Nonparametric Estimation in SDE Models Involving an Explanatory Process
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| Format: | Preprint |
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2025
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| _version_ | 1866916832748765184 |
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| author | Comte, Fabienne Marie, Nicolas |
| author_facet | Comte, Fabienne Marie, Nicolas |
| contents | 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 $(X_t,Y_t)$ ($t > 0$). 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_06098 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonparametric Estimation in SDE Models Involving an Explanatory Process Comte, Fabienne Marie, Nicolas Statistics Theory 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 $(X_t,Y_t)$ ($t > 0$). 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. |
| title | Nonparametric Estimation in SDE Models Involving an Explanatory Process |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2507.06098 |