 Black-Scholes equation with continuous stock dividends

When the underlying stock pays dividends at rate $\theta S_t$, its gain is $d{S}_t+\theta S_{t}dt$. Under the risk-neutral measure its ex-dividend drift is $(\rho -\theta )S_t$. A sufficiently smooth terminal-claim value obeys $p_t+(\rho -\theta )x{p}_x+{\sigma }^2{x}^2{p}_{xx}/2-\rho p=0$. These are dividends on the stock, not a source term paid by the claim itself.