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Ridesharing markets are complex: drivers are strategic, rider demand and driver availability are stochastic, and complex city-scale phenomena like weather induce large scale correlation across space and time. At the same time, past work has focused on a subset of these challenges. We propose a model of ridesharing networks with strategic drivers, spatiotemporal dynamics, and stochasticity. Supporting both computational tractability and better modeling flexibility than classical fluid limits, we use a two-level stochastic model that allows correlated shocks caused by weather or large public events. Using this model, we propose a novel pricing mechanism: stochastic spatiotemporal pricing (SSP). We show that the SSP mechanism is asymptotically incentive-compatible and that all (approximate) equilibria of the resulting game are asymptotically welfare-maximizing when the market is large enough. The SSP mechanism iteratively recomputes prices based on realized demand and supply, and in this sense prices dynamically. We show that this is critical: while a static variant of the SSP mechanism (whose prices vary with the market-level stochastic scenario but not individual rider and driver decisions) has a sequence of asymptotically welfare-optimal approximate equilibria, we demonstrate that it also has other equilibria producing extremely low social welfare. Thus, we argue that dynamic pricing is important for ensuring robustness in stochastic ride-sharing networks.