Pathwise uniqueness and continuous dependence for SDEs with non-regular drift

By E. Fedrizzi, Franco Flandoli. 2010

114 citations recorded by the source corpus.

A new proof of a pathwise uniqueness result of Krylov and Röckner is given. It concerns SDEs with drift having only certain integrability properties. In spite of the poor regularity of the drift, pathwise continuous dependence on initial conditions may be obtained, by means of this new proof. The proof is formulated in such a way to show that the only major tool is a good regularity theory for the heat equation forced by a function with the same regularity of the drift.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Finance

View Paper: https://www.semanticscholar.org/paper/4396a4dc56555e1609ae0e6c7d2560c87036843e

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Computational Theory and Mathematics

In Finance

Citations in the corpus

Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness (2014, background): “However, we are not able to show that this representative is jointly continuous in (t, x) (we actually do not even show joint measurability), though such joint continuity is known to be true in the subcritical case (see [34]).”
Stochastic flows for Lévy processes with Hölder drifts (2015, background): “When α ∈ (0, 1), this in particular partially solves an open problem from Priola [18].”
Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift (2011, methodology): “ Kolmogorov equations. Such regularity results, leading to continuity or smoothness of thesolutions, however, do not hold in inﬁnitedimensions.A technique diﬀerent from [33] is used in [14]; see also [5, 10] and [29]. This technique allows us to prove uniqueness for stochastic equations with time in”
Sobolev differentiable stochastic flows for SDEs with singular coefficients: Applications to the transport equation (2012, methodology): “ on a Zvonkin-type transformation [32] and estimates of solutions of an associated backward parabolic PDE. We SOBOLEV FLOWS FOR SDES WITH SINGULAR COEFFICIENTS 5 also mention the recent related works [9, 10] and [2]. For an overview of this topic, the reader may also consult the book [13].2 In this ”
Noise Prevents Singularities in Linear Transport Equations (2012, methodology): “enerates, under assumptions (1), a stochastic ﬂow of H¨older continuous homeomorphisms, with some weak formofdiﬀerentiability. The existence of an H¨older continuous stochastic ﬂow has been proved in [12], [13], [21]. A diﬀerentiability property in terms of ﬁnite increments has been given in [13]. H”
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.”