Диссертация (1137347), страница 6

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

Текст из файла (страница 6)

We reserve the notation c for constants that only depend on (A) but not on T . The valuesof C, c may change from line to line and do not depend on the considered parameter ε. Also,for given integers i, j ∈ N s.t. i < j, we will denote by [[i, j]] the set {i, i + 1, · · · , j}.We are now in position to state main results of this Chapter, which we have already mentionedin Chapter 1.Theorem 3.1.1. Fix ε > 0 and a final deterministic time horizon T > 0. Under (A) and for q > d,there exist C := C(q) ≥ 1, c := c(q) ∈ (0, 1] s.t. for all 0 ≤ s < t ≤ T, (x, y) ∈ (Rd )2 :pc (t − s, y − x)−1 |(p − pε )(s, t, x, y)| ≤ C∆ε,γ,q ,(3.5)where p(s, t, x, .), pε (s, t, x, .) respectively stand for the transition densities at time t of equations (3.1),(3.2) starting from x at time s. Also, we denote for a given c > 0 and for all (u, z) ∈ R+ × Rd ,|z|2cd/2pc (u, z) := (2πu)d/2 exp(−c 2u ).The proof will be given in Section 3.1.4.Remark 3.1.1 (About the constants).

We mention that the constant C := C(q) in (3.5) explodeswhen q ↓ d and is decreasing in q. In particular, it can be chosen uniformly as soon as q ≥ q0 > d.Before stating our results for Markov Chains we introduce two kinds of innovations in (3.3).Namely:(IG) The i.i.d. random variables (ξk )k≥1 are Gaussian, with law N (0, Id ), where Id stands for theidentity matrix of size d × d. In that case the dynamics in (3.3) correspond to the Euler discretizationof equations (3.1) and (3.2).(IP) For a given integer M > 2d + 5 + γ, the innovations (ξk )k≥1 are centred and have C 5 density fξwhich has, together with its derivatives up to order 5, at most polynomial decay of order M . Namely,for all z ∈ Rd and multi-index ν, |ν| ≤ 5:|Dν fξ (z)| ≤ CQM (z),R1dzQr (z) = 1.where we denote for all r > d, z ∈ Rd , Qr (z) := cr (1+|z|)r,Rd(3.6)Theorem 3.1.2 (Stability Control for Markov Chains).

Fix ε > 0 and a final deterministic timehorizon T > 0. For h = T /N, N ∈ N∗ , we set for i ∈ N, ti := ih. Under (A), assuming thateither(IG) or (IP) holds, and for q > d there exist C := C(q) ≥ 1, c := c(q) ∈ (0, 1] s.t. for all0 ≤ ti < tj ≤ T, (x, y) ∈ (Rd )2 :χc (tj − ti , y − x)−1 |(ph − phε )(ti , tj , x, y)| ≤ C∆ε,γ,q ,(3.7)where ph (ti , tj , x, .), phε (ti , tj , x, .) respectively stand for the transition densities at time tj of the MarkovChains Y and Y (ε) in (3.3) starting from x at time ti . Also:22- If (IG) holds:χc (tj − ti , y − x) := pc (tj − ti , y − x),with pc as in Theorem 3.1.1.- If (IP) holds:cdQM −(d+5+γ)χc (tj − ti , y − x) :=(tj − ti )d/2|y − x|(tj − ti )1/2 /c.The proof will be given in Section 3.1.5.3.1.2On Some Related Applications.Model Sensitivity for Option Prices.Assume for instance that the (log)-price of a financial asset is given by the dynamics in (3.1).

Undersuitable assumptions the price of an option on that asset at time t and when Xt = x is given byE[f (exp(XTt,x ))] up to an additional discounting factor. In the previous expression f is the pay-offfunction. For a rather large class of pay-offs, say measurable functions with polynomial growth,including irregular ones, Theorem 3.1.1 allows to specifically quantify how a perturbation of thecoefficients impacts the option prices. Precisely for a given ε > 0, under (A):t,x,(ε)))]||Eε (t, T, x, f )| := |E[f (exp(XTt,x ))] − E[f (exp(XTZ≤ C∆ε,γ,qf (exp(y))pc (T − t, x, y)dy.RdThis previous control can be as well exploited to investigate perturbations of a model which providessome closed formulas, e.g. a perturbation of the Black and Scholes model that would include astochastic volatility taking for instance σε (x) = σ + εψ(x) for some bounded Hölder continuousfunction ψ and ε small enough. In that case, assuming that the drift is known and unperturbed, wehave ∆ε,γ,∞ = |σε − σ|γ = ε|ψ|γ .For such studies, we can quote the work of Corielli et al.

[CFP10] who give estimates on optionprices through parametrix expansions by truncation. Some of their results, see e.g. their Theorem3.1, can be related to a perturbation analysis since they obtain an approximation of an option pricefor a local volatility model in terms of the Black–Scholes price and a correction term correspondingto the first order term in the parametrix series. A more probabilistic approach to similar problemscan be found in Benhamou et al. [BGM10].

However, none of the indicated works indeed deals withthe global perturbation analysis we perform here.Weak Error AnalysisIt is well known that if the coefficients b, σ in (3.1) are smooth and a satisfies the non-degeneracycondition (A2), then weak error on the densities for the approximation by the Euler scheme is wellcontrolled. Precisely, for a given time step h > 0, let us set for i ∈ N, ti := ih. Introduce now theEuler scheme X0h = x, ∀i ≥ 1, Xthi+1 = Xthi + b(ti , Xthi )h + σ(ti , Xthi )(Wti+1 − Wti ) and denote byph (ti , x, .) its density at time ti . The dynamics of the Euler scheme clearly enters the scheme (3.3).It has been established in Konakov and Mammen [KM02] (see also Bally and Talay [BT96b] for anextension to the hypoelliptic setting) that:|p − ph |(ti , tj , x, y) ≤ Chpc (tj − ti , x, y).23If the coefficients in (3.1) are not smooth, it is then possible to use a mollification procedure, takingfor x ∈ Rd , bε (t, x) := b(t, .) ? ρε (x), σε (t, x) := σ(t, .) ? ρε (x) with ? denotesthe convolution productR(along the space variable x), ρε := ε−d ρ(x/ε) and ρ ∈ C ∞ (Rd , R+ ), Rd ρ(x)dx = 1, |supp(ρ)| ⊂ Kfor some compact set K of Rd .

For the mollifying kernel ρε , one then easily checks that for γ-Höldercontinuous in space coefficients b, σ there exists C s.t.sup |b(t, .) − bε (t, .)|∞ ≤ Cεγ ,t∈[0,T ]sup |σ(t, .) − σε (t, .)|η ≤ Cεγ−η , η ∈ (0, γ).t∈[0,T ](3.8)The important aspect is that we lose a bit with respect to the sup norm when investigating the Höldernorm. We then have by Theorems 3.1.1 and 3.1.2 and their proof, that, for γ-Hölder continuous inspace coefficients b, σ and taking p = ∞, there exist c, C s.t. for all 0 ≤ s < t ≤ T, 0 ≤ ti < tj ≤T, (x, y) ∈ (Rd )2 :|(p − pε )(s, t, x, y)| ≤ CCη εγ−η pc (t − s, y − x),|(ph − phε )(ti , tj , x, y)| ≤ CCη εγ−η pc (tj − ti , y − x),where the constant Cη explodes when η tends to 0.To investigate the global weak error (p−ph )(ti , tj , x, y) = {(p−pε )+(pε −phε )+(phε −ph )}(ti , tj , x, y),it therefore remains to analyse the contribution (pε − phε )(ti , tj , x, y).

The results of [KM02] indeedapply but yield |(pε − phε )(ti , tj , x, y)| ≤ Cε hpc (tj − ti , y − x) where Cε is explosive when ε goes tozero. The global error thus writes:|(p − ph )(ti , tj , x, y)| ≤ C{Cη εγ−η + Cε h}pc (tj − ti , y − x),and a balance is needed to derive a global error bound. This is precisely the analysis which isperformed in [KM17].

In this Chapter, we extend to the densities (up to a slowly growing factor) theresults previously obtained by Mikulevičius and Platen [MP91] on the weak error, i.e. they showed|E[f (XT ) − f (XTh )]| ≤ Chγ/2 provided f ∈ C 2+γ (Rd , R).

Precisely, we obtain through a suitableanalysis of the constants Cη , Cε , which respectively depend on the behaviour of the parametrixseries and of the derivatives of the heat kernel with mollified coefficients, that |p − ph |(ti , tj , x, y) ≤Chγ/2−ψ(h) pc (tj − ti , y − x) for a function ψ(h) going to 0 as h → 0. (which is induced by the previousloss of η in (3.8)). In the quoted work, the authors also obtain some error bounds for piecewise smoothdrifts having a countable set of discontinuities. This part explicitly requires the stability result ofTheorems 3.1.1, Theorems 3.1.2 for q < +∞. The idea being that the difference between the piecewise smooth drift and its smooth approximation (actually the mollification procedure is only requiredaround the points of discontinuity), is well controlled in Lq norm, q < +∞.Extension to some Kinetic ModelsThe results of Theorems 3.1.1 and 3.1.2 should extend without additional difficulties to the case ofdegenerate diffusions of the form:dXt = b(t, Xt , Yt )dt + σ(t, Xt , Yt )dWt ,dYt = Xt dt,(3.9)under the same previous assumptions on b, σ when we consider perturbations of the non-degenerate(ε)(ε)components, i.e.

for a given ε > 0, (Xt , Yt ) where:(ε)= bε (t, Xt , Yt )dt + σε (t, Xt , Yt )dWt ,(ε)= Xt dt.dXtdYt(ε)(ε)(ε)(ε)(ε)(3.10)Indeed, under (A), the required parametrix expansions of the densities associated with the solutionsof equation (3.9), (3.10) are mentioned in Chapter 4 (see also [KMM10]).24A posteriori controls in parameter estimationLet us consider to illustrate this application a parametrized family of diffusions of the form:dXt = b(t, Xt )dt + σ(η, t, Xt )dWt ,(3.11)where η ∈ θ ⊂ Rd , the coefficients b, σ are smooth, bounded, with bounded derivatives, and thenon-degeneracy condition (A2) holds uniformly w.r.t. η. A natural practical problem consists inestimating the true parameter η from an observed discrete sample (Xtni )i∈[0,n] where the (ti )i∈[0,n]form a partition of the observation interval, i.e.

Характеристики

Список файлов диссертации