Weak error for Continuous Time Markov Chains related to fractional in time P(I)DEs

Kelbert, Mark; Konakov, Valentin; Menozzi, Stéphane

doi:10.1016/j.spa.2015.10.013

Cited by 3 publications

(3 citation statements)

References 36 publications

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“…An obvious extension widely used in the literature (see e.g. [18], [62], [81], [26] and references therein) represent various mixtures of such derivatives, both discrete and continuous,…”

Section: Generalized Fractional Operators and Markov Processesmentioning

confidence: 99%

The Probabilistic Point of View on the Generalized Fractional Partial Differential Equations

Kolokoltsov

2019

FCAA

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This paper aims at unifying and clarifying the recent advances in the analysis of the fractional and generalized fractional Partial Differential Equations of Caputo and Riemann-Liouville type arising essentially from the probabilistic point of view. This point of view leads to the path integral representation for the solutions of these equations, which is seen to be stable with respect to the initial data and key parameters and is directly amenable to numeric calculations (Monte-Carlo simulation). In many cases these solutions can be compactly presented via the wide class of operator-valued analytic functions of the Mittag-Leffler type, which are proved to be expressed as the Laplace transforms of the exit times of monotone Markov processes.

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“…An obvious extension widely used in the literature (see e.g. [18], [62], [81], [26] and references therein) represent various mixtures of such derivatives, both discrete and continuous,…”

Section: Generalized Fractional Operators and Markov Processesmentioning

confidence: 99%

The Probabilistic Point of View on the Generalized Fractional Partial Differential Equations

Kolokoltsov

2019

FCAA

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“…This representation could be useful for simulating paths of fractional Pearson diffusions. The discrete schemes for the underlying densities and their error bounds can be found in Kelbert et al (2016). For similar approaches to obtaining solutions of such SDEs we refer to Scalas & Viles (2014).…”

Section: Fractional Pearson Diffusions As Solutions Of the Sdesmentioning

confidence: 99%

Heavy-tailed fractional Pearson diffusions

Leonenko

Papić

Sikorskii

et al. 2017

Stochastic Processes and their Applications

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We define heavy-tailed fractional reciprocal gamma and Fisher-Snedecor diffusions by a non-Markovian time change in the corresponding Pearson diffusions. Pearson diffusions are governed by the backward Kolmogorov equations with space-varying polynomial coefficients and are widely used in applications. The corresponding fractional reciprocal gamma and Fisher-Snedecor diffusions are governed by the fractional backward Kolmogorov equations and have heavy-tailed marginal distributions in the steady state. We derive the explicit expressions for the transition densities of the fractional reciprocal gamma and Fisher-Snedecor diffusions and strong solutions of the associated Cauchy problems for the fractional backward Kolmogorov equation.

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“…Even more recently in [DS18] they give some exact asymptotic formulas for the Green's function of fractional evolution equations. The authors in [KKM16] study error estimates for continuous time random walk (CTRW) approximation of classical fractional evolution equations, for which the heat kernel estimates for D β u = ∆u and D β u = ψ(−i∇)u, where ψ(−i∇) generates a symmetric stable process, are obtained as a by-product.…”

Section: Introductionmentioning

confidence: 99%

Green’s Function Estimates for Time-Fractional Evolution Equations

Johnston

Kolokoltsov

2019

Fractal Fract

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We look at estimates for the Green's function of time-fractional evolution equations of the form D ν 0+ * u = Lu, where D ν 0+ * is a Caputo-type timefractional derivative, depending on a Lévy kernel ν with variable coefficients, which is comparable to y −1−β for β ∈ (0, 1), and L is an operator acting on the spatial variable. First, we obtain global two-sided estimates for the Green's function of D β 0 u = Lu in the case that L is a second order elliptic operator in divergence form. Secondly, we obtain global upper bounds for the Green's function of D β 0 u = Ψ(−i∇)u where Ψ is a pseudo-differential operator with constant coefficients that is homogeneous of order α. Thirdly, we obtain local two-sided estimates for the Green's function of D β 0 u = Lu where L is a more general non-degenerate second order elliptic operator. Finally we look at the case of stable-like operator, extending the second result from a constant coefficient to variable coefficients. In each case, we also estimate the spatial derivatives of the Green's functions. To obtain these bounds we use a particular form of the Mittag-Leffler functions, which allow us to use directly known estimates for the Green's functions associated with L and Ψ, as well as estimates for stable densities. These estimates then allow us to estimate the solutions to a wide class of problems of the form D (ν,t) 0 u = Lu, where D (ν,t) is a Caputo-type operator with variable coefficients.

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Weak error for Continuous Time Markov Chains related to fractional in time P(I)DEs

Cited by 3 publications

References 36 publications

The Probabilistic Point of View on the Generalized Fractional Partial Differential Equations

The Probabilistic Point of View on the Generalized Fractional Partial Differential Equations

Heavy-tailed fractional Pearson diffusions

Green’s Function Estimates for Time-Fractional Evolution Equations

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