= Solution
First work with the <filtration> $\mathcal G_t=\mathcal F_t^W\vee\sigma(\mathbf1_{\{u\leq\tau\}}:0\leq u\leq t)$, which records the <Brownian motion> and whether <default> has occurred. On $\{s>\tau\}$ all subsequent <defaultable stock> prices vanish. On $\{s\leq\tau\}$, the <memoryless property> of the <exponential distribution> gives survival from $s$ to $t$ with <conditional probability> $e^{-\lambda(t-s)}$. The residual survival event is <independent> of the future <independent increments> of the <Brownian motion>. Thus, for deterministic $s<t$,
$$
\mathbb E[\widehat S_t\mid\mathcal G_s]
=\mathbf1_{\{s\leq\tau\}}e^{\lambda t}
e^{-\lambda(t-s)}\mathbb E[S_t\mid\mathcal F_s^W]
=\mathbf1_{\{s\leq\tau\}}e^{\lambda s}S_s
=\widehat S_s.
$$
Also $\mathbb E\widehat S_t=e^{\lambda t}e^{-\lambda t}S_0=S_0$, so this is a true <martingale>, with no localization needed. Its <natural filtration> is contained in $\mathcal G$; applying the <tower property of conditional expectation> to the displayed identity yields \b[the requested natural-filtration martingale property]:
$$
\boxed{\mathbb E[\widehat S_t\mid\mathcal F_s^{\widehat S}]=\widehat S_s.}
$$
The printed convention keeps the <stock> alive at $t=\tau$. It is a left-continuous convention at that one random time. Replacing it by $\mathbf1_{\{t<\tau\}}$ gives the usual <càdlàg> <zero-recovery default model>; since $\mathbb P(\tau=t)=0$ at each fixed $t$, the fixed-time <martingale> calculations above are unchanged.
Back to article page
