Existence, Uniqueness and Stability of Invariant Distributions in Continuous-Time Stochastic Models

We study a dynamic stochastic general equilibrium model in continuous time. Related work has proven that optimal consumption in this model is a smooth function of state variables. This allows us to describe the evolution of optimal state variables (wealth and labour market status) by stochastic differential equations. We derive conditions under which an invariant distribution for state variables exists and is unique. We also provide conditions such that initial distributions converge to the long-run distribution.