Nonparametric Estimation in SDE Models Involving an Explanatory Process

This paper deals with the process $X = (X_t)_{t\in [0,T]}$ defined by the stochastic differential equation (SDE) $dX_t = (a(X_t) + b(Y_t))dt +\sigma(X_t)dW_1(t)$, 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 $(X_t,Y_t)$ ($t>0$). 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.

• Publication year

  2025

• Venue

  Unknown venue

• Publication date

  2025-07-08

• Fields of study

  Mathematics

• Identifiers
  arXiv 2507.06098
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

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