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Consider a particle whose position is driven by the following equation:

$$Y_t = y + t + W_t + C\min\big(1,(Y_t+1)^+\big)\Lambda_t,\quad \mbox{for all } 0\le t<\tau_*,$$

where $y>0$, $0<C<1$, $(W_t)_{t\ge 0}$ is a standard Brownian motion,

$$\Lambda_t:=\log\big(\mathbb P(\tau>t)\big),\quad \tau:=\inf\{t\ge 0: Y_t\le 0\}$$

and

$$\tau_*:=\inf\left\{t\ge 0: \mathbb P\big(\inf_{0\le s\le t} Y_s\le 0\big)=1\right\}.$$

We say the particle is absorbed once it hits zero. Could we show $\mathbb P(\tau=\infty)=\mathbb P(Y_t>0, \forall t\ge 0)>0$? Any answers, references or remarks are highly appreciated!

Personal thoughts : Consider an alternative SDE

$$X_t = y + t + W_t + C\min\big(1,(X_t+1)^+\big)\log(\lambda),\quad \mbox{for all } t\ge 0,$$

where $\lambda:=\mathbb P(X_t>0, \forall t\ge 0)$. Using a comparison argument, it can be shown that $Y_t\ge X_t$ for all $t\ge 0$. Therefore,

$$\mathbb P(Y_t>0, \forall t\ge 0)\ge \mathbb P(X_t>0, \forall t\ge 0)=\lambda.$$

It remains to show $\lambda>0$. As noted that, on the event $\left\{X_t>0, \forall t\ge 0\right\}$, it follows that $X$ has the same law as $(y + t + W_t + C\log(\lambda))_{t\ge 0}$ knowing that $Z_t>0, \forall t\ge 0$. A straightforward computation yields

$$ 1 -\lambda^{-2C}e^{-2y}=\mathbb P(Z_t>0, \forall t\ge 0)=\mathbb P(X_t>0, \forall t\ge 0)=\lambda\quad \quad \quad \quad \quad (\ast)$$

As shown that, under suitable conditions (e.g. when $y$ is large enough), $(\ast)$ always admits a strictly positive solution, which proves partially my desired result. However, for general $y>0$, I don't know how to proceed.

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  • $\begingroup$ Here we assume the existence of the solution $Y$. $\endgroup$ – Neymar Oct 13 '20 at 1:08
  • $\begingroup$ These equations aren't SDEs so it's really not clear whether one should take what you ask at face value or whether you mean something completely different (maybe with an integral over the last term on the r.h.s.). $\endgroup$ – Martin Hairer Oct 13 '20 at 8:54
  • $\begingroup$ @MartinHairer Thank you very kindly for pointing out my mistake. How should we call these equations in the literature? An idea is to study the regularity of the map $t\mapsto \Lambda_t$, however I am unable to find the associated PDE. Do we have some similar Feyman-Kac formula that connects this equation and PDE? $\endgroup$ – Neymar Oct 13 '20 at 10:47

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