i ζ ,Z ] ( nX+Y ,X,Y,Z) 1 2 3 1 2 3 { } { h i} = e2iπknxf(x)P (dx) X { } Z with f(x) = E[exp 2iπkY + iζ X + iζ Y + i ζ ,Z X = x]. The fact that 1 2 3 { h i}|{ } f.P gives the result. X { } ∈ R 6 b) Conversely, taking (k,ζ ,ζ ,ζ ) = (1,0, 2π,0) gives Pˆ (n) 0 i.e. P 1 2 3 X X − { } → (cid:3)∈ . R Let us suppose now that Y is an Rd-valued random variable, measured with an equidistant graduation corresponding to an orthonormal rectilinear coordinate system, and estimated to the nearest graduation component by component. Thus we put 1 Y = Y + θ(nY) n n with θ(y) = (1 y , , 1 y ). Let us emphasize that Y is a deterministic 2 − { 1} ··· 2 − { d} n function of Y. Theorem 4. a) If P is Rajchman and if X is Rm-valued Y d (X,n(Y Y)) = (X,(V ,...,V )) (8) n 1 d − ⇒ wherethe V ’s areindependentidenticallydistributed uniformlydistributed on( 1, 1) i −2 2 and independent of X. For all ϕ 1 lip(Rd) ∈ C ∩ d d (X,n(ϕ(Y ) ϕ(Y))) = (X, V ϕ (Y)) (9) n − ⇒ i ′i i=1 X d 1 n2E[(ϕ(Y ) ϕ(Y))2 Y =y] ϕ2(y) in L1(P ) (10) n − | → 12 ′i Y i=1 X in particular d 1 n2E[(ϕ(Y ) ϕ(Y))2] E [ ϕ2(y)]. (11) n − → Y 12 ′i i=1 X b) If ϕ is of class 2, the conditional expectation n2E[ϕ(Y ) ϕ(Y) Y = y] n C − | possesses a version n2(ϕ(y+ 1θ(ny)) ϕ(y)) independent of the probability measure n − P which converges in the sense of distributions to the function 1 ϕ. 24 △ c) If P dy on Rd, ψ L1([0,1]) Y ≪ ∀ ∈ d (X,ψ(n(Y Y))) = (X,ψ(V)). (12) n − ⇒ d) We consider the bias operators on the algebra 2 of bounded functions with Cb bounded derivatives up to order 2 with the sequence α = n2. If P and if one n Y ∈ R of the following condition is fulfilled i) i = 1,...,d the partial derivative ∂ P in the sense of distributions is a i Y ∀ measure P of the form ρ P with ρ L2(P ), Y i Y i Y ≪ ∈ ii) P = h1 dy with G open set, h H1 L (G), h > 0, Y G G ∈ ∩ ∞ | | 7 then hypotheses (H1) to (H4) are satisfied and A[ϕ] = 1 ϕ 24 △ A[ϕ] = 1 ϕ+ 1 ϕ ρ case i) 24 △ 24 ′i i A[ϕ] = 1 ϕ+ 1 1 h ϕ case ii) 24 △ 24hP ′i ′i Γe[ϕ] = 1 ϕ2. 12 ′i P e P Proof. The argument for relation (8) is similar to the one dimensional case stated in proposition 1. The relation (9) comes from the Taylor expansion ϕ(Y ) ϕ(Y) = n − = d (Y Y ) 1ϕ (Y ,...,Y ,Y +t(Y Y ),Y ,...,Y )dt i=1 n,i − i 0 ′i n,1 n,i−1 i n,i − i i+1 d and the convergence P R d (X, θ(nY )ϕ (Y)) = (X, ϕ (Y)V ) i ′i ⇒ ′i i i i X X thanks to (8) and the following approximation in L1 1 E θ(nY )ϕ (Y) θ(nY ) ϕ (...,Y +t(Y Y ),...)dt 0. i ′i − i ′i i n,i − i → (cid:12)(cid:12)Xi Xi Z0 (cid:12)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) To prove the formulas (10) and (11) let us remark that (cid:12) (cid:12) n2E[(ϕ(Y ) ϕ(Y)2 Y = y] = n − | 2 1 = E θ(nY ) ϕ (...,Y +t(Y Y ),...)dt Y = y  i ′i i n,i − i |  (cid:12)(cid:12)Xi Z0 (cid:12)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 2  (cid:12) 1 1 1 (cid:12) = θ(ny ) ϕ (y + θ(ny ),...,y +t θ(ny ),...)dt P -a.s. i ′i 1 n 1 i n i Y (cid:12)(cid:12)Xi Z0 (cid:12)(cid:12) (cid:12) (cid:12) each term(cid:12)(θ(ny ) 1ϕ (...)dt)2 converges to θ2ϕ2(y) = 1 ϕ2 in(cid:12)L1 and each term (cid:12) i 0 ′i ′i 12 ′i (cid:12) θ(ny )θ(ny ) 1... 1... goes to zero in L1 what proves the part a) of the statement. i j 0 R 0 R The part b) is obtained following the same lines with a Taylor expansion up to R R second order and an integration by part thanks to the fact that ϕ is now supposed to be 2. C In order to prove c) let us suppose first that P = 1 .dy. Considering a Y [0,1]d sequence of functions ψ tending to ψ in L1 we have the bound k b ∈ C E[eihu,Xieivψ(θ(nY))] E[eihu,Xieivψk(θ(nY))] | − | v ψ(θ(ny)) ψ (θ(ny)) dy k ≤ | | | − | = |v|R pn1−=10··· pp11+1···|ψ(θ(ny1)...)−ψk(θ(ny1)...)|dy1...dyd = v ψ(θ(x ),...) ψ (θ(x ),...) dx1 dxd | |P··· R··· | 1 − k 1 | n ··· n = v ψ ψ . k L1 | |kP− PkR R 8 This yields (12) in this case. Now if P dy then P dy on [0,1]d and the Y Y weak convergence under dy on [0,1]d impli≪es the weak co{n}ve≪rgence under P what Y { } yields the result. Ind) thepointi) isproved bytheapproachalreadyused inBouleau[3]consisting of proving that hypothesis (H3) is fulfilled by displaying the operator A thanks to an integration by parts. The point ii) is an application of the Girsanov-type theorem (cid:3) 3. e Remarks. 1) About the relations (9) (10) (11), let us note that with respect to the form 1 [ϕ] = E ϕ2 E 24 Y ′i i X when it is closable, the random variable V ϕ appears to be a gradient : if we i i ′i put ϕ# = V ϕ then a we have i i ′i P 1 P E[ϕ#2] = ϕ2 = Γ[ϕ] 12 ′i i X the square field operator associated to . We will find this phenomenon again on E the Wiener space. 2) Approximation to the nearest graduation, by excess, or by default. When the approximation is done to the nearest graduation, on the algebra 2 the four bias Cb operators are obtained in theorem 4 with the sequence α = n2, (with α = n the n n four bias operators would be zero). We would obtain a quite different result with an approximation by default or by excess because of the dominating effect of the shift. If the random variable Y is approximated by default by Y(d) = [nY] then n n 1 n(Y(d) Y) =d U and E[n(Y(d) Y)] n − ⇒ − n − → −2 as soon as Y is say bounded. With this approximation, if we do not erase the shift down proportional to 1 , andif we take α = n we obtain first order bias operators −2n n without diffusion : A[ϕ] = 1ϕ = A[ϕ] and A = 0. The same happens of course −2 ′ − with the approximation by excess. e 3) Extension to more general graduations. Let Y be an Rd-valued random variable approximated by Y = Y + ξ (Y) with a sequence α on the algebra n n n ↑ ∞ = e u,x , u Rd , the function ξ satisfying h i n D L{ ∈ } α E[ ξ 3(Y)] 0 n n | | →  α E[ϕ(Y) u,ξ (Y) 2] E [ϕ.u γu] ϕ , u Rd ( ) n h n i → Y ∗ ∀ ∈ D ∀ ∈ ∗  with γ L (P ) and ∂γij in distributions sense L2(P )  ij ∈ ∞ Y ∂xj ∈ Y  α E[ϕ(Y) u,ξ (Y) ] 0 ϕ .  n n  h i → ∀ ∈ D      9 Under these hypotheses we have Theorem 4bis. a) (H1) is satisfied and A[ϕ] = 1 γ ∂2ϕ . 2 ij ij∂xi∂xj b) If for i = 1,...,d, the partial derivative ∂ P in the sense of distributions is a i YP bounded measure of the form ρ P with ρ L2(P ) then assumptions (H1) to (H4) i Y i Y ∈ are fulfilled and ϕ ∀ ∈ D 1 ∂2ϕ ∂γ ∂ϕ ij A[ϕ] = γ + ( ( +γ ρ )) ij ij j 2 ∂x ∂x ∂x ∂x i j j i ij i j X X X e the square field operator is Γ[ϕ] = γ ∂ϕ ∂ϕ. ij ij∂xi∂xj P Proof. The argument is simple thanks to the choice of the algebra and consists D of elementary Taylor expansions to prove the existence of the bias operators. Then (cid:3) theorem 1 applies. Historical comment. In his intuitive version, the idea underlying the arbitrary functions method is ancient. The historian J. von Plato [16] dates it back to a book of J. von Kries [12]. We find indeed in this philosophical treatise the idea that if a roulette had equal and infinitely small black and white cases, then there would be an equal probability to fall on a case or on the neighbour one, hence by addition an equal probability to fall either on black or on white. But no precise proof was given. The idea remains at the common sense level. A mathematical argument for the fairness of the roulette and for the equi- distribution of other mechanical systems (little planets on the Zodiac) was proposed by H. Poincaré in his course on probability published in 1912 ([17], Chap. VIII 92 § and especially 93). In present language, Poincaré shows the weak convergence of § tX + Ymod 2π when t to the uniform law on (0,2π) when the pair (X,Y) ↑ ∞ has a density. He uses the characteristic functions. His proof supposes the density be 1 with bounded derivative in order to perform an integration by parts, but C the proof would extend to the general absolutely case if we were using instead the Riemann-Lebesgue lemma. The question is then developed without major changes by several authors, E. Borel [1] (case of continuous density), M. Fréchet [4] (case of Riemann-integrable density), B. Hostinski [9] [10] (bidimensional case) and is tackled anew by E. Hopf [6] , [7] and [8] with the more general point of view of asymptotic behaviour of dissipative dynamical systems. Hopf has shown that these phenomena are related to mixing and belong to the framework of ergodic theory. III. Infinite dimensional extensions of the arbitrary functions principle. 10