We propose a multivariate framework for modeling dependent default times that extends the classical Cox process by incorporating both common and idiosyncratic shocks. Our construction uses càdlàg, increasing processes to model cumulative intensities, relaxing the requirement of absolutely continuous compensators. Analytical tractability is preserved through the multiplicative decomposition of Azéma supermartingales under assumptions that guarantee deterministic compensators. The framework captures a wide range of dependence structures and allows for both simultaneous and non-simultaneous defaults. We derive closed-form expressions for joint survival probabilities and illustrate the flexibility of the model through examples based on Lévy subordinators, compound Poisson processes, and shot-noise processes, encompassing several well-known models from the literature as special cases. Finally, we show how the framework can be extended to incorporate stochastic continuous components, thereby unifying gradual and abrupt sources of default risk.