Approximation of SDEs by Population-Size-Dependent Galton–Watson Processes

Abstract

A certain class of stochastic differential equations, containing the Cox–Ingersoll–Ross model and the geometric Brownian motion, is considered. The corresponding solutions are approximated weakly by discrete-time population-size-dependent Galton–Watson processes with immigration. The long-time behavior of the limiting processes is also investigated.

Keywords:

• Cox–Ingersoll–Ross model
• Doob–Meyer decomposition
• Galton–Watson process
• Martingale problem
• Population-size-dependent branching
• Stochastic differential equation
• Weak convergence

Mathematics Subject Classification:

• Primary 60H10, 60J80
• Secondary 60F17