Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Solution theory of fractional SDEs in complete subcritical regimes

Lucio Galeati, Máté Gerencsér

Abstract

We consider stochastic differential equations (SDEs) driven by a fractional Brownian motion with a drift coefficient that is allowed to be arbitrarily close to criticality in a scaling sense. We develop a comprehensive solution theory that includes strong existence, path-by-path uniqueness, existence of a solution flow of diffeomorphisms, Malliavin differentiability and $ρ$-irregularity. As a consequence, we can also treat McKean-Vlasov, transport and continuity equations.

Solution theory of fractional SDEs in complete subcritical regimes

Abstract

We consider stochastic differential equations (SDEs) driven by a fractional Brownian motion with a drift coefficient that is allowed to be arbitrarily close to criticality in a scaling sense. We develop a comprehensive solution theory that includes strong existence, path-by-path uniqueness, existence of a solution flow of diffeomorphisms, Malliavin differentiability and -irregularity. As a consequence, we can also treat McKean-Vlasov, transport and continuity equations.
Paper Structure (21 sections, 41 theorems, 199 equations)

This paper contains 21 sections, 41 theorems, 199 equations.

Table of Contents

  1. Introduction
  2. Scaling heuristics and existing literature
  3. Discussion of the main results
  4. Counterexamples to uniqueness in the supercritical regime
  5. Preliminaries on fractional Brownian motion
  6. Setup and notation
  7. A priori estimates and stochastic sewing
  8. Stability
  9. Strong well-posedness for functional drift
  10. Strong well-posedness for distributional drift
  11. Flow regularity and Malliavin differentiability
  12. McKean-Vlasov equations
  13. Weak compactness and weak existence
  14. $\rho$-irregularity
  15. Applications to transport and continuity equations
  16. ...and 6 more sections

Key Result

Theorem 1.4

Assume eq:main exponent and let $x_0\in\mathbb{R}^d$, $b\in L^q_t C^\alpha_x$, $m\in[1,\infty)$. Then:

Theorems & Definitions (110)

  • Example 1.1
  • Example 1.2
  • Example 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 100 more