Consider the drifted Brownian motion $X\_t=1+\lambda(t)+W\_t$, where $\lambda: \mathbb R\to [0,\infty)$ with $1\le \lambda'(t)\le 2$ and $(W\_t)\_{\ge 0}$ denotes a Brownian motion. Define the hitting time


$$\tau:=\inf\{t\ge 0: X\_t\le 0\}$$


and further the function $p(t):=\mathbb P[\tau>t]$ for all $t\ge 0$. Can we show that $t\mapsto p(t)$ is Lipschitz?



Lipschitz continuity of $\mathbb P[\tau>t]$ with respect to $t$
I claim this is not a complete answer. I wonder whether it can be improved to obtain the Lipschitz continuity.


Define


$$\left(\frac{d\mathbb Q}{d\mathbb P}\right)\_t := \exp\left(-\int\_0^t \lambda'(s)dW\_s-\frac{1}{2}\int\_0^t(\lambda'(s))^2ds\right).$$


Then $B\_t:=W\_t+\lambda(t)$ is a Brownian motion under $\mathbb Q$. Thus,


$$p(t) = \mathbb E^{\mathbb Q}\left[\left(\frac{d\mathbb P}{d\mathbb Q}\right)\_{t}{\bf 1}\_{\{\inf\_{0\le r\le t}(1+\lambda(r)+W\_{r})>0\}}\right] = \mathbb E^{\mathbb Q}\left[\left(\frac{d\mathbb P}{d\mathbb Q}\right)\_{t}{\bf 1}\_{\{\inf\_{0\le r\le t}(1+B\_{r})>0\}}\right].$$


For every $0\le t<t+\Delta t$, it holds


$$0\le p(t)-p(t+\Delta t) = \mathbb E^{\mathbb Q}\left[\left(\frac{d\mathbb P}{d\mathbb Q}\right)\_{t+\Delta t}\left({\bf 1}\_{\{\inf\_{0\le r\le t}(1+B\_{r})>0\}}-{\bf 1}\_{\{\inf\_{0\le r\le t+\Delta t}(1+B\_{r})>0\}}\right)\right].$$


Using Holder inequality for $a, b>1$ with $1/a+1/b=1$, one has


$$p(t)-p(t+\Delta t)\le \mathbb E^{\mathbb Q}\left[\left(\frac{d\mathbb P}{d\mathbb Q}\right)\_{t+\Delta t}^a\right]^{1/a} \left(\mathbb Q\left[\inf\_{0\le r\le t}B\_{r}>-1\right]-\mathbb Q\left[\inf\_{0\le r\le t+\Delta t}B\_{r}>-1\right]\right)^{1/b}.$$


Let $\Phi:\mathbb R\to (0,1)$ be the CDF defined by


$$\Phi(x):=\int\_{-\infty}^x\frac{1}{\sqrt{2\pi}}e^{-z^2/2}dz.$$


Then we finally obtain


$$p(t)-p(t+\Delta t)\le \mathbb E^{\mathbb Q}\left[\left(\frac{d\mathbb P}{d\mathbb Q}\right)\_{t+\Delta t}^a\right]^{1/a}2^{1/b} \left( \Phi\left(\frac{1}{\sqrt{t}}\right)-\Phi\left(\frac{1}{\sqrt{t+\Delta t}}\right)\right)^{1/b}.$$


It follows from some tedious computation, we have


$$ \mathbb E^{\mathbb Q}\left[\left(\frac{d\mathbb P}{d\mathbb Q}\right)\_{t+\Delta t}^a\right]^{1/a} \le \exp(2(a-1)(t+\Delta t))$$


and


$$2^{1/b} \left( \Phi\left(\frac{1}{\sqrt{t}}\right)-\Phi\left(\frac{1}{\sqrt{t+\Delta t}}\right)\right)^{1/b} \le \left(\frac{2}{\pi t^3}\right)^{1/2b} \exp\left(-\frac{1}{2bt}\right) \Delta t^{1/b}.$$


We conclude thus the (local) $1/b-$Holder continuity. Is there any chance to have the Lipschitz continuity?


