Application of Kalman Filter in Stochastic Differential Equations

In areas such as finance, engineering, and science, we often face situations that change quickly and unpredictably. These situations are tough to handle and require special tools and methods capable of understanding and predicting what might happen next. Stochastic Differential Equations (SDEs) are renowned for modeling and analyzing real-world dynamical systems. However, obtaining the parameters, boundary conditions, and closed-form solutions of SDEs can often be challenging. In this paper, we will discuss the application of Kalman filtering theory to SDEs, including Extended Kalman filtering and Particle Extended Kalman filtering. We will explore how to fit existing SDE systems through filtering and track the original SDEs by fitting the obtained closed-form solutions. This approach aims to gather more information about these SDEs, which could be used in various ways, such as incorporating them into parameters of data-based SDE models.