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：本文针对$N$个离散观测的独立同分布随机微分方程（SDEs）系统，研究了漂移系数中参数的估计问题。该研究考虑了额外的约束条件，即仅允许公开数据被发布并用于统计推断。我们正式为随机微分方程系统引入了局部差分隐私（LDP）的概念。研究目标是通过构建基于伪似然方法的对比函数来估计漂移参数。为满足隐私要求，我们在模型中加入了适当缩放后的拉普拉斯噪声。主要研究成果包括：推导出与隐私级别相关的显式条件，并在这些条件下证明了所提估计量的一致性与渐近正态性。值得注意的是，收敛速率与隐私水平存在复杂关联，在某些情况下可能完全不同于忽略隐私约束的情形。我们的结论在离散化步长趋于零且过程数量$N$趋于无穷时成立。