Stochastic cusp catastrophe model and its Bayesian computations

Area of Science:

• Catastrophe Theory
• Stochastic Differential Equations
• Bayesian Statistics

Background:

• The classical cusp catastrophe model is vital in understanding system bifurcations.
• Statistical inference for stochastic cusp differential equations is analytically intractable due to complex transition densities.

Purpose of the Study:

• To address the unsolved statistical inference problem for the stochastic cusp differential equation.
• To develop a novel Bayesian approach for analyzing the cusp catastrophe model.

Main Methods:

• A novel Bayesian framework was developed.
• Hamiltonian Monte Carlo (HMC) was combined with Euler approximation and Hermite expansion for likelihood approximation.
• The approach was validated using simulation studies.

Main Results:

• The proposed Bayesian method successfully handles the analytically intractable likelihood function.
• Simulation studies confirmed the validity and efficacy of the novel approach.
• The method demonstrated practical applicability in analyzing real-world financial data.

Conclusions:

• The novel Bayesian approach provides a viable solution for statistical inference in stochastic cusp differential equations.
• This method offers a powerful tool for analyzing complex systems exhibiting catastrophic bifurcations.
• The approach has potential applications in econometrics and financial market analysis, such as exchange rate modeling.