I'm considering a random walk Y on $\mathbb{Z}$ with jump sizes in $[-E, E]$, defined as follow: $Y_0 = 0$ and for $n \ge 1 $: $$Y_n = \sum_{i = 1}^n X_i,$$ with $(X_i)$ a family of i.i.d random variables.
I'm interested in computing the probability that $Y$ reaches a non negative state, given that the random walk has already made negative jumps of size $k$.
In other words, and let $Y_n^- = \sum_{i=0}^n \max(0, -X_i)$ be the process counting the total size of negative jumps and $Y_n^+ = \sum_{i=0}^n \max(0, X_i)$, so that $Y_n = Y_n^+ - Y_n^-$.
Let $\tau_k = \inf \{ n \ge 0 : Y_n^- \ge k \}$. I want to compute $$p_k = \mathbb{P}(\exists n \ge \tau_k ; Y_n \ge 0 ).$$
When $E = 1$ (i.e. $\pm 1$ jumps), I can derive an analytical expression for this probability.
However, I'd like to derive an analytical expression for the probability, or at least an upper bound when the jumps are not restricted to $\pm 1$. In particular, I'm interested in the case where the direction of the jump is drawn from a Bernoulli distribution and the size of the jumps drawn from a Binomial distribution: \begin{align} \mathbb {P} (X=k) & = p {E \choose k}\,p^{k}(1-p)^{E - k} \\ \mathbb {P} (X=-k) & = (1 - p) {E \choose k}\, (1-p)^{k} p^{E - k} \end{align} when the random walk is biased towards negative jumps ($p < 0.5$).
An approach to solve this question would be to compute the probability distribution of $Y_\tau$, $P(Y_\tau)$. From there, it reduces to a standard hitting problem and one can compute the hitting probability (at least numerically) using standard tools for Markov chains.
Any hints on how to solve these kind of problems would be appreciated.