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Programming language researchers have developed a wide array of tools to reason about probabilistic programs, but it remains unclear how to apply these proof techniques to continuous-time stochastic processes. To bridge this gap, we propose a verification method for properties of continuous-time stochastic processes by reasoning about them as standard programs. To demonstrate our approach, we define a continuous semantics for a probabilistic imperative language $\mathrm{pWhile}^{\mathrm{dt}}$ by taking the Kuratowski limit of the standard semantics, which collects limit points of the standard program semantics as the time-step approaches zero. Then, we present a Hoare-style logic that uses loop invariants and decreasing functions to verify properties of continuous programs. To establish soundness, we identify a broad class of assertions that are preserved from the standard semantics to the continuous semantics. To demonstrate our approach, we verify properties of ten continuous-time stochastic processes modeled in $\mathrm{pWhile}^{\mathrm{dt}}$, including variants of Brownian motion and a stochastic hybrid system.