Probability of random crossing a specific value any time

Question

Let $x(t+1) = x(t) + e(t)$, $e(t)$ iid $\mathcal{N}(0,1)$. What is the probability of $x(s)> c$, for any $0<s<T$? Calculation for any specific $s$ is easy. But I am looking for the probability that the random walk crosses $c$ any time between now and $T$. Easy to implement approximation preferred.

Answer 1

Let $\tau$ be the minimal $t<T$ such that $x(t)>c$, if such $t$ exists; set $\tau=T-1$ if there is no such $t$. Then the event $\{x(\tau)>c\}$ is the disjoint union of two events, $A=\{x(T-1)>c\}$ and $B=\{x(\tau)>c\}\cap \{x(T-1)>c\}$. Reflecting the rest of the walk after time $\tau$ is a measure preserving bijection from $B$ to a subset of $A$. This shows that $P(B) \le P(A)$. For Gaussian increments and most parameters, $P(A)$ and $P(B)$ will be very close.