arXiv subjects

Hamdi Fathallah

Publications and source records attributed to Hamdi Fathallah.

7 recordsLinked to original sources

arXiv2026

Random Jump Intensities and Bernstein Density Estimation in Ergodic Basic Affine Jump-Diffusion Processes

This paper develops a random-effects extension of the ergodic basic affine jump-diffusion (BAJD) model for a population of independent trajectories with individual unobserved jump intensities and common structural parameters. Under continuous-time observation, each intensity is estimated by the empirical jump frequency. We establish the strong consistency and stable mixed-normal limit of this estimator as the observation horizon tends to infinity. The estimated intensities are then used as pseudo-observations to construct a Bernstein-polynomial estimator of their common density. Its bias, variance, mean integrated squared error, and pointwise asymptotic normality are derived under a sequential asymptotic framework. The theory is developed for general positive jump-size distributions with finite activity and specialized to the Gamma$(k,\lambda)$ family, including the exponential and Erlang cases. For this family, we derive a pooled estimator of the common rate parameter and establish its consistency, asymptotic normality, and first-order conditional asymptotic independence from the individual intensity estimators. We also propose a consistent plug-in estimator of the population stationary mean. The methodology is illustrated through simulations and an empirical application to financial realized-volatility data.

stat.ME↗

arXiv2025

Hybrid estimation for a mixed fractional Black-Scholes model with random effects from discrete time observations

We propose a hybrid estimation procedure to estimate global fixed parameters and subject-specific random effects in a mixed fractional Black-Scholes model based on discrete-time observations. Specifically, we consider $N$ independent stochastic processes, each driven by a linear combination of standard Brownian motion and an independent fractional Brownian motion, and governed by a drift term that depends on an unobserved random effect with unknown distribution. Based on $n$ discrete time statistics of process increments, we construct parametric estimators for the Brownian motion volatility, the scaling parameter for the fractional Brownian motion, and the Hurst parameter using a generalized method of moments. We establish their strong consistency under the two-step regime where the observation frequency $n$ and then the sample size $N$ tend to infinity, and prove their joint asymptotic normality when $H \in \big(\frac12, \frac34\big)$. Then, using a plug-in approach, we consistently estimate the random effects, and we study their asymptotic behavior under the same sequential asymptotic regime. Finally, we construct a nonparametric estimator for the distribution function of these random effects using a Lagrange interpolation at Chebyshev-Gauss nodes based method, and we analyze its asymptotic properties as both $n$ and $N$ increase. We illustrate the theoretical results through a numerical simulation framework. We further demonstrate the efficiency performance of the proposed estimators in an empirical application to crypto returns data, analyzing five major cryptocurrencies to uncover their distinct volatility structures and heterogeneous trend behaviors.

math.ST↗

arXiv2025

Asymptotic properties and drift parameter estimations of the ergodic double Heston model based on continuous-time observations

The double Heston model is one of the most popular option pricing models in financial theory. It is applied to several issues such that risk management and volatility surface calibration. This paper deals with the problem of global parameter estimations in this model. Our main stochastic results are about the stationarity and the ergodicity of the double Heston process. The statistical part of this paper is about the maximum likelihood and the conditional least squares estimations based on continuous-time observations; then for each estimation method, we study the asymptotic properties of the resulted estimators in the ergodic case.

math.ST↗

arXiv2024

Random effects estimation in a fractional diffusion model based on continuous observations

The purpose of the present work is to construct estimators for the random effects in a fractional diffusion model using a hybrid estimation method where we combine parametric and nonparametric thechniques. We precisely consider $n$ stochastic processes $\left\{X_t^j,\ 0\leq t\leq T\right\}$, $j=1,\ldots, n$ continuously observed over the time interval $[0,T]$, where the dynamics of each process are described by fractional stochastic differential equations with drifts depending on random effects. We first construct a parametric estimator for the random effects using the techniques of maximum likelihood estimation and we study its asymptotic properties when the time horizon $T$ is sufficiently large. Then by taking into account the obtained estimator for the random effects, we build a nonparametric estimator for their common unknown density function using Bernstein polynomials approximation. Some asymptotic properties of the density estimator, such as its asymptotic bias, variance and mean integrated squared error, are studied for an infinite time horizon $T$ and a fixed sample size $n$. The asymptotic normality and the uniform convergence of the estimator are investigated for an infinite time horizon $T$, a high frequency and as the order of Bernstein polynomials is sufficiently large. Some numerical simulations are also presented to illustrate the performance of the Bernstein polynomials based estimator compared to standard Kernel estimator for the random effects density function.

math.ST↗

arXiv2024

On Conditional least squares estimation for the AD(1,n) model

This paper deals with the problem of global parameter estimation of AD(1, n) where n is a positive integer which is a subclass of affine diffusions introduced by Duffie, Filipovic, and Schachermayer. In general affine models are applied to the pricing of bond and stock options, which is illustrated for the Vasicek, Cox-Ingersoll-Ross and Heston models. Our main results are about the conditional least squares estimation of AD(1, n) drift parameters based on two types of observations : continuous time observations and discrete time observations with high frequency and infinite horizon. Then, for each case, we study the asymptotic properties according to ergodic and non-ergodic cases. This paper introduces as well some moment results relative to the AD(1, n) model.

math.ST↗

arXiv2023

Asymptotic properties of AD(1, n) model and its maximum likelihood estimator

This paper deals with the problem of global parameter estimation of affine diffusions in $\mathbb{R}_+ \times \mathbb{R}^n$ denoted by $AD(1, n)$ where $n$ is a positive integer which is a subclass of affine diffusions introduced by Duffie et al in [14]. The $AD(1, n)$ model can be applied to the pricing of bond and stock options, which is illustrated for the Vasicek, Cox-Ingersoll-Ross and Heston models. Our first result is about the classification of $AD(1, n)$ processes according to the subcritical, critical and supercritical cases. Then, we give the stationarity and the ergodicity theorems of this model and we establish asymptotic properties for the maximum likelihood estimator in both subcritical and a special supercritical cases.

math.ST↗

arXiv2006

Identification d'un processus autorégressif gaussien stable par la méthode de moyennisation logarithmique

In the present work, we consider a stable one-dimensional gaussian autoregressive model in continous time. Using the limit theorems with logarithmic averaging obtained for continous local martingales, we construct then an estimator of the noise covariance $σ^{2}$ and an estimator of $θ$ different of the one of the least squares estimator. By exploiting the weighting method we ameliorate the convergence rates of these new estimators.

math.PR↗