I have a Bern$(p)$ random walk ($Y_i = 1$ with probability $p$ and Y_i = 0 with $1-p$) with two absorbing boundaries, $A: Y^i \leq t_i$ and $B:Y^i \geq d_i-t_i$. Now, both $d_i$ and $t_i$ are evolving with time. $d_i$ is a simple affine function of $i$ (say $d_i = i+c$), but I want to choose $t_i$ such that $\mathbb{P}[\tau_A > \tau_B] = \delta$. I.e. I want to ensure quick absorption, but I want the probability that $A$ absorbs to be at least $1-\delta$. I want to do this by adjusting the $t_i$ parameters.
Is this a super tricky monster problem? I was hoping there would exist some kind of optimal sequential hypothesis testing threshold theorem that I could use :p
Setting $X^i=Y^i - \frac{1}{2} d_i$ and $g_i=\frac{1}{2}d_i - t_i$ gives you $A : X^i\leq -g_i$ and $B : X^i\geq g_i$.
you can rewrite : $X^i = x+\sum_{l=1}^i \xi_i$ where $\xi_i$ are i.i.d r.v. taking in your case $\frac{1}{2}$ and $-\frac{1}{2}$ values with proba $p$ and $1-p$
If I choose $\mu$ such that $pe^{\frac{\mu}{2} }+(1-p)e^{-\frac{\mu}{2}}=1$, $Z^i=e^{\mu X^i}$ is a martingale.
I have using stopping theorem and the fact that $\mathbb{P}(\tau_A>\tau_B)=\delta$
choosing $g_i \equiv g$ a constant
$$\mathbb{E}[Z^{\min(\tau^A,\tau^B)}] = e^{-\mu g}(1-\delta)+\delta e^{\mu g}=e^{\mu x}$$
or written in an other form:
$$\delta =\frac{e^{\mu x}-e^{-\mu g}}{e^{\mu g}-e^{-\mu g}}$$
Going back to your case, you are looking for $t_i = \frac{i}{2} - g$ where $g$ is a constant.