The basic model I'm working with is a neuron that receives input from other neurons which cause instantaneous spikes in the voltage. In a nutshell, I have a differential equation that describes the voltage of a neuron, and when the voltage reaches a certain threshold, the neuron has fired and I would like to estimate the expected time to first fire. So for simplicity, I have the following:
$$dV = -(V-V_{rest})dt + \sigma_edN_e - \sigma_idN_i$$
So the equation is basically the voltage jumping but there's a term to help the neuron decay back to its resting state. Here, $dN_e$ represents the "up" jumps that increase the voltage by $\sigma_e$ (a constant) at rate $\lambda_e$, and $dN_i$ represents the "down" jumps that decrease the voltage by a level $\sigma_i$ at rate $\lambda_i$. We often assume $\lambda_e > \lambda_i$ to hopefully prevent the expected hitting time being infinity. Let me define a stopping time by the first time my voltage reaches a certain level:
$$\tau = \inf \left\lbrace t\geq 0: V(t) \geq L\right\rbrace$$
I want to estimate $E\left[ \tau\right]$ and if possible, $Var(\tau)$. Of course because of the discontinuity of the jumps, I may overshoot the level $L$, but since I'm assuming the jump sizes are somewhat small, I'm just going to approximate $V(\tau) = L$ (The goal is just to get an approximate analytical solution to the expectation; it doesn't have to be exact).
I haven't ever taken an SDE course so a lot of the formalities I have a very rough understanding of (for example, I've seen the Lévy–Khintchine formula several times, but I would have no idea how to actually work with it). I know I can kind of center the jumps to make them martingales by defining $d\tilde{N}_e = dN_e - \lambda_e dt$, or something like that, but I'm not sure if that helps for anything. I also heard from a math professor that one can approximate a poisson process by a brownian motion with drift and an error term, i.e. that $N(t)= t + B(t) + \xi(t)$ where $|\xi(t)| \leq e^\eta\log(2 \vee t)$ (where $\eta$ is a random variable) so that it would be possible to approximate the Poisson jumps by a Brownian motion with drift, but I haven't found anything in literature showing this.
Anyways, if anyone could help me estimate the expected hitting time or point me to how to solve this problem, it would be extraordinarily appreciated! Thank you.
EDIT:
The full equation I'm interested in is:
$$dV = \left[-(V-V_{rest}) + a\exp\left(\frac{V-\theta}{a}\right)\right] dt + \sigma_edN_e - \sigma_idN_i$$
for $a>0$ (but small). In fact, as $a \rightarrow 0$, this equation becomes the one at the very beginning, but before I work on this one, I'm trying to do the simpler one.