The problem of estimating a parameter in the drift coefficient is addressed for $N$ discretely observed independent and identically distributed stochastic differential equations (SDEs). This is done considering additional constraints, wherein only public data can be published and used for inference. The concept of local differential privacy (LDP) is formally introduced for a system of stochastic differential equations. The objective is to estimate the drift parameter by proposing a contrast function based on a pseudo-likelihood approach. A suitably scaled Laplace noise is incorporated to meet the privacy requirements. Our key findings encompass the derivation of explicit conditions tied to the privacy level. Under these conditions, we establish the consistency and asymptotic normality of the associated estimator. Notably, the convergence rate is intricately linked to the privacy level, and is some situations may be completely different from the case where privacy constraints are ignored. Our results hold true as the discretization step approaches zero and the number of processes $N$ tends to infinity.

翻译：针对$N$个离散观测的独立同分布随机微分方程（SDEs），本文研究了漂移系数中参数的估计问题。在额外约束条件下（仅允许发布和使用公共数据进行推断），正式引入了随机微分方程系统的局部差分隐私（LDP）概念。通过提出基于伪似然方法的对比函数来实现漂移参数的估计，并加入适当尺度的拉普拉斯噪声以满足隐私要求。核心发现包括：推导了与隐私水平相关的显式条件。在该条件下，证明了相应估计量的一致性和渐近正态性。值得注意的是，收敛速度与隐私水平密切相关，在某些情况下可能完全不同于忽略隐私约束时的情形。当离散化步长趋近于零且过程数量$N$趋于无穷时，本文结论仍然成立。