Pathwise uniqueness for singular SDEs driven by stable processes

By Enrico Priola. 2010

163 citations recorded by the source corpus.

We prove pathwise uniqueness for stochastic differential equations driven by non-degenerate symmetric $\alpha$-stable L\'evy processes with values in $\R^d$ having a bounded and $\beta$-H\"older continuous drift term. We assume $\beta > 1 - \frac{\alpha}{2} $ and $\alpha \in [ 1, 2)$. The proof requires analytic regularity results for associated integro-differential operators of Kolmogorov type. We also study differentiability of solutions with respect to initial conditions and the homeomorphism property.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Finance

View Paper: https://www.semanticscholar.org/paper/b7def285bda37bffbfe061038766a9eb4d3cfb47

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Citations in the corpus

Stochastic flows for Lévy processes with Hölder drifts (2015, methodology): “In the subcritical case, under condition (1.8) for ν = ν1 and condition (1.9) below for some γ > α/2 (which is Hypothesis 2 in [18]), Priola [18] derived Hölder estimate for solutions of (1.4) which enabled him to show that SDE (1.1) has a unique strong solution.”
Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift (2011, background): “izations of [33] to unbounded drifts in [23, 34] and also to the references therein; see [17, 18]. We note that [32] also includes the case of α-stable noise, α≥1, which in turn was extended to Rd in [29]. Explicit cases of parabolic stochastic partial diﬀerential equations, with space–time white no”
Jump type stochastic differential equations with non-Lipschitz coefficients: Non-confluence, Feller and strong Feller properties, and exponential ergodicity (2017, methodology): “Indeed, similar assumptions are used in Priola and Wang (2006), Qiao (2014), Wang (2010) to obtain the strong Feller property. Proposition 5.4 further relaxes this condition to a “local” one. In case that the diffusion matrix is degenerate, one needs to place certain conditions on the jumps to obtai”
Lp-maximal regularity of nonlocal parabolic equations and applications (2013, unknown): “…κ t 1 n (cid:13) = 1 n ∈ R nd i ∈ R d (cid:13) = 1 n ∈ ∈ R , (cid:21) denotes the Dirac measure in R ( and the generator of L is given We need the following simple result about the smoothness of the distribution density of Lévy process (see [16, Lemma 3.1] for the symmetric case) Proposition 2.”
On the differentiability of the solution to an equation with drift and fractional diffusion (2010, background): “For the case s ≥ 1/2, a related result has recently been obtained in [15], where a similar regularity for an eigenvalue problem is established.”
Ergodicity of stochastic differential equations with jumps and singular coefficients (2017, methodology): “[19] Flandoli F., Gubinelli M. and Priola E.: Well-posedness of the transport equation by stochastic perturbation.”
Exponential ergodicity and regularity for equations with Lévy noise (2011, methodology): “To prove the existence and pathwise uniqueness of solutions, we only need to modify a little bit the method established in [27].”
Regularity of Stochastic Kinetic Equations (2016, background): “According to a general scheme (see [36], [24], [19], [14], [18] [30], [15], [16], [2], [6], [37], [38]) to study regularity properties of the stochastic characteristics one first needs to establish precise regularity results for solutions to associated Kolmogorov equations.”
Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises (2016, background): “The reader can refer to [4, 11, 12, 16, 21] and references therein for recent studies on the existence and uniqueness of strong solution to (1.”
Noiseless regularisation by noise (2020, unknown): “α-stable and log regular processes) [1, 35, 41, 45, 16].”