We analyze a stochastic dynamic finite-horizon economic model with climate change, in which the social planner faces uncertainty about future climate change and its economic damages. Our model (SDICE*) incorporates, possibly heavy-tailed, stochasticity in Nordhaus' deterministic DICE model. We develop a regression-based numerical method for solving a general class of dynamic finite-horizon economy climate models with potentially heavy-tailed uncertainty and general utility functions. We then apply this method to SDICE* and examine the effects of light- and heavy-tailed uncertainty. The results indicate that the effects can be substantial, depending on the nature and extent of the uncertainty and the social planner's preferences.