Диссертация (1137347), страница 7
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if T = 1, 0 = tn0 < tn1 < · · · < tnn = 1.Introducing the contrast:U n (η) :=ni1 Xhlog(det(a(η, tni−1 ))) + ha−1 (η, tni−1 , Xtni−1 )Xin , Xin i ,n i=1Xtn − Xtni−1,∀i ∈ [1, n], Xin := pi nti − tni−1and denoting by η̂n √the corresponding minimizer, it was shown by Genon-Catalot and Jacod[GCJ93] that under Pη , n(η̂n − η) converges in law towards a mixed normal variable S which is,conditionally to F1 := σ[(Xs )s∈[0,1] ], centred and Gaussian. For a precise expression of the covariancewhich depends on the whole path of (Xt )t∈[0,1] we refer to Theorem 3 and its proof in [GCJ93].This means that, when n is large, conditionally to F1 , we have on a subset Ω̄ ⊂ Ω which has highprobability, that |η̂n − η| ≤ √Cn for a certain threshold C.
Setting εn = n−1/2 , σεn (t, x) := σ(η̂n , t, x)and with a slight abuse of notation σ(t, x) := σ(η, t, x), one gets that, on Ω̄:|σ(t, x) − σεn (t, x) − (σ(t, y) − σεn (t, y))| ≤ |x − y| ∧ Cn−1/2⇒ |σ − σε |ϑ ≤ (Cn−1/2 )1−ϑ , ϑ ∈ (0, 1].We can then invoke our Theorem 3.1.2 to compare the densities of the diffusions with the estimatedparameter and the exact one in (3.11).3.1.3Derivation of formal series expansion for densitiesParametrix Representation of the Density for DiffusionsIn the following, for given (s, x) ∈ R+ × Rd , we use the standard Markov notation (Xts,x )t≥s to denotethe solution of (3.1) starting from x at time s.Assume again that (Xts,x )t≥s has for all t > s a smooth density p(s, t, x, .) (which is the case ifadditionally to (A) the coefficients are smooth see e.g.
Friedman [Fri64]). We would like to estimatethis density at a given point y ∈ Rd . To this end, we again use the parametrix expansion with respectto the density of the frozen process with spatial variable frozen at y. For all (s, x) ∈ [0, T ] × Rd , t ≥ swe set for this Chapter:ZtX̃ty = x +σ(u, y)dWu .sIts density p̃y readily satisfies the Kolmogorov Backward equation:∂u p̃y (u, t, z, y) + L̃yu p̃y (u, t, z, y) = 0, s ≤ u < t, z ∈ Rd ,p̃y (u, t, ., y) → δy (.),u↑t25(3.12)where for all ϕ ∈ C02 (Rd , R), z ∈ Rd :L̃yu ϕ(z) =1Tr σσ ∗ (u, y)Dz2 ϕ(z) ,2stands for the generator of X̃ y at time u.On the other hand, since we have assumed the transition density of X to be smooth, it must satisfythe Kolmogorov forward equation (see e.g. Dynkin [Dyn65]).
For a given starting point x ∈ Rd attime s,∂u p(s, u, x, z) = L∗u p(s, u, x, z) = 0, s < u ≤ t, z ∈ Rd ,(3.13)p(s, u, x, .) → δx (.),u↓swhere L∗u stands for the formal adjoint (which is again well defined if the coefficients in (3.1) aresmooth) of the generator of (3.1) which for all ϕ ∈ C02 (Rd , R), z ∈ Rd writes:Lu ϕ(z) =1Tr σσ ∗ (u, z)Dz2 ϕ(z) + hb(u, z), Dz ϕ(z)i.2Equations (3.12), (3.13) yield the formal expansion below which is initially due to McKean andSinger [MS67].Zy(p − p̃ )(s, t, x, y) =tZdu∂usZdzp(s, u, x, z)p̃y (u, t, z, y)RdtZ=dudz (∂u p(s, u, x, z)p̃y (u, t, z, y) + p(s, u, x, z)∂u p̃y (u, t, z, y))sRdZ t Z=dudz L∗u p(s, u, x, z)p̃y (u, t, z, y) − p(s, u, x, z)L̃yu p̃y (u, t, z, y)sRdZ t Z=dudzp(s, u, x, z)(Lu − L̃yu )p̃y (u, t, z, y),(3.14)Rdsusing the Dirac convergence for the first equality, equations (3.13) and (3.12) for the second one.
Weeventually take the adjoint for the last equality. Note carefully that the differentiation under theintegral is also here formal since we would need to justify that it can actually be performed usingsome growth properties of the density and its derivatives which we a priori do not know.Let us remind the notation from the Chapter 2:Z t Zf ⊗ g(s, t, x, y) =dudzf (s, u, x, z)g(u, t, z, y)sRdfor the time-space convolution and let us define p̃(s, t, x, y) := p̃y (s, t, x, y), that is in p̃(s, t, x, y) weconsider the density of the frozen process at the final point and observe it at that specific point. Wealso remind the notationn of the parametrix kernel:H(s, t, x, y) := (Ls − L̃s )p̃(s, t, x, y) := (Ls − L̃ys )p̃y (s, t, x, y).(3.15)This yields to iterated convolutions of the kernel and leads to the formal expansion:p(s, t, x, y) =∞Xp̃ ⊗ H (r) (s, t, x, y),r=026(3.16)where p̃ ⊗ H (0) = p̃, H (r) = H ⊗ H (r−1) , r ≥ 1.We now come to the main difference from the proof in Chapter 2.
Since we are working underthe Assumption (A3), namely the Hölder continuity of the diffusion coefficients, it is not possible torefer directly to previously obtained results for the existence of the density or parametrix expansion.However, both facts are still true.Proposition 3.1.3. Under the sole assumption (A), for t > s, the density of Xtx,s solving (3.1)exists and can be written as in (3.16).Proof.
The proof can already be derived from a sensitivity argument. We first introduce two parametrixseries of the form (3.16). Namely,p(s, t, x, y) := pe(s, t, x, y) +∞Xp̃ ⊗ H (r) (s, t, x, y)(3.17)p̃ε ⊗ Hε(r) (s, t, x, y).(3.18)r=1andpε (s, t, x, y) := peε (s, t, x, y) +∞Xr=1Let us point out that, at this stage, p and pε are defined as sum of series. The purpose is then to(ε),s,xat point y.identify those sums with the densities of the processes Xts,x , XtThe convergence of the series (3.17) and (3.18) is in some sense standard (see e.g. [Men11] orFriedman [Fri64]) under (A).
We recall for the sake of completeness the key steps for (3.17).From direct computations, there exist c1 ≥ 1, c ∈ (0, 1] s.t. for all T > 0 and all multi-indexα, |α| ≤ 8,∀0 ≤ u < t ≤ T, (z, y) ∈ (Rd )2 , |Dzα pe(u, t, z, y)| ≤wherec1pc (t − u, y − z),(t − u)|α|/2(3.19)cd/2c |y − z|2pc (t − u, y − z) =exp −,2 t−u(2π(t − u))d/2stands for the usual Gaussian density in Rd with 0 mean and covariance (t − u)c−1 Id . From (3.19),the boundedness of the drift and the Hölder continuity in space of the diffusion matrix we readily getthat there exists c1 ≥ 1, c ∈ (0, 1],|H(u, t, z, y)| ≤c1 (1 ∨ T (1−γ)/2 )pc (t − u, z − y).(t − u)1−γ/2(3.20)Now the key point is that the control (3.20) yields an integrable singularity giving a smoothing effectin time once integrated in space in the time-space convolutions appearing in (3.17) and (3.18).
Itfollows by induction that:rYrγγγB( , 1 + (i − 1) )pc (t − s, y − x)(t − s) 222i=1rγ((1 ∨ T (1−γ)/2 )c1 )r+1 Γ( 2 )rγ=pc (t − s, y − x)(t − s) 2 .Γ(1 + r γ2 )|ep ⊗ H (r) (s, t, x, y)| ≤ ((1 ∨ T (1−γ)/2 )c1 )r+1(3.21)These bounds readily yield the convergence of the series as well as a Gaussian upper-bound. Namelyp(s, t, x, y) ≤ c1 exp((1 ∨ T (1−γ)/2 )c1 [(t − s)γ/2 ])pc (t − s, y − x).27(3.22)An important application of the stability of the perturbation consists in considering coefficientsbε := b ? ζε , σ := σ ? ζε in (3.18), where ζε is a mollifier in time and space.
For mollified coefficientswhich satisfy the non-degeneracy assumption (A2), the existence and smoothness of the density pε forthe associated process X (ε) in (3.2) can be derived from [IKO62]. Observe carefully that the previousGaussian bounds also hold for pε uniformly in ε and independently of the mollifying procedure. Thistherefore gives thatpε (s, t, x, y) −→ p(s, t, x, y),(3.23)ε→0boundedly and uniformly. Thus, for every continuous bounded function f we derive from the boundedconvergence theorem and (3.22) that for all 0 ≤ s < t, x ∈ Rd :ZZ(ε)Es,x [f (Xt )] =f (y)pε (s, t, x, y)dy −→f (y)p(s, t, x, y)dy.(3.24)ε→0RdRdRIn particular, taking f = 1 gives that Rd p(s, t, x, y)dy = 1 and the uniform convergence in (3.23)gives that p(s, t, x, .
. . ) is non negative. We therefore derive that p(s, t, x, ·) is a probability densityon Rd .On the other hand, under (A), we can derive from Theorem 11.3.4 of [SV79] that (Xsε )s∈[0,T ] ⇒ lawε→0(Xs )s∈[0,T ] . This gives that for any bounded continuous function f :(ε),s,xE[f (Xt)] −→ E[f (Xts,x )].ε→0This convergence and (3.24) then yield that the random variable Xts,x admits p(s, t, x, ·) as a density.We can thus now conclude that the processes X, X (ε) in (3.1), (3.2) have transition densities givenby the sum of the series (3.17),(3.17).Parametrix for Markov ChainsOne of the main advantages of the formal expansion in (3.16) is that it has a direct discrete counterpartin the Markov chain setting.
Indeed, denote by (Yttji ,x )j≥i the Markov chain with dynamics (3.3)starting from x at time ti . Observe first that if the innovations (ξk )k≥1 have a density then so doesthe chain at time tk .Let us now introduce its generator at time ti , i.e. for all ϕ ∈ C02 (Rd , R), x ∈ Rd :i ,xLhti ϕ(x) := h−1 E[ϕ(Ytti+1) − ϕ(x)].In order to give a representation of the density of ph (ti , tj , x, y) of Yttji ,x at point y for j > i, weintroduce similarly to the continuous case, the Markov chain (or inhomogeneous random walk) withcoefficients frozen in space at y.
For given (ti , x) ∈ [0, T ] × Rd , tj ≥ ti we set:Ỹttji ,x,y1/2:= x + hj−1Xσ(tk , y)ξk+1 ,k=ih,yand denote its density p̃(ti , tj , x, .). Its generator at time ti writes for all ϕ ∈ C02 (Rd , R), x ∈ Rd :−1i ,x,yL̃h,yE[ϕ(Ỹtti+1) − ϕ(x)].ti ϕ(x) = hUsing the notation p̃h (ti , tj , x, y) := p̃h,y (ti , tj , x, y), we introduce now for 0 ≤ i < j ≤ N theparametrix kernel:hH h (ti , tj , x, y) := (Lhti − L̃h,yti )p̃ (ti + h, tj , x, y).Analogously to Lemma 3.6 in [KM00], which follows from a direct algebraic manipulation, we derivethe following representation for the density which can be viewed as the Markov chain analogue ofProposition 3.1.3.28Proposition 3.1.4 (Parametrix Expansion for the Markov Chain).
Assume (A) is in force. Then,for 0 ≤ ti < tj ≤ T ,j−iXph (ti , tj , x, y) =p̃h ⊗h H h,(r) (ti , tj , x, y),r=0where the discrete time convolution type operator ⊗h is defined byf ⊗h g(ti , tj , x, y) =j−i−1XZhk=0f (ti , ti+k , x, z)g(ti+k , tj , z, y)dz.RdAlso g⊗h H h,(0) = g and for all r ≥ 1, H h,(r) := H h ⊗h H h,(r−1) denotes the r-fold discrete convolutionof the kernel H h .3.1.4Stability of Parametrix Series.We will now investigate more specifically the sensitivity of the density w.r.t. the coefficients throughthe difference of the series.
For a given fixed parameter ε, under (A) the densities p(s, t, x, .), pε (s, t, x, ·)at time t of the processes in (3.1), (3.2) starting from x at time s both admit a parametrix expansionof the previous type.Stability for DiffusionsLet us consider the difference between two parametrix expansions:|p(s, t, x, y) − pε (s, t, x, y)| = |∞Xp̃ ⊗ H (r) (s, t, x, y) −r=0≤ |(p̃ − p̃ε )(s, t, x, y)| + |∞X∞Xp̃ε ⊗ Hε(r) (s, t, x, y)|r=0p̃ ⊗ H (r) (s, t, x, y) −r=1∞Xp̃ε ⊗ Hε(r) (s, t, x, y)|.(3.25)r=1The strategy to study the above difference, using some well known properties of the Gaussian kernelsand their derivatives recalled in (3.19), consists in first studying the difference of the main terms.We have the following Lemma.Lemma 3.1.5 (Difference of the first terms and their derivatives).
Under (A), there exist c1 ≥1, c ∈ (0, 1] s.t. for all 0 ≤ s < t, (x, y) ∈ (Rd )2 and all multi-index α, |α| ≤ 4,|Dxα p̃(s, t, x, y) − Dxα p̃ε (s, t, x, y)| ≤c1∆ε,σ,γ pc (t − s, y − x).(t − s)|α|/2Proof. Let us first consider |α| = 0 and introduce some notations. Set:Z tZ tΣ(s, t, y) :=a(u, y)du, Σε (s, t, y) :=aε (u, y)du.s(3.26)sLet us now identify the columns of the matrices Σ(s, t, y), Σε (s, t, y) with d-dimensional column vectors, i.e. for Σ(s, t, y):Σ(s, t, y) = Σ1 Σ2 · · · Σd (s, t, y).We now rewrite:p̃(s, t, x, y) = fx,y (Θ(s, t, y)), Θ(s, t, y) = ((Σ1 )∗ , · · · , (Σd )∗ )∗ (s, t, y),p̃ε (s, t, x, y) = fx,y (Θε (s, t, y)), Θε (s, t, y) = ((Σ1ε )∗ , · · · , (Σdε )∗ )∗ (s, t, y),29with2fx,y : Rd → Rwhere Γ := 111:d −1exp − h(Γ ) (y − x), y − xi ,Γ 7→ fx,y (Γ) =2(2π)d/2 det(Γ1:d )1/2Γ1Γ2 ..
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