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Estimates for exponential functionals of continuous Gaussian processes with emphasis on fractional Brownian motion

José Alfredo López-Mimbela, Gerardo Pérez-Suárez

Abstract

Our aim in this article is to provide explicit computable estimates for the cumulative distribution function (c.d.f.) and the $p$-th order moment of the exponential functional of a fractional Brownian motion (fBM) with drift. Using elementary techniques, we prove general upper bounds for the c.d.f. of exponential functionals of continuous Gaussian processes. On the other hand, by applying classical results for extremes of Gaussian processes, we derive general lower bounds. We also find estimates for the $p$-th order moment and the moment-generating function of such functionals. As a consequence, we obtain explicit lower and upper bounds for the c.d.f. and the $p$-th order moment of the exponential functionals of a fBM, and of a series of independent fBMs. In addition, we show the continuity in law of the exponential functional of a fBM with respect to the Hurst parameter.

Estimates for exponential functionals of continuous Gaussian processes with emphasis on fractional Brownian motion

Abstract

Our aim in this article is to provide explicit computable estimates for the cumulative distribution function (c.d.f.) and the -th order moment of the exponential functional of a fractional Brownian motion (fBM) with drift. Using elementary techniques, we prove general upper bounds for the c.d.f. of exponential functionals of continuous Gaussian processes. On the other hand, by applying classical results for extremes of Gaussian processes, we derive general lower bounds. We also find estimates for the -th order moment and the moment-generating function of such functionals. As a consequence, we obtain explicit lower and upper bounds for the c.d.f. and the -th order moment of the exponential functionals of a fBM, and of a series of independent fBMs. In addition, we show the continuity in law of the exponential functional of a fBM with respect to the Hurst parameter.
Paper Structure (14 sections, 31 theorems, 182 equations, 7 figures)

This paper contains 14 sections, 31 theorems, 182 equations, 7 figures.

Table of Contents

  1. Introduction and background
  2. Exponential functionals of continuous Gaussian processes
  3. Upper bounds
  4. Lower bounds
  5. Moment and moment-generating function estimates
  6. Exponential functionals of fBM
  7. Case $T\in(0,\infty)$
  8. The case $T=\infty$
  9. The case $H=1$
  10. The case $H=1/2$
  11. Continuity in law of exponential functionals of fBMs
  12. Exponential functional of a series of independent fBMs
  13. Numerical examples and discussion
  14. Comparison with previous results

Key Result

Lemma 2.1

Let $f\in \mathcal{M}_T^{X}$. The random variable $J_{T}^{X}(f)$ has normal distribution with mean $m_T^{X}(f)$ and variance $(s_T^{X}(f))^2$, and it satisfies

Figures (7)

  • Figure 1: Upper bounds for the c.d.f. of $I_{1}^{1,1,3/4}$ obtained in Theorem \ref{['cor-upper-bound-finite']} with $\lambda=-1,0.5,0,0.5,1$. The red line corresponds to the case $\lambda=0$.
  • Figure 2: Upper bounds (blue lines) are derived from Theorem \ref{['cor-upper-bound-finite']} with $\lambda=0$. Lower bounds (red lines) are derived from dung. The e.c.d.f. of $I_T^{\mu,\sigma,H}$ (green lines) was plotted with $1000$ simulations of $I_T^{\mu,\sigma,H}$.
  • Figure 3: Upper bounds (blue lines) are derived from Theorem \ref{['cor-upper-bound-finite']} with $\lambda=0$. Lower bounds (red lines) are derived from Theorem \ref{['lower-bound-finite-ii']} with $\lambda=0$. The e.c.d.f. of $I_T^{\mu,\sigma,H}$ (green lines) was plotted with $1000$ simulations of $I_T^{\mu,\sigma,H}$.
  • Figure 4: Upper bounds (blue lines) and lower bounds (red lines) for $\mathbb{E}[\exp(-\lambda I_T^{\mu,\sigma,H})]$ are derived from Corollary \ref{['cor-gmf-finite']}.
  • Figure 5: Upper bounds (blue lines) are derived from Theorem \ref{['bounds_infinity']} (iii). Lower bounds (red lines) are derived from Theorem \ref{['pro-lowe-bound-func-infinite']}. The exact c.d.f. of $I_\infty^{\mu,\sigma,1/2}$ (orange lines) is given by (\ref{['func_exp_brow']}).
  • ...and 2 more figures

Theorems & Definitions (75)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • ...and 65 more