Research Paper#Stochastic Processes, Brownian Motion, Random Walks🔬 ResearchAnalyzed: Jan 3, 2026 08:45
Boundary Random Walks Converge to Feller's Brownian Motions
Analysis
This paper establishes a connection between discrete-time boundary random walks and continuous-time Feller's Brownian motions, a broad class of stochastic processes. The significance lies in providing a way to approximate complex Brownian motion models (like reflected or sticky Brownian motion) using simpler, discrete random walk simulations. This has implications for numerical analysis and understanding the behavior of these processes.
Key Takeaways
- •Establishes an invariance principle connecting boundary random walks and Feller's Brownian motions.
- •Provides a method for approximating a wide range of Brownian motion models using simpler random walks.
- •The behavior at the boundary is characterized by a quadruple (p1, p2, p3, p4), encompassing various classical models.
- •Offers insights into the numerical simulation and understanding of complex stochastic processes.
Reference
“For any Feller's Brownian motion that is not purely driven by jumps at the boundary, we construct a sequence of boundary random walks whose appropriately rescaled processes converge weakly to the given Feller's Brownian motion.”