Stability of Densities for Perturbed Diffusions and Markov Chains

Abstract

We study the sensitivity of the densities of non degenerate diffusion processes and related Markov Chains with respect to a perturbation of the coefficients. Natural applications of these results appear in models with misspecified coefficients or for the investigation of the weak error of the Euler scheme with irregular coefficients. Résumé. Nousétudions la sensibilité des densités de processus de diffusion non dégénérés et desChaînes de Markov associées par rapportà une perturbation des coefficients. Ces résultats trouvent des applications naturelles dans l'étude de modèles avec incertitude sur les coefficients ou pour l'analyse de l'erreur faible du schéma d'Eulerà coefficients irréguliers.

Weak error for the Euler scheme approximation of diffusions with non-smooth coefficients

“…The paper is organized as follows. We first introduce a suitable mollification procedure of the coefficients in Section 2 and derive from the stability results of Konakov et al [KKM16] how the error of the mollifying procedure is then reflected on the densities. This allows to control the terms p − p ε and p h ε − p h in (1.5).…”

“…Proof. Equation (2.10) readily follows from Theorems 1 and 2 in [KKM16]. The point is here to specify the control (2.11) on the constant appearing in (2.10).…”

“…The point is here to specify the control (2.11) on the constant appearing in (2.10). Lemma 3 in [KKM16], quantifies the explosive contributions for each term of the parametrix series giving the difference of the densities. It holds for both the diffusion and the Euler scheme, see Section 3.2 of [KKM16] for details, and yields:…”

Probability density function of SDEs with unbounded and path-dependent drift coefficient

“…The existence and regularity of a probability density function (pdf) of X x t with respect to Lebesgue measure have been studied by many authors. If the drift b : [0, ∞) × R d → R d is bounded Hölder continuous and diffusion matrix σ is bounded, uniformly elliptic and Hölder continuous, then it is well-known that by using Levi's parametrix method (see, [15]), there exists the fundamental solution of parabolic type partial differential equations (Kolmogorov equation), and by Feynman-Kac formula, it is a pdf of a solution of associated SDEs (see also, [35,36,39,45]). Note that the parametrix method can be applied to the case of L p ([0, T ] × R d )-valued drift with p ≥ d + 2 [53], Hölder continuous (unbounded) drift [11], Brownian motion with signed measure belonging to the Kato class [29] and Hyperbolic Brownian motion with drift [26].…”

Stability of Densities for Perturbed Degenerate Diffusions

“…Since we consider perturbations of the densities with respect to the non-degenerate component, following the same steps as in [KKM15] one can show that the Lemma below holds:…”