Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients

By Alessio Figalli. 2008

259 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/c1f79712b03b769080aed3eaca93339460c6c21e

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In Applied Mathematics

In Finance

Citations in the corpus

Renormalized Solutions for Stochastic Transport Equations and the Regularization by Bilinear Multiplicative Noise (2010, background): “The paper [12] deals with only L -solutions v of the parabolic equation (7) and proves uniqueness, but again assuming div b ∈ L , the generalization of which is one of our main purposes, otherwise the theory would be equal to the deterministic one.”
Well-posedness of Multidimensional Diffusion Processes with Weakly Differentiable Coefficients (2015, methodology): “The problem arising from the condition σ∇u ∈ L(2)t (L(2)x), which prevents a Lagrangian theory, is well understood in [19], where much effort is put in showing, for the bounded elliptic case, uniqueness in the class of narrowly continuous probability densities u ∈ L(2)t (L 2 x) [19, Theorem 4.”
From nonlinear Fokker–Planck equations to solutions of distribution dependent SDE (2018, result): “It turns out that, once one has solved (1.2), which is in general a hard task, and if one can prove some mild integrability properties for the solutions, a recent version of the socalled superposition principle by Trevisan in [30] (generalizing earlier work by Figalli [15]), in connection with a cla”
Existence and uniqueness of solutions for Fokker–Planck equations on Hilbert spaces (2009, background): “[1], [2], [21], [22], [19], [20] and the references therein and also the fundamental paper [18]).”
Weak Solutions to Fokker–Planck Equations and Mean Field Games (2015, background): “Moreover, which is also to be stressed, rigourous connections between PDE’s results and the associated stochastic flow were proved in this setting ([20], [25]).”
Well-posedness and large deviation for degenerate SDEs with Sobolev coefficients (2010, background): “In [14], Figalli has proved the well-posedness of martingale solutions for the SDE (1.”
Well-posedness of distribution dependent SDEs with singular drifts (2018, background): “2), which is originally due to Figalli [9] and Trevisan [23].”
Existence and Uniqueness of Solutions to Fokker–Planck Type Equations with Irregular Coefficients (2008, unknown): “The authors are grateful to A. Figalli for communicating a preprint version of [8].”
Quantitative estimates of propagation of chaos for stochastic systems with W-1,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{d (2017, background): “We refer to [31] for an example of such study.”
Superposition and mimicking theorems for conditional McKean–Vlasov equations (2020, methodology): “The results of [33] have been extended to a wide range of SDEs with unbounded coefficients in [66] (see also [12, 61] and the comparison with the previous work [58] in [66, Remark 3.”
Stochastic ODEs and stochastic linear PDEs with critical drift: regularity, duality and uniqueness (2014, background): “b ∈ L(2), if diffuse initial conditions with suitable density are considered; see [38, 13].”
Well-posedness of the transport equation by stochastic perturbation (2008, background): “See also generalizations to transport-diffusion equations and the associated stochastic differential equations by C. Le Bris and P.L. Lions [27] and A. Figalli [12] (in a direction different from the one of the present paper).”