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Central limit theorems for martingales-I : continuous limits

Bruno Rémillard, Jean Vaillancourt

Abstract

When the limiting compensator of a sequence of martingales is continuous, we obtain a weak convergence theorem for the martingales; the limiting process can be written as a Brownian motion evaluated at the compensator and we find sufficient conditions for both processes to be independent. Examples of applications are provided, notably for occupation time processes and statistical estimators of financial volatility measures.

Central limit theorems for martingales-I : continuous limits

Abstract

When the limiting compensator of a sequence of martingales is continuous, we obtain a weak convergence theorem for the martingales; the limiting process can be written as a Brownian motion evaluated at the compensator and we find sufficient conditions for both processes to be independent. Examples of applications are provided, notably for occupation time processes and statistical estimators of financial volatility measures.
Paper Structure (11 sections, 10 theorems, 45 equations)

This paper contains 11 sections, 10 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Examples of application to occupation times
  4. Occupation times for Brownian motion
  5. Occupation times for random walks
  6. Scaling limit for the random comb
  7. Volatility modeling and estimation
  8. Conclusion
  9. Auxiliary results
  10. Some useful results
  11. Expression for $\mu_v$

Key Result

Theorem 2.1

Assume that Hypothesis hyp:an_unbounded holds with $A$ continuous everywhere; that $J_t(M_n) \stackrel{Law}{\to} 0$ for any $t>0$; that there exists an $\mathbb{F}$-adapted sequence of $D$-valued square integrable martingales $B_n$ started at $B_n(0)=0$ so that Then $(M_n, A_n,B_n) \stackrel{\mathcal{C}}{\rightsquigarrow} (M,A,B)$ holds, where $M$ is a continuous square integrable ${\mathcal{F}}_

Theorems & Definitions (25)

  • Theorem 2.1
  • proof
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Remark 3.1
  • Lemma 4.1
  • proof
  • Remark 4.1
  • Theorem 4.2: Numerical scheme
  • ...and 15 more