Defaultable stock (${\hat{S}}_t$)

A stock model whose price can be killed or reduced at a default time. The zero-recovery default model combines continuous price dynamics before default with a jump to zero.

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.