The mentioned paper can be found at this link: http://www.columbia.edu/~sk75/sinica.pdf . I will also mention another theorem in another paper which can be found here: https://link.springer.com/article/10.1007/BF00531747.
In the first paper, the most important theorem is proved solely based on the theorem 3.1 which states
Theorem: For any constants $b\geq y$ and $b>0$, as $m\rightarrow\infty$, $$\mathbb{P}\Big(\ U_{m}<y\sqrt{m}\ ,\ \tau'(b,U)\leq m\ \Big)\\ =\mathbb{P}\Big(\ U(1)\leq y\ ,\ \tau(b+\beta/\sqrt{m},U)\leq1\ \Big)+o(m^{-1/2})$$ where the constant $\beta=-(\zeta(\frac{1}{2})/\sqrt{2\pi})$. Where $\zeta$ is the Riemann zeta function.
Here:
- $\tau'(b,U):=\inf\{n\geq1:U_{n}\geq b\sqrt{m}\}$,
- $\tau(b,U):=\inf\{t\geq0:U(t)\geq b\}$ and
- $U_{n}=\sum_{i=1}^{n}\left(Z_{i}+\frac{v}{\sqrt{m}}\right)$ where $Z_i$ is a standard random variable and $U(t):=vt+B(t)$ where $B(t)$ is the standard Brownian motion.
Now on this paper there is no proof but there is a reference to the second paper by siegmund yuh with the link above. And there the theorem is really similar but the results do not match and I cannot see how to go from one result to another. The theorem on the other paper is:
Theorem: Suppose $\mu =0$, $\mathbb{E}x^2_1=1$, and $\gamma=\mathbb{E}x^3_1$ is finite. Let $b=\zeta m^{1/2}$.
If the distribution of $x_1$ is strongly non lattice in the sense that $$\limsup_{|t|\rightarrow\infty}|\mathbb{E}[\exp(itx_1)]|<1$$ then for each $x>0$ as $m\rightarrow\infty$ $$\mathbb{P}\left[\ \tau<m\ ,\ s_m<(\zeta -x)m^{1/2}\ \right] \\ =1 - \Phi(\zeta +x) - m^{-1/2}\phi(\zeta+x)\left[2\beta +(\gamma/6)(x^2-\zeta^2-1)\right]\\ +o(m^{-1/2})\ . $$ Here $\beta=\mathbb{E}(s^2_{\tau+}/2\mathbb{E}s_{\tau+}$ if $\zeta>0$ and $\beta=\mathbb{E}s_{\tau+}$ if $\zeta =0$. And $\Phi,\phi$ denote standard normal density and distribution functions. Here $\tau=\inf\{n:S_n>b\}$ is a stopping time and $S_n=\sum_{i=1}^nx_i$. Also $x_i$ are iid with mean $\mu$.
You see that the theorems are practically the same but the results are different and I am incapable of relating them. Since the proof exists for second theorem, if I can relate these two, then I could use that proof. Hopefully I provided all the necessary details. Please don't hesitate to ask if you need more explanation. Thank you so much for you help.
If anyone is curious, I came up with an answer.
Lemma: define $b$ and $y$ via the relations $H=S_{0}e^{b\sigma\sqrt{T}}$ and $K=S_{0}e^{y\sigma\sqrt{T}}$ $\mathbb{P}(S(T)\leq K,\tau(H,S)\leq T)=1-\Phi(2b-c)$
The main idea is to notice that the following events are equivalent $\{\tau(H,S)\leq T\}=\{\inf\{t\geq0:S(t)\geq H\}\}=\{\sup_{s\leq T}S(t)\geq H\}$
\begin{align} \mathbb{P}(S(T)\leq K,\tau(H,S)\leq T) & =\mathbb{P}(\log S(T)\leq y\sigma\sqrt{T},\sup_{s\leq T}\log S(t)\geq b\sigma\sqrt{T})\nonumber \\ & =\mathbb{P}(\log S(T)\geq(2b-c)b\sigma\sqrt{T})\\ & =1-\Phi(2b-c) \end{align}
This is the application of the reflection principle. Then to establish the connection with the third statement we notice that $$ \Phi\left(2\left(b+\frac{\beta}{\sqrt{m}}\right)-c\right)=\Phi\left(2b-c\right)+\frac{2\beta}{\sqrt{m}}\phi(2b-c)+o(m^{-1/2}) $$ where the LHS is obtained by the Taylor polynomial. And we are done because $\gamma=0$ in the special case $x_i$ standard normal random variables