Диссертация (1137347), страница 14

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Setting ∀s ∈ [0, tj ], ãφ(s) = σ̃φ(s) σ̃φ(s),recall from (AD2) condition that ãφ(s) is symmetric, one can finally obtain that the covariance matrixΣhtj of the vector (X̃thj , Ỹthj ) is equal to!R tjR tjãφ(s) ds(tj − s)ãφ(s) dsh00R tjRtΣt j =.(tj − s)ãφ(s) ds 0 j (tj − s)2 ãφ(s) ds0The frozen process also depends on tj through an additional term in the diffusion coefficient.0 0From now on, p̃h,tj0 ,(x ,y ) denotes the transition density of the discretization scheme (4.24) and let us0 0emphasize that for the frozen coefficients, we will denote for simplicity p̃h,tj ,(x ,y ) (tj 0 , (x, y), (·, ·)) =:p̃h (tj 0 , (x, y), (·, ·)) - the transition density between times 0 and tj 0 ≤ tj of the frozen Markov chain.Let us now introduce the discrete counterpart of the parametrix kernel considered for the con2dtinuous objects in (4.18).

To this end, for a sufficientlyfunction smooth ψ : R → R and fixed0 0(x0 , y 0 ) ∈ R2d , j ∈ (0, N ] define operators Lh and L̃h ≡ L̃h,tj ,(x ,y )where:h00L f (tj , (x, y), (x , y ))−1"Z= hph (h, (x, y), (u, v))f (tj − h, (u, v), (x0 , y 0 ))dudvR2d#−h00L̃ f (tj , (x, y), (x , y ))f (tj − h, (x, y), (x0 , y 0 )) ,−1"Z= hp̃h (h, (x, y), (u, v))f (tj − h, (u, v), (x0 , y 0 ))dudvR2d#−f (tj − h, (x, y), (x0 , y 0 )) .Define the discrete kernel Hh by0 0hhHh (tj , (u, v), (x , y )) = L − L̃ p̃h (tj − h, (u, v), (x0 , y 0 )), 0 ≤ j ≤ N.55(4.25)From the previous definition, for all 0 ≤ j ≤ NZ Hh (tj , (u, v), (x0 , y 0 )) = h−1ph − p̃h (h, (u, v), (w, z))p̃h (tj − h, (w, z), (x0 , y 0 ))dwdz.R2dAnalogously to Lemma 3.6 in [KM00], which follows from a direct algebraic manipulation, it hasbeen derived in [LM10] that the transition density of the scheme admits the following representation.Proposition 4.2.3 (Parametrix Expansion for the Euler scheme).

Assume that the assumptions(AD) are in force. Then0h0p (tj , (x, y), (x , y )) =j Xp̃(r)⊗h Hh(tj , (x, y), (x0 , y 0 )),(4.26)r=0for the discrete time convolution type operator ⊗h defined by(g ⊗h f )(tj , (x, y), (x0 , y 0 )) =j−1 ZXhi=0(0)g(ti , (x, y), (u, v))f (tj − ti , (u, v), (x0 , y 0 ))dudv,R2d(r)(r−1)where g ⊗h Hh := g, and for all r ≥ 1, Hh = Hh ⊗h Hhdenotes the r−fold discrete convolution(r)of the kernel Hh . W.r.t. the above definition, we use the convention that p̃h ⊗h Hh (0, (x, y), (x0 , y 0 )) =0, r ≥ 1.4.3Stability resultsIn this section we are going to study the sensitivity of the transition densities of some Kolmogorov likedegenerate diffusion processes with respect to a perturbation of the coefficients of the non-degeneratecomponent.4.3.1Stability for perturbed diffusionsWe now introduce a perturbed version of (4.9) with dynamics:((ε)(ε)(ε)(ε)(ε)dXt = bε (Xt , Yt )dt + σε (Xt , Yt )dWt ,(ε)(ε)dYt = Xt dt, t ∈ [0, T ],(4.27)where bε : R2d → Rd , σε : R2d → Rd ⊗ Rd satisfy at least the same assumptions as b, σ and are insome sense meant to be close to b, σ for small values of ε > 0.

In particular, from Proposition 4.2.3(ε)(ε)we have that (Xt , Yt ) admits a density.The goal of this Section is to investigate how the closeness of (bε , σε ) and (b, σ) is reflected on therespective densities of the associated processes.In many applications (misspecified volatility models or calibration procedures) it can be useful toknow how the controls on the differences |b − bε |, |σ − σε | (for suitable norms) impact the differencepε − p of the densities corresponding respectively to the dynamics with the perturbed parameters andthe one of the model.Let us now introduce, under (AD), the quantities that will bound the difference of the densitiesin our main results below.

Set for ε > 0:∀q ∈ (1, +∞], ∆dε,b,q := |b(., .) − bε (., .)|Lq (Rd ) .56Since σ, σε are both γ-Hölder continuous, see (A3), we also define∆dε,σ,γ := |σ(., .) − σε (., .)|d,γ ,γwhere γ ∈ (0, 1], |.|d,γ stands for the Hölder norm in space on Cb,d(Rd , Rd ⊗ Rd ), which denotes thespace of Hölder continuous bounded functions with respect to the distance d defined as follows:∀(x, y), (x0 , y 0 ) ∈ (Rd )2 , d (x, y), (x0 , y 0 ) := |x − x0 | + |y 0 − y|1/3 .(4.28)γNamely, a measurable function f is in Cb,d(Rd , Rd ⊗ Rd ) if|f |d,γ := sup |f (x)| + [f ]d,γ , [f ]d,γ :=x∈Rdsup(x,y)6=(x0 ,y 0 )∈R2d|f (x, y) − f (x0 , y 0 )|γ < +∞.d (x, y), (x0 , y 0 )The previous control in particular implies for all ((x, y), (x0 , y 0 )) ∈ (R2d )2 :|a(x, y) − a(x0 , y 0 ) − aε (x, y) + aε (x0 , y 0 )|≤ 22−γ (K + κ)∆dε,σ,γ dγ (x, y), (x0 , y 0 ) .We eventually set ∀q ∈ (1, +∞],∆dε,γ,q := ∆dε,σ,γ + ∆dε,b,q ,which will be the key quantity governing the error in our results.Theorem 4.3.1 (Stability Control).

Fix T > 0. Under (AD), for q ∈ (4d, +∞], there existsC := C(q) ≥ 1, c ∈ (0, 1] s.t. for all 0 < t ≤ T, ((x, y), (x0 , y 0 )) ∈ (R2d )2 :|(p − pε )(t, (x, y), (x0 , y 0 ))| ≤ C∆dε,γ,q pc,K (t, (x, y), (x0 , y 0 )),where p(t, (x, y), (., .)), pε (t, (x, y), (., .)) respectively stand for the transition densities at time t ofequations (4.9), (4.27) starting from (x, y) at time 0.Proof. We will now investigate more specifically the sensitivity of the density w.r.t. the coefficientsperturbation through the difference of the series. From Proposition 4.2.2 , for a given fixed parameterε, under (AD) the densities p(t, (x, y), (·, ·)), pε (t, (x, y), (·, ·)) at time t of the processes in (4.9), (4.27)starting from (x, y) at time 0 both admit a parametrix expansion of the previous type.Let us consider the difference between the two parametrix expansions for (4.9) and (4.27) in theform (4.19):|p(t, (x, y), (x0 , y 0 )) − pε (t, (x, y), (x0 , y 0 ))|≤+∞X|p̃ ⊗ H (r) (t, (x, y), (x0 , y 0 )) − p̃ε ⊗ Hε(r) (t, (x, y), (x0 , y 0 ))|.r=0Since we consider perturbations of the densities with respect to the non-degenerate component,following the same steps as in [KKM17] one can show that the Lemma below holds:Lemma 4.3.2 (Difference of the first terms and their derivatives).

There exist c1 ≥ 1, c ∈ (0, 1] s.t.for all 0 < t, (x, y), (x0 , y 0 ) ∈ R2d and all multi-index α, |α| ≤ 4,|Dxα p̃(t, (x, y), (x0 , y 0 )) − Dxα p̃ε (t, (x, y), (x0 , y 0 ))| ≤57c1 ∆dε,σ,γ pc,K (t, (x, y), (x0 , y 0 )).t|α|/2Lemma 4.3.3 (Control of the one-step convolution). For all 0 < t, (x, y), (x0 , y 0 ) ∈ R2d and 4d <q ≤ ∞ it holds:|p̃ ⊗ H (1) (t, (x, y), (x0 , y 0 )) − p̃ε ⊗ Hε(1) (t, (x, y), (x0 , y 0 ))|!nγ γ(1−γ)/2 2dd2) [∆ε,σ,γ + Iq=+∞ ∆ε,b,+∞ ]B 1,≤ c1 (1 ∨ Tt22!o1d+Iq∈(4d,+∞) ∆ε,b,q B+ α(q), α(q) tα(q) pc,K (t, (x, y), (x0 , y 0 )),2where c1 , c are as in Lemma 4.3.2 and for q ∈ (4d, +∞) we set α(q) =12−(4.29)2dq .Proof. Let us write:|p̃ ⊗ H (1) (t, (x, y), (x0 , y 0 )) − p̃ε ⊗ Hε(1) (t, (x, y), (x0 , y 0 ))|(4.30)0 00 0≤ |(p̃ − p̃ε ) ⊗ H(t, (x, y), (x , y ))| + |p̃ε ⊗ H − Hε (t, (x, y), (x , y ))| := I + II.From Lemma 4.3.2 and (4.20) we readily get for all q ∈ (4d, +∞]:I ≤ ((1 ∨ T(1−γ)/2)c1 )2∆dε,γ,q pc2 ,K (t, (x, y), (x0 , y 0 ))B!γ γ1,t2 .2(4.31)To estimate (II) let us first consider H − Hε more precisely:(H − Hε )(t − u, (w, z), (x0 , y 0 ))10 000 00= Tr a(w, z) − a(x , y − x (t − u)) − aε (w, z) + aε (x , y − x (t − u)22×Dwp̃(t − u, (w, z), (x0 , y 0 ))10 002+ Tr aε (w, z) − aε (x , y − x (t − u)) Dw (p̃ − p̃ε ) (t − u, (w, z), (x0 , y 0 ))2+hb(w, z) − bε (w, z), Dw p̃(t − u, (w, z), (x0 , y 0 ))i+hbε (w, z), Dw (p̃ − p̃ε )(t − u, (w, z), (x0 , y 0 ))i!:=∆1ε H + ∆2ε H (t − u, (w, z), (x0 , y 0 ))+hb(w, z) − bε (w, z), Dw p̃(t − u, (w, z), (x0 , y 0 ))i+hbε (w, z), (Dw p̃ − Dw p̃ε )(t − u, (w, z), (x0 , y 0 ))i.Since functions a(w, z), aε (w, z) are Hölder uniformly continuous and (4.17) holds:|∆1ε H|(t − u, (w, z), (x0 , y 0 ))|γγ/2c∆dε,γ,∞ |w − x0 | + |z − y 0 + x0 (t − u)|pc,K (t − u, (w, z), (x0 , y 0 ))≤(t − u)0 0pc ,K (t − u, (w, z), (x , y ))≤ c∆dε,γ,∞ 2.(t − u)1−γ/258(4.32)From Lemma 4.3.2 and the Hölder uniform continuity of the function aε (x, y) it follows:|∆2ε H|(t − u, (w, z), (x0 , y 0 ))γγ/3c∆dε,γ,∞ |w − x0 | + |z − y 0 + x0 (t − u)|pc,K (t − u, (w, z), (x0 , y 0 ))≤(t − u)0 0p(t−u,(w,z),(x,y))c̃ ,K≤ c∆dε,γ,∞ 2.1−γ/2(t − u)Thus, the fact that |b(w, z) − bε (w, z)| ≤ c∆dε,b,γ and (4.17) give the control for q = +∞.

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