I am not an expert in this area. But I believe that the word "Deterministic" is for "Policy" in the "Deterministic Policy" Gradient. It does not mean deterministic environment.
Stochastic policy: Probabilistic(random) action choice for a given state.
Deterministic policy: one action is chosen for a given state.
Deterministic Policy Gradient algorithm still can handle a stochastic (and continuous, of cause) environment, but the policy will be deterministic.
"in DDPG, the Actor directly maps states to actions (the output of the network directly the output) instead of outputting the probability distribution across a discrete action space" -towards Data Science
Deterministic Policy Gradient Algorithms by Silver et al. PDF