Nesrine Chebli | Searcharxiv

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.

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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.

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