Preprint announcement no.1
Happy to announce the release of my first preprint on HAL!
I am happy to announce that I have submitted my first preprint to HAL. You can find the paper, entitled “Nonparametric estimation of an integral transform of the transition density for diffusion processes” under the following link: https://hal.science/hal-05024650.
Abstract
We consider the observation of $N$ independent diffusion processes on a time interval $[0,2T]$. Given a fixed time $t$, we construct a nonparametric estimator of $F_t(x) := \int_{-\infty}^{\infty} g(y) p_t (x,y) \,dy$, an integral transform of the transition density $p_t(x,y)$ with a known, real map $g$. The estimator is defined as the minimizer of a least-squares regression contrast over a finite-dimensional subspace of $\mathbb{L}^2(A, dx)$. We prove risk bounds in reference norms under various assumptions on g and the estimation interval and study the bias- and variance terms to assess the rate of convergence in the standard $\mathbb{L}^2(A)$-norm. For this, we assume $F_t$ lies in a given regularity space, specifically the Sobolev(-Hermite) ellipsoids. We propose a model selection procedure and show that the corresponding least-square estimator achieves the bias-variance compromise. Estimating $F_t$ has applications in financial mathematics, for instance in option pricing. We conclude by illustrating the performance of our estimator for various diffusion processes and candidates for g, and finally by applying it to an option pricing scenario.
