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Extreme Value theory and Poisson statistics for discrete time samplings of stochastic differential equations

F. Flandoli, S. Galatolo, P. Giulietti, S. Vaienti

Abstract

We investigate the distribution and multiple occurrences of extreme events stochastic processes constructed by sampling the solution of a Stochastic Differential Equation on $\mathbb{R}^n$. We do so by studying the action of an annealead transfer operators on ad-hoc spaces of probability densities. The spectral properties of such operators are obtained by employing a mixture of techniques coming from SDE theory and a functional analytic approach to dynamical systems.

Extreme Value theory and Poisson statistics for discrete time samplings of stochastic differential equations

Abstract

We investigate the distribution and multiple occurrences of extreme events stochastic processes constructed by sampling the solution of a Stochastic Differential Equation on . We do so by studying the action of an annealead transfer operators on ad-hoc spaces of probability densities. The spectral properties of such operators are obtained by employing a mixture of techniques coming from SDE theory and a functional analytic approach to dynamical systems.
Paper Structure (18 sections, 23 theorems, 156 equations)

This paper contains 18 sections, 23 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Main Results
  4. Organization of the paper
  5. Transfer operators, Banach spaces and regularization lemmas.
  6. The Kolmogorov operator and the transfer operator
  7. Perturbed operators
  8. Functional Spaces: quasi-Hölder space on $\mathbb{R}^d$
  9. Regularization properties for the transfer operator.
  10. Regularization for the perturbed operators
  11. Rare Events Via Transfer Operator
  12. Perturbative hypotheses
  13. Sufficient conditions to check assumptions R1 and R2
  14. Verifying the perturbative assumptions in our case
  15. Proof of Theorem \ref{['thm:gen2']}
  16. ...and 3 more sections

Key Result

Theorem 2

Let $h,\tau>0$ and let $X_t$ be the solution of eq:system1 at time $t$. Let $u_n$ be a real sequence such that where $B_{n}$ is the ball $B\left( x_{0},e^{ -u_{n}} \right)$. Let $t_{k}:=kh$. Then

Theorems & Definitions (51)

  • Remark 1
  • Theorem 2
  • Remark 3
  • Remark 4
  • Definition 5
  • Theorem 6
  • Theorem 7: MPZ, Theorem 1.2 and Remark 1.3.
  • Definition 8
  • Definition 9: transfer operator
  • Lemma 10
  • ...and 41 more