Lions and Musiela give sufficient conditions to verify when a stochastic exponential of a continuous local martingale is a martingale or a uniformly integrable martingale. Blei and Engelbert and Mijatović and Urusov give necessary and sufficient conditions in the case of perfect correlation (ρ=1). For financial applications, such as checking the martingale property of the stock price process in correlated stochastic volatility models, we extend their work to the arbitrary correlation case (−1≤ρ≤1). We give a complete classification of the convergence properties of both perpetual and capped integral functionals of time‐homogeneous diffusions and generalize results in Mijatović and Urusov with direct proofs avoiding the use of separating times (concept introduced by Cherny and Urusov and extensively used in the proofs of Mijatović and Urusov).