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 Ft(x) := ∫-∞∞ g(y)pt(x, y) dy, an integral transform of the transition density pt(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 𝕃2(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 𝕃2(A)-norm. For this, we assume Ft lies in a given regularity space, specifically the Sobolev(-Hermite) ellipsoids. We propose a model selection procedure and show that the corresponding least squares estimator achieves the bias-variance compromise. Our choice to estimate Ft 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.