Диссертация (1137347), страница 13

Просмтор этого файла доступен только зарегистрированным пользователям. Но у нас супер быстрая регистрация: достаточно только электронной почты!

Текст из файла (страница 13)

The non degeneratecomponent has at time t the usual diffusive scale in t1/2 corresponding to the self-similarity index51or typical scale of the Brownian motion, whereas the degenerate one has, Rin small time, a "faster"ttypical behaviour in t3/2 corresponding to the standard scale of the integral 0 Ws ds. By "faster", wemean that the time normalization in the exponential deviation bounds appearing in (4.15) are biggerin small time, i.e.

t−3/2 ≥ t−1/2 for the typical scales or standard deviations.From direct computations on Gaussian density, it follows that for any indexes α, β, such that|α| ≤ 4, |β| ≤ 2, ∃C > 0 such that:00,y|Dxα Dyβ p̃t,x(t, (x, y), (x0 , y 0 ))| ≤ηCpc,K (t, (x, y), (x0 , y 0 )).t|α|/2+3|β|/2(4.17)T,(x0 ,y 0 )We adopt the following convention: p̃η (T −s, (x, y), (x0 , y 0 )) stands for p̃η(T −s, (x, y), (x0 , y 0 )).The key quantity in the parametrix method is the kernel function which writes similarly as in thenon-degenerate case:00∀η ∈ [0, 1] Hη (t, (x, y), (x0 , y 0 )) := (Lη − L̃t,(x ,y ),η )p̃η (t, (x, y), (x0 , y 0 )),(4.18)where Lη denotes the same operator as in (4.8), but with mollified coefficients bη and ση .Note carefully that in the above kernel Hη , because of the linear structure of the degeneratecomponent in the model, the most singular terms, i.e.

those involving derivatives w.r.t. y, i.e. thefast variable, vanish.Let us now remind the notationZ t Zf ⊗ g(t, (x, y), (x0 , y 0 )) =dudzdwf (u, (x, y), (w, z))g(t − u, (w, z), (x0 , y 0 ))R2d0as the time-space convolution.Using the standard mollification argument and applying forward and backward Kolmogorov equations one can deriveZ=0(pη − p̃η )(t, (x, y), (x0 , y 0 )) = pη ⊗ Hη (t, (x, y), (x0 , y 0 ))Ztdupη (u, (x, y), (w, z))Hη (t − u, (w, z), (x0 , y 0 ))dwdz,R2dand after the iteration procedure one get the formal expansion:00pη (t, (x, y), (x , y )) =∞Xp̃η ⊗ Hη(r) (t, (x, y), (x0 , y 0 )),(4.19)r=0Obtaining estimates on pη from the formal expression (4.19) requires to have good controls onthe right-hand side. Precisely thanks to (4.17), we first get that there exist c1 > 1, c > 0 s.t.

for allu ∈ [0, t),|Hη (t − u, (w, z), (x0 , y 0 ))|≤12T r {aη (w, z) − aη (x0 , y 0 − x0 (t − u))} Dwp̃η (t − u, (w, z), (x0 , y 0 ))2+hbη (w, z), Dw p̃η (t − u, (w, z), (x0 , y 0 ))iC|w − x0 |γ + |z − (y 0 − x0 (t − u))|γ/3C≤+2(t − u)(t − u)1/2×pc,K (t − u, (w, z), (x0 , y 0 )) p (t − u, (w, z), (x0 , y 0 ))c,K≤ c1 1 ∨ T (1−γ)/2.(t − u)1−γ/2We can establish by induction the following key result.52(4.20)Lemma 4.2.1. There exist constants C ≥ 1, c > 0 s.t.

for all η ∈ [0, 1] one has for all r ∈N∗ , (t, (x, y), (x0 , y 0 )) ∈ (0, T ] × (R2d )2 :p̃η ⊗ Hη(r) (t, (x, y), (x0 , y 0 )) γγ γ(r − 1)γ γr+1 rγ/2≤CtB 1,×B 1+ ,× ··· × B 1 +,22 222×pc,K (t, (x, y), (x0 , y 0 )),(r)recalling that Hη(r−1):= Hη⊗ Hη .Proof. The result (4.17) in particular yields that ∃C2 > 0, such that ∀u ∈ (0, t], p̃η (t−u, (x, y), (w, z)) ≤C2 pc,K (t − u, (x, y), (w, z)) uniformlyw.r.t. η ∈ [0, 1].Setting C := c1 1 ∨ T (1−γ)/2 ∨ C2 , we also obtain uniformly in η|p̃η ⊗ Hη (t, (x, y), (x0 , y 0 ))|Z≤tZduZtZp̃η (u, (x, y), (w, z))|Hη (t − u, (w, z), (x0 , y 0 ))|dwdz,R2d01pc,K (t − u, (w, z), (x0 , y 0 ))dwdz(t − u)1−γ/20R2d γ≤ C 2 tγ/2 B 1,pc,K (t, (x, y), (x0 , y 0 )),2R1using the semigroup property (4.16) in the last inequality and where B(p, q) = 0 up−1 (1 − u)q−1 dudenotes the β−function.

By induction on r:p̃η ⊗ Hη(r) (t, (x, y), (x0 , y 0 )) γγ γ(r − 1)γ γr+1 rγ/2×B 1+ ,× ··· × B 1 +,≤CtB 1,22 222×pc,K (t − s, (x, y), (x0 , y 0 )), r ∈ N∗ ,≤duC 2 pc,K (u, (x, y), (w, z))which means that the sum of the series (4.19) is uniformly controlled w.r.t. η ∈ [0, 1].These bounds imply that the series representing the density of the initial process pη (t, (x, y), (x0 , y 0 ))could be expressed as (4.19) yield and the following bound uniformly in η ∈ [0, 1]: pη (t, (x, y), (x0 , y 0 )) ≤c1 pc,K (t, (x, y), (x0 , y 0 )).From the bounded convergence theorem one can derive thatpη (t, (x, y), (x0 , y 0 )) −→η→0∞Xp̃ ⊗ H (r) (t, (x, y), (x0 , y 0 )) := p(t, (x, y), (x0 , y 0 )),r=0(4.21)uniformly in (t, (x, y), (x0 , y 0 )), where p̃(u, (x, y), (w, z)) := p̃0 (u, (x, y), (w, z)) and H (r) (t−u, (w, z), (x0 , y 0 )) :=(r)H0 (t − u, (w, z), (x0 , y 0 )).Due to the uniform convergence in η (which implies the uniqueness in law):ZZf (z, w)pη (t, (x, y), (w, z))dwdz −→f (z, w)p(t, (x, y), (w, z))dwdz,R2dη→0R2dfor all continuous and bounded f .

The well-posedness of the martingale problem and the sametechnique as in Theorem 11.4.2 from [SV79] then give that the process (Xt , Yt ) has the transitiondensity which is exactly the sum of the parametrix series p(t, (x, y), (x0 , y 0 )).53Remark 4.2.2. Although our model (4.9) does not totally fulfill the framework of Theorem 11.4.2 from[SV79]: one can derive the same result following every step of the proof.

Namely, the only Theoremwhich is used in the proof by Stroock and Varadhan and which is not so clear in our framework isTh. 9.2.12 in [SV79].Namely, in our case, we need to derive that ∀η ∈ [0, 1], t ∈ [0, T ], (x, y) ∈ R2dand h = (h1 , h2 ) ∈ R2d small:Zlim sup|pη (t, (x, y), (w, z)) − pη (t, (x, y), (w + h1 , z + h2 ))|dwdz = 0.(4.22)|h|→0ηR2dThe equation (4.22) can be proved in the same technique as in [SV79] taking into account the facttransition densities of the SDEs with mollified coefficients bη , ση are smooth and the limit of theparametrix sum is a continues function.Thus, we have proved the below proposition.Proposition 4.2.2. Under the sole assumption (AD), for t > 0, the transition density of the process(Xt , Yt ) solving (4.9) exists and can be written as the series in (4.19) with η = 0.Parametrix expansion. SchemeLet us introduce the approximation scheme for (4.9).

For a fixed N and T > 0 we define a time grid{0, t1 , . . . , tN } with a given step h := T /N , i.e. ti = ih, for i = 0, . . . , N and the scheme(RtRthhhhXth = x + 0 b(Xφ(s), Yφ(s))ds + 0 σ(Xφ(s), Yφ(s))dWs ,R(4.23)thhYt = y + 0 Xs ds.where φ(t) = ti ∀t ∈ [ti , ti+1 ). Observe that the above scheme is in fact well defined even thoughthe non degenerate component of the scheme itself appears in the integral. On every time-step theincrements of (Xth , Yth )t∈[ti ,ti+1 ] , i ≥ 0 are actually Gaussian. They indeed correspond to a suitablerescaling of the Brownian increment and its integral on the considered time step, see also Remark4.2.3.Remark 4.2.3. The specific version of the Euler scheme we perform for the model (4.9) gives us directlythe existence of the one step Gaussian transition density whereas in the generic Markov settings of[KMM10] some "aggregations" are needed.

Namely, under assumptions (A) the discretization scheme(4.23) admits a Gaussian transition density: for all (x, y) ∈ R2d , 0 < j ≤ N, A ∈ B(R2d ) (where B(R2d )stands for the Borel σ−field of R2d ) we get:P[(Xthj , Ythj ) ∈ A|(X0h , Y0h ) = (x, y)]Zph (h, (x, y), (x1 , y1 ))ph (h, (x1 , y1 ), (x2 , y2 )) × . . .=(R2d )j−1 ×A×ph (h, (xj−1 , yj−1 ), (xj , yj ))dx1 dy1 dx2 dy2 . . . dxj dyjZ=:ph (tj , (x, y), (xj , yj ))dxj dyj ,Awhere the notation ph (h, (xi , yi ), (xi+1 , yi+1 )), i ∈ [0, N − 1], stands for the density of a Gaussianxi + b(xi , yi )hrandom variable with meanand non degenerate covariance matrixyi + xi h + b(xi , yi )h2 /2a(xi , yi )ha(xi , yi )h2 /2. Two-sided Gaussian bounds of Kolmogorov type for the scheme2a(xi , yi )h /2 a(xi , yi )h3 /3transition density ph (tj , (x, y), (xj , yj )) have been established in [LM10].54As for the diffusion density, we would like to take the advantage of applying the parametrixtechnique to the discretization scheme transition density (4.23) as in [LM10].

We first need tointroduce the frozen version for the scheme (4.23) and the discrete counterpart of the time-spaceconvolution kernel. From this we can derive the parametrix representation for the density of thediscretization scheme.For fixed points (x, y), (x0 , y 0 ) ∈ R2d , the fixed final time tj ,0 ≤ j ≤ j 0 ≤ N we define h,(x0 ,y 0 ,tj )h,(x0 ,y 0 ,tj )hhX̃t , Ỹt≡ X̃t, Ỹtt∈[0,tj ]t∈[0,tj ]by X̃0h , Ỹ0h = (x, y), and ∀t ∈ (0, tj ) :(RtX̃th = x + 0 σ(x0 , y 0 − x0 (tj − φ(s)))dWs ,RtRtRvỸth = y + 0 X̃vh dv = y + xt + 0 0 σ(x0 , y 0 − x0 (tj − φ(s)))dWs dv,(4.24)where φ(t) = ti , ∀t ∈ [ti , ti+1 ).RtRvRtLet us emphasise that 0 0 σ(x0 , y 0 −x0 (tj −φ(s)))dWs dv = 0 (t−v)σ(x0 , y 0 −x0 (tj −φ(s)))dWs =:Rt(t − v)σ̃φ(s) dWs (the equality means that processes are equal in distributions) as two Gaussian0∗processes with zero-means and the same covariance matrices.

Характеристики

Список файлов диссертации