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Euler scheme for stochastic functional differential equations driven by fractional Brownian motion

Johanna Garzón, Jorge A. León, Jorge Lozada, Soledad Torres

Abstract

In this paper, we apply rough paths techniques to provide an approximation of the solution of stochastic functional differential equations driven by fractional Brownian motion with Hurst parameter $H>1/2$. Here, the involved stochastic integral is the Young one and the coefficient is evaluated in the set of $λ$-Hölder continuous functions on $[-τ,0]$, for some suitable $τ>0$ and $λ\in(1/2,H)$. The rate of convergence of our scheme is $1/n^γ$, for any $γ<2λ-1$. Also, numerical simulations are provided to illustrate our theoretical results.

Euler scheme for stochastic functional differential equations driven by fractional Brownian motion

Abstract

In this paper, we apply rough paths techniques to provide an approximation of the solution of stochastic functional differential equations driven by fractional Brownian motion with Hurst parameter . Here, the involved stochastic integral is the Young one and the coefficient is evaluated in the set of -Hölder continuous functions on , for some suitable and . The rate of convergence of our scheme is , for any . Also, numerical simulations are provided to illustrate our theoretical results.

Paper Structure

This paper contains 7 sections, 6 theorems, 155 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hölder continuous function and Young integral
  4. Stochastic functional differential equations
  5. Euler Approximation
  6. Convergence of the Euler scheme
  7. Simulations

Key Result

Proposition 2.1

Let $\alpha, \ \beta \in (0, 1]$ such that $\alpha+\beta> 1$, $h\in C^\alpha([0, T]; \mathbb{R})$ and $g\in C^\beta([0, T]; \mathbb{R})$. Then, for $(s,t)\in{\mathcal{S}}_{2}([0,T])$, the Young integral $\int_{s}^{t} h(r) \, dg(r)$ is defined as the limit of Riemann sums. That is, where $\pi_{s,t}=\{s=t_0<t_1,\ldots<,t_m=t\}$ is a partition of $[s,t]$. Furthermore, there exists a constant $c_{\a

Figures (1)

  • Figure :

Theorems & Definitions (14)

  • Proposition 2.1: Young integral for Hölder continuous functions
  • Lemma 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 4 more