Nanoscale filament is the active area in oxygen vacancy type resistance random access memory (ReRAM), which is formed stochastically during electric test after being fabricated in a clean room. That is, the filament dimension cannot be controlled with a designed mask pattern. Here, we introduce a formula to describe the current cycle-to-cycle trajectories based on a stochastic differential equation (SDE) with microscopic structure parameters: filament dimension and oxygen vacancy concentration. Since ReRAM conduction follows hopping, the filament can be described by a random resistance network (RRN). The stochastic configuration of an RRN follows the Brownian motion, which is the key parameter in the diffusion of SDE. The formula provides a practically quantitative filament characterization method, which is verified by direct observation of the filament in actual devices. Based on the formula, we can predict ReRAM endurance with the given microscopic structure parameters.