A prototypical fluid-structure interaction problem is that of an closed elastic filament immersed in 2D Stokes flow, where the fluids inside and outside the closed filament have equal viscosity. This problem was probably first introduced in the context of Peskin's immersed boundary method, and is often used to test computational methods for FSI problems. Here, we study the well-posedness and qualitative behavior of this problem. We show local existence and uniqueness of solution with initial configuration in the Holder space $C^{1,\alpha}$, $0<\alpha<1$, and show furthermore that the solution is smooth for positive time. We show that the circular configurations are the only stationary configurations, and show exponential asymptotic stability with an explicit rate. Finally, we identify a scalar quantity that goes to infinity if and only if the solution ceases to exist. If this quantity is bounded for all time, we show that the solution must converge exponentially to a circle. This is joint work with Analise Rodenberg and Dan Spirn.