We study the effect of price cap regulation on investment in new capacity in an oligopolistic (Cournot) industry, using a continuous time model with stochastic demand. A price cap has two mutually competing effects on investment under demand uncertainty: it makes the option of deferring investment very valuable, but it also reduces the interest of strategic underinvestment to raise prices. We show that there exists an optimal price cap that maximizes investment incentives. As with deterministic demand, the optimal price cap is the clearing price of the competitive market. However, unlike the deterministic case, we show that such a price cap does not restore the competitive equilibrium; there is still under-investment. Sensitivity analyses and Monte Carlo simulations show that the efficiency of price cap regulation depends critically on demand volatility and that errors in the choice of the price cap can have detrimental consequences on investment and average prices.