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Application of Kalman Filter in Stochastic Differential Equations

Wencheng Bao, Shi Feng, Kaiwen Zhang

TL;DR

The paper demonstrates how Kalman filtering and its extensions (EKF and Particle-EKF) can be applied to stochastic differential equations to estimate parameters and track hidden states when closed-form solutions are unavailable. By formulating parameter estimation within a state-space and likelihood framework, it compares linear, nonlinear, and non-Gaussian filtering approaches across OU, OU with jumps, Heston, and Bates models. The results show accurate state tracking and substantial time savings for parameter estimation relative to MLE in many linear or near-linear settings, with EKF offering reasonable performance for nonlinear models and Particle-EKF enabling tracking in more complex, jump-diffusion dynamics. These findings underscore the practical potential of Kalman-filter-based methods for real-time SDE inference and motivate future work on more complex models and predictive applications.

Abstract

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.

Application of Kalman Filter in Stochastic Differential Equations

TL;DR

The paper demonstrates how Kalman filtering and its extensions (EKF and Particle-EKF) can be applied to stochastic differential equations to estimate parameters and track hidden states when closed-form solutions are unavailable. By formulating parameter estimation within a state-space and likelihood framework, it compares linear, nonlinear, and non-Gaussian filtering approaches across OU, OU with jumps, Heston, and Bates models. The results show accurate state tracking and substantial time savings for parameter estimation relative to MLE in many linear or near-linear settings, with EKF offering reasonable performance for nonlinear models and Particle-EKF enabling tracking in more complex, jump-diffusion dynamics. These findings underscore the practical potential of Kalman-filter-based methods for real-time SDE inference and motivate future work on more complex models and predictive applications.

Abstract

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.
Paper Structure (24 sections, 53 equations, 12 figures, 5 tables, 3 algorithms)

This paper contains 24 sections, 53 equations, 12 figures, 5 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Technique Simulation Review
  3. Maximum Likelihood Estimation (MLE) Framework
  4. Likelihood Function
  5. Conditional Density for Itô Diffusion
  6. Marginal likelihood
  7. Kalman Filter Framework
  8. Linear Kalman Regression
  9. Extended Kalman Filter (EKF)
  10. Particle Extended Kalman Filter (Particle-EKF)
  11. SDE Simulation Results
  12. Ornstein–Uhlenbeck (OU) Process
  13. MLE of OU Process
  14. Kalman Filter of OU Process
  15. OU Process with Jump
  16. ...and 9 more sections

Figures (12)

  • Figure 1: Neural Network to Approach a Heston Model
  • Figure 2: Simulation of Ornstein–Uhlenbeck Process with original and MLE
  • Figure 3: Simulation of Black–Karasinski Model with original and MLE
  • Figure 4: OU Process Tracking with Kalman Filter
  • Figure 5: Simulation of Ornstein–Uhlenbeck Process with jump and MLE
  • ...and 7 more figures