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Let

$$Y_t:=1+\int_0^t b_sds + W_t,\quad\forall t\ge 0,$$

where $(b_t)_{t\ge 0}$ is a bounded adapted process and $(W_t)_{t\ge 0}$ is a standard Brownian motion. Denote $\tau:=\{t\ge 0: Y_t\le 0\}$ and $X_t:=Y_{t\wedge \tau}$. It is known that the law of $X_t$, denoted by $\mu_t$, has the following decomposition:

$$\mu_t(dx) = \alpha(t)\delta_0(dx) + p_t(x)dx,\quad \forall t>0.$$

In the previous post Does the density of a stopped drifted Brownian motion vanish at zero? Mateusz has shown $p_t(0+):=\lim_{x\to 0+}p_t(x)=0$ for $t>0$ when the drift $b$ is deterministic. Can we generalize this result to adapted processes?

PS : Mateusz claims that using Chapman-Kolmogorov equation and a comparison argument allows to conclude that the density $X_t$ goes to $0$ at zero, but I don't know the details.

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