We introduce a flexible method to simultaneously infer both the drift and volatility functions of a discretely observed scalar diffusion. We introduce spline bases to represent these functions and develop a Markov chain Monte Carlo algorithm to infer, a posteriori, the coefficients of these functions in the spline basis. A key innovation is that we use spline bases to model transformed versions of the drift and volatility functions rather than the functions themselves. The output of the algorithm is a posterior sample of plausible drift and volatility functions that are not constrained to any particular parametric family. The flexibility of this approach provides practitioners a powerful investigative tool, allowing them to posit a variety of parametric models to better capture the underlying dynamics of their processes of interest. We illustrate the versatility of our method by applying it to challenging datasets from finance, paleoclimatology, and astrophysics. In view of the parametric diffusion models widely employed in the literature for those examples, some of our results are surprising since they call into question some aspects of these models.

翻译：我们提出了一种灵活方法，可同时推断离散观测标量扩散过程的漂移函数和波动函数。通过引入样条基函数表示这些函数，我们开发了马尔可夫链蒙特卡洛算法，在后验框架下推断函数在样条基中的系数。一项关键创新在于：我们使用样条基对漂移函数和波动函数的变换版本而非原函数进行建模。该算法输出一组合理的后验样本漂移函数与波动函数，其不局限于任何特定参数族。这种方法的灵活性为研究者提供了强大的探索工具，可提出多种参数模型以更精确地捕捉目标过程的动态特性。我们通过金融学、古气候学和天体物理学领域的挑战性数据集验证了该方法的广泛适用性。鉴于上述案例文献中广泛采用的参数扩散模型，部分结果令人惊讶——它们对现有模型的某些假设提出了质疑。