{"ID":2897128,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.06098","arxiv_id":"2507.06098","title":"Nonparametric Estimation in SDE Models Involving an Explanatory Process","abstract":"This paper deals with the process $X = (X_t)_{t\\in [0,T]}$ defined by the stochastic differential equation (SDE) $dX_t = (a(X_t) + b(Y_t))dt +σ(X_t)dW_1(t)$, where $W_1$ is a Brownian motion and $Y$ is an exogenous process. The first task - of probabilistic nature - is to properly define the model, to prove the existence and uniqueness of the solution of such an equation, and then to establish the existence and a suitable control of a density with respect to the Lebesgue measure of the distribution of $(X_t,Y_t)$ ($t \u003e 0$). In the second part of the paper, a risk bound and a rate of convergence in specific Sobolev spaces are established for a copies-based projection least squares estimator of the $\\mathbb R^2$-valued function $(a,b)$. Moreover, a model selection procedure making the adequate bias-variance compromise both in theory and practice is investigated.","short_abstract":"This paper deals with the process $X = (X_t)_{t\\in [0,T]}$ defined by the stochastic differential equation (SDE) $dX_t = (a(X_t) + b(Y_t))dt +σ(X_t)dW_1(t)$, where $W_1$ is a Brownian motion and $Y$ is an exogenous process. The first task - of probabilistic nature - is to properly define the model, to prove the existen...","url_abs":"https://arxiv.org/abs/2507.06098","url_pdf":"https://arxiv.org/pdf/2507.06098v1","authors":"[\"Fabienne Comte\",\"Nicolas Marie\"]","published":"2025-07-08T15:39:21Z","proceeding":"math.ST","tasks":"[\"math.ST\"]","methods":"[]","has_code":false}