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Statistical Inference for Fractional Diffusions

Pablo Ramses Alonso-Martin, Horatio Boedihardjo, Anastasia Papavasiliou

Abstract

This is a review of statistical inference methodology for stochastic differential equations driven by fractional Brownian motion, otherwise called fractional diffusions. The first section reviews the theory needed to rigorously define them. The second section reviews existing theory of statistical inference for fractional diffusions, identifies remaining challenges and introduces a novel approach. The final section discusses results for the case where fractional diffusions result as a homogenisation limit.

Statistical Inference for Fractional Diffusions

Abstract

This is a review of statistical inference methodology for stochastic differential equations driven by fractional Brownian motion, otherwise called fractional diffusions. The first section reviews the theory needed to rigorously define them. The second section reviews existing theory of statistical inference for fractional diffusions, identifies remaining challenges and introduces a novel approach. The final section discusses results for the case where fractional diffusions result as a homogenisation limit.

Paper Structure

This paper contains 32 sections, 17 theorems, 260 equations.

Table of Contents

  1. Introduction to Fractional Diffusions
  2. Introduction
  3. Differential equations driven by fractional Brownian motion
  4. $H>\frac{1}{2}$ case: Young integration
  5. $\frac{1}{4}<H\leq\frac{1}{2}$ case: rough path regime
  6. Iterated integrals of fractional Brownian motion
  7. Tensor products
  8. Properties of iterated integrals for fractional Brownian motion
  9. Integrals against $\alpha$-Hölder geometric rough paths
  10. Integrals of controlled rough paths
  11. Solutions to differential equations driven by fractional Brownian motion
  12. Statistical Inference for Fractional Diffusions
  13. Literature review
  14. Fractional Ornstein--Uhlenbeck model
  15. Fractional Vasicek model
  16. ...and 17 more sections

Key Result

Lemma 1.2

If $y:[0,1]\to\mathbb{R}$ and $z:[0,1]\to\mathbb{R}$ are $\alpha$-Hölder continuous and $\beta$-Hölder continuous, respectively, with $\alpha+\beta>1$, then the limit exists. Here, $\lim_{|\mathcal{P}|\to 0}$ denotes the limit as the mesh size of partitions $\mathcal{P}=(t_0,\ldots,t_n)$ of $[s,t]$ tends to zero. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (34)

  • Definition 1.1
  • Lemma 1.2: Young36
  • Definition 1.3
  • Definition 1.4
  • Remark 1.5
  • Lemma 1.6: LyonsCaruanaLevy
  • Definition 1.7: GubControlRP
  • Lemma 1.8: GubControlRP
  • Theorem 1.9: GubControlRP
  • Theorem 2.1
  • ...and 24 more