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Weak error for the Euler scheme approximation of diffusions with non-smooth coefficients
Electronic Journal of Probability. 2017. Vol. 22. P. 1-47.
Konakov V., Menozzi S.
We study the weak error associated with the Euler scheme of non degenerate diffusion processes with non smooth bounded coefficients. Namely, we consider the cases of Hölder continuous coefficients as well as piecewise smooth drifts with smooth diffusion matrices.
Keywords: diffusion processesдиффузионные процессыparametrixпараметриксEuler schemeсхема Эйлеракоэффициенты ГёльдераHölder coefficients Piecewise smooth bounded drifts
Publication based on the results of:
Konakov V., Menozzi S., Journal of Theoretical Probability 2011 Vol. 24 No. 2 P. 454-478
Consider a multidimensional stochastic differential equation governed by a symmetric stable process. Under suitable assumptions on the coefficients, the unique strong solution of the above equation admits a density with respect to Lebesgue measure, and so does its Euler scheme. Using a parametrix approach, we derive an error expansion with respect to a time step ...
Added: December 4, 2012
Konakov V., Kozhina A., Menozzi S., ESAIM: Probability and Statistics 2017 Vol. 21 P. 88-112
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. ...
Added: April 14, 2017
Kozhina A., Фундаментальная и прикладная математика 2018 Т. 22 № 3 С. 89-117
We study the weak error associated with the Euler scheme of Kolmogorov like degenerate diffusion processes with non-smooth bounded coefficients. Precisely, we consider the case H ̈older continuous homogeneous coefficients. ...
Added: October 28, 2018
Konakov V., Mammen E., Probability Theory and Related Fields 2000 Vol. 117 No. 4 P. 551-587
We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Local limit theorems for transition
densities are proved. ...
Added: October 15, 2012
Lorik Huang, Menozzi S., Annales de l’Institut Henri Poincaré 2016 Vol. 52 No. 4 P. 1925-1975
We consider a stable driven degenerate stochastic differential equation, whose coefficients satisfy a kind of weak Hörmander condition. Under mild smoothness assumptions we prove the uniqueness of the martingale problem for the associated generator under some dimension constraints. Also, when the driving noise is scalar and tempered, we establish density bounds reflecting the multi-scale behavior ...
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L.Huang, Frikha N., Stochastic Processes and their Applications 2015 No. 125 P. 4066-4101
We obtain an expansion of the implicit weak discretization error for the target of stochastic approximation
algorithms introduced and studied in Frikha (2013). This allows us to extend and develop the
Richardson–Romberg extrapolation method for Monte Carlo linear estimator (introduced in Talay and
Tubaro (1990) and deeply studied in Pag`es (2007)) to the framework of stochastic optimization by ...
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Konakov V., Menozzi S., Molchanov S., Annales de l’Institut Henri Poincaré 2010 Vol. 46 No. 4 P. 908-923
For a class of degenerate diffusion processes of rank 2, i.e. when only Poisson brackets of order one are needed to span the whole space, we obtain a parametrix representation of McKean-Singer type for the density. We there from derive an explicit Gaussian upper bound and a partial lower bound that characterize the additional singularity ...
Added: December 4, 2012
Konakov V., Mammen E., Probability Theory and Related Fields 2009 Vol. 143 No. 1 P. 137-176
We consider triangular arrays of Markov chains that converge weakly to a diffusion process. Second order Edgeworth type expansions for transition densities are proved. The paper differs from recent results in two respects. We allow nonhomogeneous diffusion limits and we treat transition densities with time lag converging to zero. Small time asymptotics are motivated by ...
Added: December 4, 2012
Konakov V., Mammen E., Bernoulli: a journal of mathematical statistics and probability 2005 Vol. 11 No. 4 P. 591-641
We consider triangular arrays of Markov chains that converge weakly to a diffusion process. We prove Edgeworth-type expansions of order o(n-1-δ),δ>0, for transition densities. For this purpose we apply the parametrix method to represent the transition density as a functional of densities of sums of independent and identically distributed variables. Then we apply Edgeworth expansions ...
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Бородин А. Н., Switzerland : Birkhauser/Springer, 2017
This book provides a rigorous yet accessible introduction to the theory of stochastic processes. A significant part of the book is devoted to the classic theory of stochastic processes. In turn, it also presents proofs of well-known results, sometimes together with new approaches. Moreover, the book explores topics not previously covered elsewhere, such as distributions of ...
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Poisson equation in the whole space was studied earlier for so called ergodic generators L corresponding to homogeneous Markov diffusions. Solving this equation is one of the main tools for diffusion approximation in the theory of stochastic averaging and homogenisation. Here a similar equation with a potential is considered, firstly because it is natural for ...
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Shaposhnikov S., Manita O. A., Journal of Dynamics and Differential Equations 2016 Vol. 28 No. 2 P. 493-518
We study the Cauchy problem for Fokker–Planck–Kolmogorov equations for finite measures with unbounded and degenerate coefficients. Sufficient conditions for the existence and uniqueness of solutions are given. ...
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We consider a non-local operator Lα which is the sum of a fractional Laplacian △ α/2 , α ∈ (0, 1), plus a first order term which is measurable in the time variable and locally β-Hölder continuous in the space variables. Importantly, the fractional Laplacian ∆ α/2 does not dominate the first order term. We ...
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Konakov V., Markova A., / Cornell University. Series math "arxiv.org". 2014. No. 1412.1607v1.
We consider a sequence of Markov chains weakly convergent to a diffusion. We suppose that a drift term contains a linearly increasing component. The usual parametrix method fails because of this unbounded drift term. We show how to modify the parametrix method to obtain local theorems for this case. ...
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Chaudru de Raynal P., Menozzi S., Honoré I., Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 2018
We provide here some sharp Schauder estimates for degenerate PDEs of Kolmogorov type when the coefficients lie in some suitable anisotropic Hölder spaces and the first order term is non-linear and unbounded. We proceed through a perturbative approach based on forward parametrix expansions. Due to the low regularizing properties of the degenerate variables, for the ...
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It is well known that every module M over the algebra ℒ(X) of operators on a finite-dimensional space X can be represented as the tensor product of X by some vector space E, M ≅ = E ⊗ X. We generalize this assertion to the case of topological modules by proving that if X is a stereotype space with the stereotype approximation property, then for each stereotype module M over the ...
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Kalyagin V.A., Koldanov A.P., Koldanov P.A. et al., Physica A: Statistical Mechanics and its Applications 2014 Vol. 413 No. 1 P. 59-70
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Sequential state discrimination is a strategy for quantum state discrimination of a sender’s quantum
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