A baseline sovereign debt projection leaves the dispersion and path dependence of future debt burdens unspecified. This paper develops a conditional projection framework for user-defined debt thresholds. An Itô stochastic differential equation for the debt-to-GDP ratio is derived from debt and output dynamics. A log transformation and stochastic integrating factor give an exact pathwise representation and, under constant parameters, an inverse-gamma stationary distribution with explicit existence and moment conditions. Monte Carlo simulation and a backward equation provide distributional envelopes and finite-horizon threshold-crossing probabilities. Illustrative experiments quantify the separation between terminal exceedance and first passage, and the sensitivity of first-passage risk to the volatility-adjusted interest-growth differential. Bounded primary deficit adjustments have diminishing proportional influence at high debt, while debt-responsive fiscal rules can alter the asymptotic drift. Users supply the economic calibration and the interpretation of threshold entry.