In this technical note we study continuous-time stochastic control of a dynamic production and replenishment system characterized by bounded control and an additive type of uncertainty. The study is motivated by problems arising in supply chains involving periodic exchange of information between a manufacturing system (supplier) and a customer (retailer). As a result, the inventories are only observed periodically while the replenishment is possible at any point of time. We identify replenishment policies for different operational conditions and show that, even for one-product-type system, the consideration of random demand over multiple update periods leads to a non-intuitive, and nontrivial, optimal production control.