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On the positivity of the density of stochastic delay differential equations driven by a fractional Brownian motion

Òscar Burés, Carles Rovira

Abstract

In this paper, we consider a Stochastic Delay Differential Equation with constant delay $r>0$ and, under the same conditions on the coefficients needed to ensure the smoothness of the density plus an ellipticity condition on the diffusion term, we prove that the density function of the solution is strictly positive in its support. In order to prove it, we give a Gaussian-type lower bound for the density of the solution combining the Nourdin and Viens' density bounding method together with Kohatsu-Higa's method.

On the positivity of the density of stochastic delay differential equations driven by a fractional Brownian motion

Abstract

In this paper, we consider a Stochastic Delay Differential Equation with constant delay and, under the same conditions on the coefficients needed to ensure the smoothness of the density plus an ellipticity condition on the diffusion term, we prove that the density function of the solution is strictly positive in its support. In order to prove it, we give a Gaussian-type lower bound for the density of the solution combining the Nourdin and Viens' density bounding method together with Kohatsu-Higa's method.

Paper Structure

This paper contains 12 sections, 17 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Wiener space associated to the fractional Brownian motion
  4. Malliavin Calculus tools
  5. Lower bound for the density
  6. The case $t \in (0,r]$
  7. A general bounding technique
  8. The case $t > r$
  9. Lower bound for $J_{1,n}$
  10. Upper bound for $|J_{2,n}|$
  11. Proof of the lower bound
  12. Real Analysis tools

Key Result

Proposition 2.3

Let $\mathbb{D}^{1,2}_W$ be the Malliavin-Sobolev space associated to $W$. Then, $\mathbb{D}^{1,2} = (K_H^*)^{-1} \mathbb{D}^{1.2}_W$ and for every $F \in \mathbb{D}_W^{1,2}$, we have whenever both sides of the equality are well defined.

Theorems & Definitions (33)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Remark 1
  • Remark 2
  • Theorem 3.1
  • Theorem 3.2
  • Corollary 3.3
  • ...and 23 more