Parametrix construction of the transition probability density of the solution to an SDE driven by $\alpha$-stable noise

Knopova, Victoria; Kulik, Alexei

doi:10.1214/16-aihp796

Cited by 44 publications

(94 citation statements)

References 44 publications

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“…We present here only the rigorous step-by-step exposition without additional discussion of the heuristics behind the method; for such a discussion e.g. [10], [19].…”

Section: Possible Extensionsmentioning

confidence: 99%

“…In our setting p t (x, y) satisfies (5.6) in a weaker approximate sense; however, the classical PMPbased argument admits an extension which is well applicable in such an approximate setting. This extended argument is essentially due to [10,Section 4]. For the reader's and further reference convenience, here we give a systematic version of this argument, based on the notion of approximate harmonic functions.…”

Section: The Positive Maximum Principle and The Semigroup Propertiesmentioning

confidence: 99%

“…The error terms R f rozent (x, y), R f rozen t (x, y)under the corresponding conditions satisfy analogues of (3.16), (3.18), and (3.20) with δ changed to δ ∧ ζ. Note that δ < ζ/α; that is, for α ≥ 1 the bounds actually remain unchanged.3.5 Some related resultsWe do not give a wide overview of the related results in this extensively developing domain, referring an interested reader to[10],[21], and a survey paper[13] for such reviews. Instead, we focus on a discussion of references directly related to the particular issues treated in the current paper.1.…”

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confidence: 99%

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Approximation in law of locally $\alpha $-stable Lévy-type processes by non-linear regressions

Kulik¹

2019

Electron. J. Probab.

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We study a real-valued Lévy-type process X, which is locally α-stable in the sense that its jump kernel is a combination of a 'principal' (state dependent) α-stable part with a 'residual' lower order part. We show that under mild conditions on the local characteristics of a process (the jump kernel and the velocity field) the process is uniquely defined, is Markov, and has the strong Feller property. We approximate X in law by a non-linear regression Xwith a deterministic regressor term f t (x) and α-stable innovation term U x t , and provide error estimates for such an approximation. A case study is performed, revealing different types of assumptions which lead to various choices of regressor/innovation terms and various types of the estimates. The assumptions are quite general, cover the super-critical case α < 1, and allow non-symmetry of the Lévy kernel and unboundedness of the drift coefficient.

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“…We present here only the rigorous step-by-step exposition without additional discussion of the heuristics behind the method; for such a discussion e.g. [10], [19].…”

Section: Possible Extensionsmentioning

confidence: 99%

Section: The Positive Maximum Principle and The Semigroup Propertiesmentioning

confidence: 99%

mentioning

confidence: 99%

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Approximation in law of locally $\alpha $-stable Lévy-type processes by non-linear regressions

Kulik¹

2019

Electron. J. Probab.

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“…Notice that the right hand side of (1.3) is smaller than the one in (1.2) when x goes to ∞. We mention that gradient estimate (1.3) plays an important role in [6,Proposition 3.2].…”

Section: Introductionmentioning

confidence: 97%

Optimal gradient estimates of heat kernels of stable-like operators

Zhang

2019

Proc. Amer. Math. Soc.

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In this note we show the optimal gradient estimate for heat kernels of stable-like operators by providing a counterexample. then there is a smooth fundamental solution p κ (t, x) to the operator L κ satisfying (see [3,4,5])Moreover, p κ (t, x) enjoys the following two-sided estimates: for some K 1 = K 1 (α, d, K 0 ) 1, K −1 1 t(t 1/α + |x|) −d−α p κ (t, x) K 1 t(t 1/α + |x|) −d−α , and gradient estimate: for some K 2 = K 2 (α, d, K 0 ) 1,(1.2)The above estimates can be found in [4,5]. Notice that for λ > 0, if we let κ λ (y) := κ(λ 1/α y), then p κ has the following scaling property:

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“…An atypical method based on optimization is instead introduced in [3,4]. Different approaches, based on an explicit representation of the density (and so giving typically results as lower and upper bounds on the density, rather than regularity) have been given in [25,28,27]. Finally, [33] proves Malliavin differentiability of solutions of stochastic equations with non-differentiable coefficients.…”

Section: Introductionmentioning

confidence: 99%

A simple method for the existence of a density for stochastic evolutions with rough coefficients

Romito¹

2018

Electron. J. Probab.

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Parametrix construction of the transition probability density of the solution to an SDE driven by $\alpha$-stable noise

Cited by 44 publications

References 44 publications

Approximation in law of locally $\alpha $-stable Lévy-type processes by non-linear regressions

Approximation in law of locally $\alpha $-stable Lévy-type processes by non-linear regressions

Optimal gradient estimates of heat kernels of stable-like operators

A simple method for the existence of a density for stochastic evolutions with rough coefficients

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