Диссертация (1137347), страница 9
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To explain the bound in (3.40) let us observe that the mth moment ofξ is finite for m < M − d.Equations (3.40) and (3.41) give the first part of the lemma. Still from the proof of Theorem 19.3in [BR76], one gets, under (A), that there exists C > 0 s.t.
for all multi-indexes ᾱ, |ᾱ| ≤ 4, β̄, |β̄| ≤m ≤ M − d − 5 for all j > k:Zno|ζ ᾱ | |Dζβ̄ q̂j−k (ζ)| + |Dζβ̄ q̂j−k,ε (ζ)| dζ ≤ C,(3.42)Rdwhere q̂j−k (ζ), q̂j−k,ε (ζ) stand for the respective characteristic functions of the random variablesy,(ε)yZ̃k,j, Z̃k,j at point ζ.To investigate the quantity |Dxα p̃h (tk , tj , x, y) − Dxα p̃hε (tk , tj , x, y)| thanks to (3.39) define now forall α, |α| ≤ 4, β, |β| ≤ m ≤ M − d − 5:∀z ∈ Rd , Θj−k,ε (z) := z β Dzα (qj−k (z) − qj−k,ε (z)) ,b j−k,ε (ζ) := (−i)|α|+|β| Dβ (ζ α {q̂j−k (ζ) − q̂j−k,ε (ζ)}) .∀ζ ∈ Rd , Θζ(3.43)Let us now estimate the difference between the characteristic functions. From the Leibniz formula,we are led to investigate for all multi-indexes β̄, ᾱ, |β̄| ≤ |β|, |ᾱ| ≤ |α| quantities of the form:(iβ̄ )−1 ζ ᾱ (Dζβ̄ q̂j−k (ζ) − Dζβ̄ q̂j−k,ε (ζ))ihy,(ε)y,(ε)yy β̄] − (Z̃k,j )β̄ exp[iζ · Z̃k,j ] .= ζ ᾱ E (Z̃k,j) exp[iζ · Z̃k,jyyyyyAssume first that j > k + 1.
In that case, set now Z̃k,j,1:= Z̃k,d(j+k)/2e, Z̃k,j,2:= Z̃k,j− Z̃k,j,1.y,(ε)y,(ε)y,(ε)y,(ε)Denoting similarly Z̃k,j,1 := Z̃k,d(j+k)/2e , Z̃k,j,2 := Z̃k,jy,(ε)− Z̃k,j,1 for the perturbed process, we get:(iβ̄ )−1 ζ ᾱ (Dζβ̄ q̂j−k (ζ) − Dζβ̄ q̂j−k,ε (ζ)) =(ihyyyyζ ᾱ E (Z̃k,j,1+ Z̃k,j,2)β̄ exp[iζ · Z̃k,j,1] exp[iζ · Z̃k,j,2]−Ehy,(ε)(Z̃k,j,1+(exp[iζ ·y,(ε)Z̃k,j,1 ] exp[iζ·y,(ε)Z̃k,j,2 ]i)hi hiyyyyCβ̄l E (Z̃k,j,1)l exp[iζ · Z̃k,j,1] E (Z̃k,j,2)β̄−l exp[iζ · Z̃k,j,2]X= ζ ᾱy,(ε)Z̃k,j,2 )β̄l,|l|≤|β̄|)hi hiy,(ε) ly,(ε)y,(ε) β̄−ly,(ε)(Z̃k,j,1 ) exp[iζ · Z̃k,j,1 ] E (Z̃k,j,2 )exp[iζ · Z̃k,j,2 ]X−Cβ̄l El,|l|≤|β̄|(=ζᾱXCβ̄lnh hihii hiy,(ε)y,(ε)yyyyE (Z̃k,j,1)l exp[iζ · Z̃k,j,1] − E (Z̃k,j,1 )l exp[iζ · Z̃k,j,1 ] E (Z̃k,j,2)β̄−l exp[iζ · Z̃k,j,2]l,|l|≤|β̄|)hih hihiioy,(ε) ly,(ε)y,(ε) β̄−ly,(ε)yyβ̄−l+ E (Z̃k,j,1 ) exp[iζ · Z̃k,j,1 ] E (Z̃k,j,2 )exp[iζ · Z̃k,j,2 ] − E (Z̃k,j,2 )exp[iζ · Z̃k,j,2,35where in the above expression we considered the binomial expansion for multi-indexes denoting byβ̄!Cβ̄l := (β̄−l)!l!with the corresponding definitions for factorials (see the proof of Lemma 3.1.5).
Introduce now, for a multi-index l, |l| ∈ [[0, |β̄|]], the functions:hihiy,(ε) ly,(ε)β̄−lyyᾱΨᾱ,(ζ) := ζ ᾱ E (Z̃k,j,2)β̄−l exp[iζ · Z̃k,j,2] , Ψᾱ,l12 (ζ) := ζ E (Z̃k,j,1 ) exp[iζ · Z̃k,j,1 ] ,andh hihiiy,(ε)y,(ε)yyE1,l (ζ) := E (Z̃k,j,1)l exp[iζ · Z̃k,j,1] − E (Z̃k,j,1 )l exp[iζ · Z̃k,j,1 ] ,h hihiiy,(ε)y,(ε)yyE2,β̄−l (ζ) := E (Z̃k,j,2)β̄−l exp[iζ · Z̃k,j,2] − E (Z̃k,j,2 )β̄−l exp[iζ · Z̃k,j,2 .Thus, we can rewrite from the previous computations:(iβ̄ )−1 ζ ᾱ (Dζβ̄ q̂j−k (ζ) − Dζβ̄ q̂j−k,ε (ζ)) =on)(ζ).Cβ̄l (E1,l Ψ1ᾱ,β̄−l )(ζ) + (E2,β̄−l Ψᾱ,l2X(3.44)l,|l|≤|β̄|Recall from (3.42) that we already have integrability for the contributions Ψ1ᾱ,β̄−l (ζ) and Ψᾱ,l2 (ζ).
Letus thus start with the control of E1,l (ζ), E2,β̄−l (ζ). We only give details for E1,l (ζ), the contributionE2,β̄−l can be handled similarly. We also consider |l| ≥ 2, since the cases |l| ≤ 2 can be handled moredirectly. Write:|E1,l (ζ)| ≤yE[|(Z̃k,j,1)ly,(ε)(Z̃k,j,1 )l |]y,(ε)E[|(Z̃k,j,1 )l || exp(iζyZ̃k,j,1)y,(ε)−+·− exp(iζ · Z̃k,j,1 )|]noy,(ε)y,(ε)y,(ε)y,(ε)yyy≤ C E[|Z̃k,j,1− Z̃k,j,1 |(|Z̃k,j,1||l|−1 + |Z̃k,j,1 ||l|−1 ] + E[|Z̃k,j,1 ||l| |ζ||Z̃k,j,1− Z̃k,j,1 |] .Apply now Hölder’s inequality with p1 = |l|, q1 = |l|/(|l| − 1) for the first term and p2 = (|l| +1)/|l|, q2 = |l| + 1 for the second one so that all the contribution appear with the same power (inorder to equilibrate the constraints concerning the integrability conditions).
One gets:|E1,l (ζ)| ≤ny,(ε) |l| 1/|l|y,(ε) |l| (|l|−1)/|l|yy|l| (|l|−1)/|l|C E[|Z̃k,j,1 − Z̃k,j,1 | ]{E[|Z̃k,j,1 | ]+ E[|Z̃k,j,1 | ]}+oy,(ε)y,(ε)y|ζ|E[|Z̃k,j,1 ||l|+1 ]|l|/(|l|+1) E[|Z̃k,j,1− Z̃k,j,1 ||l|+1 ]1/(|l|+1) .(3.45)The point is now to prove, since we have assumed m ≤ M − d − 5 ⇐⇒ m + 1 ≤ M − d − 4, thatthere exists c s.t. for all r ≤ m + 1,y,(ε)y,(ε)yyE[|Z̃k,j,1− Z̃k,j,1 |r ]1/r ≤ c∆ε,σ,γ , E[|Z̃k,j,1|r ]1/r + E[|Z̃k,j,1 |r ]1/r ≤ c.(3.46)Let us establish the pointfor the difference, the other bounds can be derived similarly. Define for√ Pi−1all i ∈ [[k, j]], M̃i := h r=k (σ − σε )(tr , y)ξr+1 . The process (M̃i )i∈[[k,j]] is a square integrablemartingale (in discrete time, w.r.t.
Fi := Σ(ξr , r ≤ i), Σ-field generated by the innovation up toPi−1the current time). Its quadratic variation writes [M̃ ]i = h r=k |(σ − σε )(tr , y)|2 |ξr+1 |2 and theBurkholder-Davis-Gundy inequalities, see e.g. Shiryaev [Shi96], give for all r ≤ M − d − 4:rE[ sup |M̃i | ] ≤i∈[[k,j]]r/2cr E[[M̃ ]j ]r/2= cr hj−1XE[(|(σ − σε )(ti , y)|2 |ξi+1 |2 )r/2 ].i=k36(3.47)If r = 2 one readily gets:j−1y,(ε)yE[|Z̃k,j,1− Z̃k,j,1 |2 ] ≤Xc2c2 hE[ sup |M̃i |2 ] ≤∆2ε,σ,γE[|ξi+1 |2 ] ≤ c̄2 ∆2ε,σ,γ .(tj − tk ) i∈[[k,j]](tj − tk )i=kLet us thus assume r > 2 and derive from (3.47)y,(ε)yE[|Z̃k,j,1− Z̃k,j,1 |r ] ≤≤crE[ sup |M̃i |r ](tj − tk )r/2 i∈[[k,j]]j−1j−1XXcr hr/2rrE[(|(σ−σ)(t,y)||ξ|)(1)r/2(1−2/r) ],εii+1(tj − tk )r/2i=ki=kapplying Hölder’s inequality for the counting measure with p = r/2, q = r/(r − 2) for the lastinequality. This finally gives:j−1yE[|Z̃k,j,1−y,(ε)Z̃k,j,1 |r ]Xcr hr/2r/2−1 r(j−k)∆E[|ξi+1 |r ] ≤ c̄r ∆rε,σ,γ .≤ε,σ,γ(tj − tk )r/2i=kSince we have assumed r ≤ m + 1 ≤ M − d − 4, this gives the first control in (3.46).
The other onereadily follows replacing σ − σε by σ or σε .From equations (3.45), (3.46) and similar controls for E2,β̄−l (ζ) we finally derive:|E1,l (ζ)| + |E2,β̄−l (ζ)| ≤ C1 ∆ε,σ,γ (1 + |ζ|).As a result we have from (3.43) and (3.44):|Dζβ (ζ α (q̂j−k (ζ) − Dζβ q̂j−k,ε (ζ)))|)(≤ C∆ε,σ,γXXβ̄, |β̄| ≤ |β|ᾱ = α − (β − β̄).l,|l|≤|β̄|(|Ψ1ᾱ,β̄−l (ζ)| + |Ψᾱ,l2 (ζ)|)(1 + |ζ|) .We finally derive from (3.43) and (3.42) (which thanks to the smoothness assumption on QM in(IP) holds as well for a multi-index ᾱ, |ᾱ| = 5):Z1|Θj−k,ε (z)| ≤|Θ̂j−k,ε (ζ)|dζ ≤ c∆ε,σ,γ .(3.48)(2π)d RdFrom (3.39) this concludes the proof for j > k + 1.
If j = k + 1 the previous arguments can besimplified and lead to the same results.Comparison of the parametrix kernelsThis step is crucial and actually the key to obtain the result for the Markov chains. We focus forsimplicity on the case q = +∞, for which pointwise controls for the differences between the driftcoefficients are available, and which already emphasizes all the difficulties. The case q ∈ (d, +∞) forthe drifts could be handled as in Lemma 3.1.6, using similar Hölder inequalities.We actually have the following Lemma.Lemma 3.1.9 (Control of the One-Step Convolution for the Chain.).
There exists c1 , c s.t. for allq = +∞ and for 0 ≤ tk < tj ≤ T, (z, y) ∈ (Rd )2 :|(H h − Hεh )(tk , tj , z, y)| ≤∆ε,γ,∞Φc,c1 (tj − tk , z − y),(tj − tk )1−γ/2with37- Φc,c1 (tj − tk , z − y) = ψc,c1 (tj − tk , z − y) under (IG),γ- Φc,c1 (tj − tk , z − y) = ψc,c1 (tj − tk , z − y) 1 + (tj|z−y|, under (IP) ,−tk )1/2where ψc,c1 is defined according to the assumptions on the innovations in Lemma 3.1.8.Proof.
The case j = k + 1 involves directly differences of densities and could be treated more directlythan the case j > k + 1. We thus focus on the latter. Introduce for k ∈ [[0, N ]], (x, w) ∈ (Rd )2 the onestep transitions:T h (tk , x, w) := b(tk , x)h + h1/2 σ(tk , x)w, Tεh (tk , x, w) := bε (tk , x)h + h1/2 σε (tk , x)w,hT0h (tk , x, w) := h1/2 σ(tk , x)w, T0,ε(tk , x, w) := h1/2 σε (tk , x)w.(3.49)From the definition of H h , Hεh , recalling that fξ stands for the density of the innovation, the differenceof the kernels writes:= h−1ZRd(H h − Hεh )(tk , tj , z, y)nodwfξ (w) p̃h (tk+1 , tj , z + T h (tk , z, w), y) − p̃h (tk+1 , tj , z + T0h (tk , y, w), y)"#nohhhh− p̃ε (tk+1 , tj , z + Tε (tk , z, w), y) − p̃ε (tk+1 , tj , z + T0,ε (tk , y, w), y) .(3.50)Let us now perform a Taylor expansion at order 2 with integral rest.
To this end, let us first introducefor λ ∈ [0, 1] the mappings:ϕhλ : Rd × Rd−→(T1 , T2 )7−→ϕhλ,ε : Rd × Rd−→(T1 , T2 )7−→RTr Dz2 p̃h (tk+1 , tj , z + λT1 , y)[T2 T2∗ ] ,RTr Dz2 p̃hε (tk+1 , tj , z + λT1 , y)[T2 T2∗ ] ,(3.51)where T2 is viewed as a column vector and T2∗ denotes its transpose. Recalling as well that ξ iscentered we get:="D∆H h,ε (tk , tj , z, y) := (H h − Hεh )(tk , tj , z, y)#E DEhhDz p̃ (tk+1 , tj , z, y), b(tk , z) − Dz p̃ε (tk+1 , tj , z, y), bε (tk , z)+h−1ZRd"×Z1dλ(1 − λ)dwfξ (w)0noϕhλ (T h (tk , z, w), T h (tk , z, w)) − ϕhλ (T0h (tk , y, w), T0h (tk , y, w))#nohhhhhh− ϕλ,ε (Tε (tk , z, w), Tε (tk , z, w)) − ϕλ,ε (T0,ε (tk , y, w), T0,ε (tk , y, w))=: (∆1 H h,ε + ∆2 H h,ε )(tk , tj , z, y),38(3.52)where for i ∈ {1, 2}, ∆i H h,ε is associated with the terms of order i. The idea is now to make ∆ε,γ,∞appear explicitly. The term ∆1 H h,ε is the easiest to handle.
We can indeed readily write:="D∆1 H h,ε (tk , tj , z, y)#E DEDz p̃h (tk+1 , tj , z, y), [b(tk , z) − bε (tk , z)] − (Dz p̃hε − Dz p̃h )(tk+1 , tj , z, y), bε (tk , z) .From Assumption (A3), equation (3.4) and Lemma 3.1.8 we derive for q = +∞:|∆1 H h,ε (tk , tj , z, y)| ≤C∆ε,γ,∞ψc,c1 (tj − tk , y − z).(tj − tk )1/2(3.53)The term ∆2 H h,ε is trickier to handle. Define to this end:nohhhhhh∆ϕh,ελ (tk , z, y, w) := ϕλ (T (tk , z, w), T (tk , z, w)) − ϕλ (T0 (tk , y, w), T0 (tk , y, w))nohh− ϕhλ,ε (Tεh (tk , z, w), Tεh (tk , z, w)) − ϕhλ,ε (T0,ε(tk , y, w), T0,ε(tk , y, w)) .Let us then decompose:"∆ϕh,ελ (tk , z, y, w):=noϕhλ (T h (tk , z, w), T h (tk , z, w)) − ϕhλ (T h (tk , z, w), T0h (tk , y, w))#nohhhhhh− ϕλ,ε (Tε (tk , z, w), Tε (tk , z, w)) − ϕλ,ε (Tε (tk , z, w), T0,ε (tk , y, w))"+noϕhλ (T h (tk , z, w), T0h (tk , y, w)) − ϕhλ (T0h (tk , y, w), T0h (tk , y, w))#nohhhhhh− ϕλ,ε (Tε (tk , y, w), T0,ε (tk , y, w)) − ϕλ,ε (T0,ε (tk , z, w), T0,ε (tk , y, w))h,ε=: (∆1 ϕh,ελ + ∆2 ϕλ )(tk , z, y, w),(3.54)and write from (3.52):∆2 H h,ε (tk , tj , z, y)==:h−1ZZdwfξ (w)Rdh,ε(∆21 H+ ∆22 H1h,εdλ(1 − λ)(∆1 ϕh,ελ + ∆2 ϕλ )(tk , z, y, w)0h,ε)(tk , tj , z, y),(3.55)for the associated contributions in ∆2 H h,ε .
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