Диссертация (1137347), страница 8
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and each (Γi )i∈[[1,d]] belongs to Rd . Also, we have denoted:. ΓdΓ1:d := Γ1 Γ2 · · · Γd ,(3.27)the d × d matrix formed with the entries (Γi )i∈[[1,d]] , each entry being viewed as a column.The multidimensional Taylor expansion now gives:|(p̃ − p̃ε )(s, t, x, y)| = |fx,y (Θ(s, t, y)) − fx,y (Θε (s, t, y))| XDν fx,y (Θ(s, t, y)){(Θε − Θ)(s, t, y)}ν= |ν|=1(3.28)X {(Θε − Θ)(s, t, y)}ν Z 1ν+2(1 − δ)D fx,y ([Θ + δ(Θε − Θ)](s, t, y))dδ ,ν!0|ν|=2Pd22where for a multi-index ν := (ν1 , · · · , νd2 ) ∈ Nd , we denote by |ν| :=i=1 νi the length of the22QdQd2multi-index, ν! = i=1 νi ! and for h ∈ Rd , hν := i=1 hνi i (with the convention that 00 = 1).
Withthese notations, from (3.26), (3.27), (3.28) and Assumption (A4) we get: X|Dν fx,y (Θ(s, t, y))|∆ε,σ,γ (t − s)|fx,y (Θ(s, t, y)) − fx,y (Θε (s, t, y))| ≤ c|ν|=1+ ∆2ε,σ,γ (t − s)2 maxδ∈[0,1]X|Dν fx,y ([Θ + δ(Θε − Θ)](s, t, y))| .|ν|=2(3.29)Since fx,y in (3.27) is a Gaussian density in the parameters x, y, we recall from Cramer and2Leadbetter [CL04] (see eq. (2.10.3) therein), that for all Γ ∈ Rd and any multi index ν, |ν| ≤ 2: 2d2Y∂1)νi fx,y (Γ) ,Dν fx,y (Γ) = |ν| (∂∂2xxi−1i−1bc+1i−bcdi=1ddwhere b·c stands for the integer part.
Hence, taking from (3.29), for all δ ∈ [0, 1], Γε,δ (s, t, y) :=[Θ + δ(Θε − Θ)](s, t, y) yields, thanks to the non-degeneracy conditions (see equation (3.19)):|Dν fx,y (Γε,δ (s, t, y))| ≤c̄1c̄1fx,y c̄Γε,δ (s, t, y) ≤pc̄ (t − s, y − x),|ν|(t − s)(t − s)|ν|for some c̄1 ≥ 1, c̄ ∈ (0, 1].Thus, from (3.27), (3.28), equations (3.29) and (3.30) give:|p̃(s, t, x, y) − p̃ε (s, t, x, y)| ≤ c̄1 ∆ε,σ,γ pc̄ (s, t, x, y).30(3.30)Up to a modification of c̄1 , c̄ or c1 , c in (3.19) we can assume that the statement of the lemma and(3.19) hold with the same constants c1 , c. The bounds for the derivatives are established similarlyusing the controls of (3.19). This concludes the proof.Remark 3.1.2. Observe from equation (3.28) that the previous Lemma still holds with ∆ε,σ,γ replacedby ∆ε,σ,∞ := supt∈[0,T ] |σ(t, .) − σε (t, .)|∞ .
The Hölder norm is required to control the differences ofthe parametrix kernels.The previous lemma quantifies how close are the main parts of the expansions. To proceed we needto consider the difference between the one-step convolutions. Combining the estimates of Lemmas3.1.5 and 3.1.6 below will yield by induction the result stated in Theorem 3.1.1.Lemma 3.1.6 (Control of the one-step convolution).
For all 0 ≤ s < t ≤ T, (x, y) ∈ (Rd )2 and forq ∈ (d, +∞]:|p̃ ⊗ H (1) (s, t, x, y) − p̃ε ⊗ Hε(1) (s, t, x, y)|nγγ≤ c21 pc (s, t, x, y) 2(1 ∨ T (1−γ)/2 )2 [∆ε,σ,γ + Iq=∞ ∆ε,b,+∞ ]B 1,(t − s) 22o1+Iq∈(d,+∞) ∆ε,b,q B+ α(q), α(q) (t − s)α(q) ,2(3.31)where c1 , c are as in Lemma 3.1.5 and for q ∈ (d, +∞] we set α(q) = 21 (1 − dq ). The above controlthen yields for a fixed q ∈ (d, +∞]:γ≤ 2C̄ 2 ∆ε,γ,q pc (s, t, x, y)(t − s) 2 ∧α(q) B 1,γ2!|p̃ ⊗ H (1) (s, t, x, y) − p̃ε ⊗ Hε(1) (s, t, x, y)|!1∨B+ α(q), α(q), C̄ = c1 (1 ∨ T (1−γ)/2 ),2(3.32)which will be useful for the iteration (see Lemma 3.1.7).Proof. Let us write:|p̃ ⊗ H (1) (s, t, x, y) − p̃ε ⊗ Hε(1) (s, t, x, y)| ≤|p̃ − p̃ε | ⊗ |H|(s, t, x, y) + p̃ε ⊗ |H − Hε |(s, t, x, y) =: I + II.(3.33)From Lemma 3.1.5 and (3.19) we readily get for all q ∈ (d, +∞]:|p̃ − p̃ε | ⊗ |H|(s, t, x, y)| ≤ ((1 ∨ T(1−γ)/2γγ(t − s) 2 .)c1 ) ∆ε,γ,q pc (t − s, y − x)B 1,22(3.34)Now we will establish that for all 0 ≤ u < t ≤ T, (z, y) ∈ (Rd )2 and q = +∞:|(H − Hε )(u, t, z, y)| ≤ ∆ε,γ,∞(1 ∨ T (1−γ)/2 )c1pc (t − u, y − z).γ(t − u)1− 2(3.35)Equations (3.35) and (3.18) give that II can be handled as I which yields the result for q = +∞.
Ittherefore remains to prove (3.35). Let us write with the notations of (3.27):12(H − Hε )(u, t, z, y) =Tr (a(u, z) − a(u, y))Dz fz,y Θ(u, t, y) + hb(u, z), Dz fz,y Θ(u, t, y) i212− Tr (aε (u, z) − aε (u, y))Dz fz,y Θε (u, t, y) + hbε (u, z), Dz fz,y Θε (u, t, y) i .231Thus, 1Tr (a(u, z) − a(u, y)){Dz2 fz,y Θ(u, t, y) − Dz2 fz,y Θε (u, t, y) }2− Tr [(aε (u, z) − aε (u, y) − (a(u, z) − a(u, y))]Dz2 fz,y Θε (u, t, y)+hb(u, z), {Dz fz,y Θ(u, t, y) − Dz fz,y Θε (u, t, y) }i− h[(bε (u, z) − b(u, z))], Dz fz,y Θε (u, t, y) i .(3.36)(H − Hε )(u, t, z, y)=Observe now that, similarly to (3.30) one has for all i ∈ {1, 2}:c̃1pc̃ (t − u, y − z),(t − u)i/2c̃1 ∆ε,σ,γpc̃ (t − u, y − z).Θε (u, t, y) | ≤(t − u)i/2|Dzi fz,y Θ(u, t, y) | + |Dzi fz,y Θε (u, t, y) | ≤|Dzi fz,y Θ(u, t, y) − Dzi fz,yAlso,|(aε (u, z) − aε (u, y) − (a(u, z) − a(u, y))| ≤ c∆ε,σ,γ |z − y|γ ,|b(u, z) − bε (u, z)| ≤ c∆ε,b,∞ .Thus, provided that c1 , c have been chosen large and small enough respectively in Lemma 3.1.5, thedefinition in (3.4) gives:|(H − Hε )(u, t, z, y)| ≤(1 ∨ T (1−γ)/2 )c1 ∆ε,γ,∞pc (t − u, y − z).(t − u)1−γ/2This establishes (3.35) for q = +∞.
For q ∈ (d, +∞) we have to use Hölder’s inequality in thetime-space convolution involving the difference of the drifts (last term in (3.36)). Set:Z t ZD(s, t, x, y) :=dup̃ε (s, u, x, z)h[(bε (u, z) − b(u, z))], Dz fz,y Θε (u, t, y) idz.RdsDenoting by q̄ the conjugate of q, i.e. q, q̄ > 1, q −1 + q̄ −1 = 1, we get from (3.18) and for q > dthat:|D(s, t, x, y)|≤ c21ZstZo1/q̄kb(u, .) − bε (u, .)kLq (Rd ) du n[pc (u − s, z − x)pc (t − u, y − z)]q̄ dz1/2(t − u)RdnRo1/q̄Z t cdp (u − s, z − x)pcq̄ (t − u, y − z)dzduRd cq̄≤ c21 ∆ε,b,q1d11d1(2π)d(1− q̄ ) (cq̄)d/q̄ (u − s) 2 (1− q̄ ) (t − u) 2 + 2 (1− q̄ ) d (1− q̄1 )Z tdc(t − s) 2du≤ c21q̄ − 2q̄ ∆ε,b,q pc (t − s, y − x)d1d1 .(1− q̄1 )2π2s (u − s)(t − u) 2 + 2 (1− q̄ )sNow, the constraint d < q < +∞ precisely gives that 1 < q̄ < d/(d − 1) ⇒that the last integral is well defined.
We therefore derive:12+ d2 (1 − 1q̄ ) < 1 so1d1d1 1 d1|D(s, t, x, y)| ≤ c21 (t − s) 2 − 2 (1− q̄ ) ∆ε,b,q pc (t − s, y − x)B(1 − (1 − ), − (1 − )).2q̄ 2 2q̄32In the case d < q < +∞, recalling that α(q) = 21 (1 − dq ), we eventually get :1p̃ε ⊗ |H − Hε |(s, t, x, y) ≤ c21 pc (t − s, y − x){∆ε,b,q (t − s)α(q) B( + α(q), α(q))2+2∆ε,σ,γ (1 ∨ T (1−γ)/2 )(t − s)γ/2 B(1, γ/2)}.(3.37)The statement now follows in whole generality from (3.33), (3.34), equations (3.35), (3.18) forq = ∞ and (3.37) for d < q < +∞.The following Lemma associated with Lemmas 3.1.5 and 3.1.6 allows to complete the proof ofTheorem 3.1.1.Lemma 3.1.7 (Difference of the iterated kernels). For all 0 ≤ s < t ≤ T, (x, y) ∈ (Rd )2 and for allq ∈ (d, +∞], r ∈ N:rC̄ r+1 Γ( γ2 ∧ α(q))γ|(p̃ ⊗ H (r) − p̃ε ⊗ Hε(r) )(s, t, x, y)| ≤ (r + 1)∆ε,γ,qpc (t − s, y − x)(t − s)r( 2 ∧α(q)) .Γ(1 + r( γ2 ∧ α(q)))(3.38)where c, c1 are as in Lemma 3.1.5 and C̄ as in Lemma 3.1.6.Proof.
Observe that Lemmas 3.1.5 and 3.1.6 respectively give (3.38) for r = 0 and r = 1. Let usassume that it holds for a given r ∈ N∗ and let us prove it for r + 1.(r)Let us denote for all r ≥ 1, ηr (s, t, x, y) := |(p̃ ⊗ H (r) − p̃ε ⊗ Hε )(s, t, x, y)|. Writeηr+1 (s, t, x, y) ≤ |[p̃ ⊗ H (r) − p̃ε ⊗ Hε(r) ] ⊗ H(s, t, x, y)| + |p̃ε ⊗ Hε(r) ⊗ (H − Hε )(s, t, x, y)|≤ ηr ⊗ |H|(s, t, x, y) + |p̃ε ⊗ Hε(r) | ⊗ |(H − Hε )|(s, t, x, y).(r)Recall now that under (A), the terms |H|(s, t, x, y) and |p̃ε ⊗ Hε | satisfy respectively and uniformlyin ε the controls of equations (3.19), (3.20).
The result then follows from the proof of Lemma 3.1.6(see equation (3.35) for q = ∞ and (3.37) for q ∈ (d, +∞)) and the induction hypothesis.Theorem 3.1.1 now simply follows from the controls of Lemma 3.1.7, the parametrix expansions(3.17) and (3.18) of the densities p, pε and the asymptotic of the gamma function.3.1.5Stability for Markov Chains.In this Section we prove Theorem 3.1.2. The strategy is rather similar to the one of Section 3.1.4thanks to the series representation of the densities of the chains given in Proposition 3.1.4.Recall first from Section 3.1.3 that we have the following representations for the density ph , phε ofthe Markov chains Y, Y (ε) in (3.3).
For all 0 ≤ ti < tj ≤ T, (x, y) ∈ (Rd )2 :ph (ti , tj , x, y) =j−iXp̃h ⊗h H h,(r) (ti , tj , x, y),r=0phε (ti , tj , x, y) =j−iXp̃hε ⊗h Hεh,(r) (ti , tj , x, y).r=033Comparison of the frozen densitiesThe first key point for the analysis with Markov chains is the following Lemma.Lemma 3.1.8 (Controls and Comparison of the densities and their derivatives). There exist c, c1s.t. for all 0 ≤ ti < tj ≤ T, (x, y) ∈ (Rd )2 and for all multi-index α, |α| ≤ 4:1ψc,c1 (tj − ti , y − x),(tj − ti )|α|/2∆ε,σ,γ|Dxα p̃h (ti , tj , x, y) − Dxα p̃hε (ti , tj , x, y)| ≤ψc,c1 (tj − ti , y − x),(tj − ti )|α|/2|Dxα p̃h (ti , tj , x, y)| + |Dxα p̃hε (ti , tj , x, y)| ≤where- Under (IG):ψc,c1 (tj − ti , y − x) := c1 pc (tj − ti , y − x),- Under (IP):c1 cdQM −d−5ψc,c1 (tj − ti , y − x) :=(tj − ti )d/2|y − x|(tj − ti )1/2 /c.Proof.
Note first that under (IG) the statement has already been proved in Lemma 3.1.5. We thusassume that (IP) holds. Introduce first the random vectors with zero mean:yZ̃k,j:=j−1j−1XX√√11y,(ε)σ(t,y)hξ,Z̃:=σε (tl , y) hξl+1 .ll+1k,j1/21/2(tj − tk )(tj − tk )l=kl=kDenoting by qj−k , qj−k,ε their respective densities, one has:Dxα p̃h (tk , tj , x, y)=Dxα p̃hε (tk , tj , x, y)=1(−1)|α| Dzα qj−k (z)|z= y−x ,(tj − tk )(d+|α|)/2(tj −tk )1/21(−1)|α| Dzα qj−k,ε (z)|z= y−x .(tj − tk )(d+|α|)/2(tj −tk )1/2(3.39)From the Edgeworth expansion of Theorem 19.3 in Bhattacharya and Rao [BR76], for qj−k , qj−k,ε , onereadily derives under (A), for |α| = 0 that there exists c1 s.t. for all 0 ≤ tk < tj ≤ T, (x, y) ∈ (Rd )2 ,p̃h (tk , tj , x, y) + p̃hε (tk , tj , x, y) ≤c1(tj − tk )d/2 1 +1|x−y|(tj −tk )1/2m ,(3.40)for all integer m < M − d, where we recall that M stands for the initial decay of the density fξ ofthe innovations bounded by QM (see equation (3.6)).We can as well derive similarly to the proof of Theorem 19.3 in [BR76], see also Lemma 3.7 in[KM00], that for all α, |α| ≤ 4:|Dxα p̃h (tk , tj , x, y)| + |Dxα p̃hε (tk , tj , x, y)| ≤c1(tj − tk )(d+|α|)/2 1 +1|x−y|(tj −tk )1/2m ,(3.41)for all m < M − d − 4.
Note indeed that differentiating in Dxα the density and the terms of theEdgeworth expansion corresponds to a multiplication of the Fourier transforms involved by ζ α , ζstanding for the Fourier variable. Hence, from our smoothness assumptions in (IP), after obvious34modifications, the estimates of Theorem 9.11 and Lemma 14.3 from [BR76] apply for these derivatives.With these bounds, one then simply has to copy the proof of Theorem 19.3. Roughly speaking, takingderivatives deteriorates the concentration of the initial control in (3.40) up to the derivation order. Onthe other hand, the bound in (3.40) is itself deteriorated w.r.t. the initial concentration condition in(3.6). The key point is that the techniques of Theorem 19.3 in [BR76] actually provide concentrationbounds for inhomogeneous sums of random variables with concentration as in (3.6) in terms of themoments of the innovations.
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