Диссертация (1137347), страница 11

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. . , An denote acollection of smooth vector fields on Rd , regarded also as first order differential operators, one candefine the second-order differential operator:k1X 2L :=A + A0 , k < d.2 i=1 i(4.2)Assume that the vector fields A1 , . . . , Ak , [Al , Am ](l,m)∈[0,k]2 , [Al , [Am , An ]](l,m,n)∈[0,k]3 , . . . where[·, ·] stands for the Lie brackets, span Rd . In this case, Hörmander proved that the operator L ishypoelliptic.454.1.2Kolmogorov’s exampleAs we have already mentioned, in 1934 A. Kolmogorov published the paper [Kol34] in which heexplicitly found the fundamental solution for the parabolic operator:2Lx = k∂xx+ b∂x + x∂y , b ∈ R, k ∈ R+ ,for scalar variables x, y, which precisely writes as:√03|y 0 − y − x 2+x t|23|x0 − x − bt|2exp −−p̃(t, (x, y), (x , y )) =22πkt4ktkt300!.(4.3)With the modern language of stochastic calculus it can be readily seen that p̃(t, (x, y), (x0 , y 0 ))corresponds to the transition density of the Gaussian process with dynamics:(s,(x,y)Xt= x + b(t − s) + (2k)1/2 (Wt − Ws ),R t s,(x,y)(4.4)s,(x,y)Yt= y + s Xudu,In Hörmander’s form, with the notations of the previous paragraph N = 2, L = 21 A21 + A0 , A1 = b∂ 0 (2k)1/2 ∂xx, A0 =so that [A1 , A0 ] =and thus, A1 , [A1 , A0 ] have together rank 2.x∂y∂y0The corresponding dynamics in (4.4) equivalently rewrites asXtXtd= A0dt + A1 dWt .YtYt(4.5) 2k 0 is a degenerate diffusion matrix on R2 ⊗ R2 .

This is the reason0 0why such systems as (4.4) are usually called degenerate.It is clear that A1 A∗1 =4.1.3Degeneracy and Hörmander conditionsThere are two main families of degenerate diffusions which are considered in modern analysis, theones who do fulfill the strong Hörmander condition, namely those for which the iterated Lie bracketsof the diffusive vector fields, i.e. those with indexes in [1, k] in (4.2), span the whole space (like e.g.the Brownian motion on the Heisenberg group see Gaveau [Gav77]), and the ones who satisfy the socalled weak Hörmander condition, for which the drift vector field, A0 in (4.2), needs to be consideredin the Lie bracketing to span the whole underlying space. As emphasized above, the Kolmogorovdiffusions belong to the second class.

Roughly speaking "strong Hörmander" means that the noisepropagates inside the system through the diffusive part only. In contrast, under the weak Hörmanderconditions the drift has a key role in the noise propagation.A striking fact is that the weak Hörmander framework intrinsically leads to multi-scale behavioursof the underlying diffusion.

This is already clearly seen in (4.3), which exhibits the two characteristictime scales of the Brownian motion and its integral ( namely t1/2 and t3/2 for the standard deviationsrespectively).On the other hand, we recall that, in the strong Hörmander setting we have a separation betweenspace and time. It is known from Kusuoka and Stroock [KS85] that for this family of diffusions, twosided bounds with the usual parabolic scaling in t1/2 holds for off-diagonal terms in the heat kernelestimates when considering the spatial distance induced by the Carnot metric associated with thevector fields.46Our research here provides a way how to adapt the parametrix expansion technique to Kolmogorovtype degenerate diffusions, even for non-smooth Hölder coefficients in the dynamics of the SDE, andto the corresponding numerical approximations through suitable Euler scheme discretizations.

Wedevelop analogously to the continuous case the way to express transition densities of the degeneratediffusion and the Euler scheme in terms of densities of frozen processes. However in the degenerateframework we have to take into account the problem of different scales mentioned in the paragraphabove. The frozen processes we use also differ from the non-degenerate case: the argument in thesecond variable includes the transport of the frozen point with a time reversal.We will in this chapter investigate the so-called weak error for various classes of test functions(namely Hölder continuous ones and Dirac masses). We refer to the introduction chapter and toSection 3.1.2 and Section 4.4 for details.Although Bally and Talay [BT96a], [BT96b] have already investigated the weak error behaviorfor the Euler scheme approximation in a general hypoelliptic setting (weak or strong) for time homogeneous coefficients in the SDE, they only considered the case of smooth coefficients which lead to ausual convergence rate of order h, with h being the time discretization step of the scheme.However, we will focus on rough coefficients.

In this framework, we rely on controls for the densityof the underlying SDE. Precisely, we need to establish sharp heat-kernel and gradient bounds. Suchbounds have naturally been obtained for smooth coefficients through Malliavin calculus techniques,see [KS84], [KS85]. In the current setting of Hölder coefficients, the parametrix approach seems moreadapted, since we cannot hope to handle tangent flows or Malliavin covariance matrices. To performthe analysis we will establish in the Kolmogorov case the analogue to the heat kernel and gradientestimates achieved in [IKO62] for the non-degenerate case in the Hölder setting for the coefficients.To derive convergence rates for the weak error we also establish some stability results for thediffusion and scheme transition densities with respect to small perturbations of the coefficients.

Theresult is of interest in itself. It is in the current context crucial in the sense that our main controlson the derivatives of the underlying heat kernel (see Theorem 4.4.2) only provide gradient bounds inthe non-degenerate directions. The smoothing procedure of the coefficients (mollification) allows toexhibit some underlying PDE which kills the first order terms in the error analysis.4.1.4General modelsOne of the most general model which allow to apply Gaussian bounds to the transition density hasbeen studied by F. Delarue and S.

Menozzi in their paper [DM10]. Precisely, it has a form of:dXt1= F1 (t, Xt1 , . . . , Xtn )dt + σ(t, Xt1 , . . . , Xtn )dWt ,dXt2= F2 (t, Xt1 , . . . , Xtn )dt,dXt3= F3 (t, Xt2 , . . . , Xtn )dt,...dXtn= Fn (t, Xtn−1 , Xtn )dt,(4.6)where Wt stands for the d-dimensional Brownian motion and ∀i, 1 ≤ i ≤ n, (Xti )t≥0 ∈ Rd . A typicalexample for (4.6) is a system of n coupled oscillators, each moving vertically and being connected tothe nearest neighbours directly, the first oscillator being forced by a random noise.In the general case when n ≥ 2, such systems appear in heat conduction models (see for examplethe original papers by Eckmann et al. [EPRB99] and Rey-Bellet and Thomas [RBL00] when thechain is forced by two heat baths; see also the paper by Bodineau and Lefevere [BL08] ).For the case of smooth coefficients in (4.6) the existence of a density for (Xt1 , .

. . , Xtn ) may be seenas a consequence of Hörmander’s theorem. Understanding the structure of the density under generalhypoelliptic conditions (i.e. for more general systems than (4.6)) is something very difficult. The47reason may be explained as follows: there may be many ways for the underlying noise to propagateinto the whole system, and therefore, many different time scales for the propagation phenomenon.In the article [DM10] the authors considered uniformly Lipschitz continuous (Fi )i∈[1,n] (or supposefor i = 1 that the drift of the non degenerate component F1 is measurable and bounded) anduniformly Hölder continuous in space uniformly elliptic diffusion matrix.

Under such assumptionsthey established Gaussian Aronson like estimates for the density of (4.6) over compact time intervalfor Hölder index γ greater than 1/2. To derive this estimate a "formal" parametrix expansion has beenused, considering a sequence of equations with smooth coefficients for which Hörmander’s theoremguarantees the existence of the density, see e.g. Hörmander [Hör67].The estimates did not depend onthe derivatives of the mollified coefficients but only on the γ- Hölder continuity assumed. Howeverto pass to the limit in the described procedure some uniqueness in law is needed.

It was preciselyderived in [DM10] through viscosity type techniques which do not exploit the underlying smoothingeffects of the parametrix kernel. This is what led to the restriction on the Hölder exponent.Exploiting such a smoothing effect, S. Menozzi proved in [Men11] the well posedness of themartingale problem for the generator associated with the SDE (4.6) in the Hölder setting withoutany restriction on the Hölder index γ. Thus, together with results from [DM10] it gives that for allγ in (0, 1] the unique weak solution for (4.6) exists and admits for all t > 0 a density that satisfiesAronson like bounds.4.1.5Preliminary results from [KMM10]For simplicity we would like to come back to another model, considered by V. Konakov, S.

Molchanovand S. Menozzi in [KMM10] and describe the parametrix derivation in details. The parametrixrepresentation achieved in [KMM10] allows to give a local limit theorem with the usual convergencerate for the associated Euler-Maruyama approximation. We anyhow mention that, the generic Markovchain approximation setting leads to additional technicalities, namely an aggregation procedure ofthe randomness is needed in order to ensure that the transitions at the considered aggregated timehave a density. We refer to the quoted work for additional details.Namely, in [KMM10] the following diffusions has been studied:(RtRtXt = x + 0 b(Xs , Ys )ds + 0 σ(Xs , Ys )dWs ,Rt(4.7)Yt = y + 0 Xs ds,with the generator such that: ∀φ ∈ C02 (R2d ), ∀(x, y) ∈ R2d ,12Lφ(x, y) = Tr a(x, y)Dx φ(x, y) + hb(x, y), ∇x φ(x, y)i + hx, ∇y φ(x, y)i,2(4.8)where (Wt )t≥0 is a standard d− dimensional Brownian motion defined on some filtered probabilityspace (Ω, F, (Ft )t≥0 , P) satisfying the usual assumptions.Exactly this model can be used for dealing with Asian options, Xt can be associated with thedynamics of the underlying asset and its integral Yt is involved in the option payoff.

Typically, theprice of such options writes Ex [ψ(XT , T −1 YT )], where for the put (resp. call) option the functionψ(x, y) = (x − y)+ (resp.(y − x)+ ), see [BPV01]. It is, thus, useful in this framework to specificallyquantify how a perturbation of the coefficients impacts the option prices.The cross dependence of the dynamics of Xt in Yt is also important for handling kinematic models2or Hamiltonian systems.

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