Nonparametric estimation of an integral transform of the transition density for diffusion processes

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. Our choice to estimate $F_t$ directly is motivated by applications in financial mathematics and a connection to kernel mean embeddings of conditional distributions. We conclude by applying the estimator in a simple option pricing scenario and illustrate its performance for various diffusion processes and candidates for $g$.