Climate physics predicts that the intensity of natural disasters will increase in the future due to climate change. One of the biggest challenges for economic modeling is the inherent uncertainty of climate events, which crucially affects consumption, investment, and abatement decisions. We present a stochastic model of a growing economy where natural disasters are multiple and random, with damages driven by the economy's polluting activity. We provide a closed-form solution and show that the optimal path is characterized by a constant growth rate of consumption and the capital stock until a shock arrives, triggering a downward jump in both variables. Optimum mitigation policy consists of spending a constant fraction of output on emissions abatement. This fraction is an increasing function of the arrival rate, polluting intensity of output, and the damage intensity of emissions. A sharp response of the optimum growth rate and the abatement share to changes in the arrival rate and the damage intensity justifies more stringent climate policies as compared to the expectation-based scenario. We subsequently extend the baseline model by adding climate-induced fluctuations around the growth trend and stock-pollution effects, demonstrating robustness of our results. In a quantitative assessment of our model we show that the optimal abatement expenditure at the global level may represent 0.9% of output, which is equivalent to a tax of $71 per ton carbon.