Zero-recovery default model (${\hat{S}}_t=1_{\{t<\tau \}}{e}^{\lambda t}{S}_t$)

For an independent exponential distribution default time of rate $\lambda$ and a continuous nonnegative martingale $S$, this process is a martingale. The compensating pre-default factor $e^{\lambda t}$ balances the survival probability $e^{-\lambda t}$. The strict survival inequality gives a càdlàg price process.