We study a class of Stochastic Differential Equations (SDEs) with jumps modeling multistage Michaelis--Menten enzyme kinetics, in which a substrate is sequentially transformed into a product via a cascade of intermediate complexes. These networks are typically high-dimensional and exhibit multiscale behavior with a strong coupling between different components, posing substantial analytical and computational challenges. In particular, the problem of statistical inference of reaction rates is significantly difficult and becomes even more intricate when direct observations of system states are unavailable and only a random sample of product formation times is observed. We address this problem in two stages. First, in a suitable scaling regime consistent with the Quasi-Steady State Approximation (QSSA), we rigorously establish a stochastic averaging principle yielding a reduced model for the product-substrate dynamics. Guided by the reduced-order dynamics, we next construct a novel Interacting Particle System (IPS) that approximates the product-substrate process at the particle level. This IPS plays a pivotal role in the inference methodology, and we prove several non-asymptotic bounds and limiting results for this system. These results facilitate the construction of an estimator based on a product-form approximate likelihood requiring only a random sample of product formation times. This approach does not need access to the system states, and we rigorously prove consistency of the estimator.

翻译：研究一类描述多阶段米氏酶动力学的带跳随机微分方程，其中底物通过一系列中间复合物级联反应逐步转化为产物。这类网络通常具有高维特征，且因不同组分间强耦合呈现多尺度行为，为分析与计算带来显著挑战。特别地，当系统状态不可直接观测、仅能获取产物生成时间的随机样本时，反应速率的统计推断问题尤为困难。我们分两阶段解决该问题：首先，在与准稳态近似相容的适当标度条件下，严格建立随机平均原理，得到产物-底物动力学的降阶模型；其次，基于降阶动力学，构建能近似产物-底物过程粒子层面的新型交互粒子系统。该粒子系统在推断方法中发挥关键作用，我们证明了该系统的若干非渐近界及极限结果。这些结论支撑了基于乘积形式近似似然的估计量构建——该估计量仅需产物生成时间的随机样本，无需访问系统状态，并严格证明了估计量的相合性。