To provide physical insight into the recently observed photoluminescence saturation behavior in single-walled carbon nanotubes implying the existence of an upper limit of exciton densities, we have performed a time-dependent theoretical study of diffusion-limited exciton-exciton annihilation in the general context of reaction-diffusion processes, for which exact treatments exist. By including the radiative recombination decay as a Poissonian process in the exactly solvable problem of one-dimensional diffusion-driven two-particle annihilation, we were able to correctly model the dynamics of excitons as a function of time with different initial densities, which in turn allowed us to reproduce the experimentally observed photoluminescence saturation behavior at high exciton densities. We also performed Monte Carlo simulations of the purely stochastic, Brownian diffusive motion of one-dimensional excitons, which validated our analytical results. Finally, by considering the diameter, chirality, and temperature dependence of this diffusion-limited exciton-exciton annihilation, we point out that high exciton densities in single-walled carbon nanotubes may be achieved at low temperature in an external magnetic field.