Large deviation principle for the mean reflected stochastic differential equation with jumps.

Area of Science:

• Stochastic Analysis
• Probability Theory
• Mathematical Physics

Background:

• Stochastic differential equations (SDEs) are fundamental in modeling complex systems.
• Mean reflected SDEs introduce boundary conditions that complicate analysis.
• Understanding the tail behavior of SDEs is critical for risk assessment.

Purpose of the Study:

• To establish a large deviation principle for mean reflected SDEs.
• To analyze systems driven by both Brownian motion and Poisson random measures.
• To provide a rigorous mathematical framework for the tail probabilities of these processes.

Main Methods:

• Utilizing the weak convergence method.
• Applying techniques from large deviation theory.
• Developing novel approaches for reflected diffusion processes.

Main Results:

• A precise large deviation principle is established for the considered mean reflected SDE.
• The analysis accounts for the interplay between continuous (Brownian) and jump (Poisson) noise.
• The results quantify the exponential rate of divergence for the system's trajectories.

Conclusions:

• The weak convergence method is effective for analyzing large deviations in mean reflected SDEs.
• This work extends existing large deviation results to a more general class of SDEs.
• The findings have implications for financial mathematics and other fields utilizing stochastic modeling.