First passage time distribution via Monte Carlo simulation

Question

The problem:

I want to assess the first passage time distribution via Monte Carlo Simulation, where the first passage time is defined as:

$$\tau=\inf\left\{t: X_t > l\right\}$$

where $l$ is the barrier Suppose further that I know how to simulate $X_t$.

This is what I did:

1. Decide a time step and the horizon. Suppose horizon is 1 and time step is 1/250. I have 250 steps.

2. Simulate for example 10000 path $X_t$ starting at $x_0$

3. I have now a matrix of 10000 sample path.

4. For each sample path I keep track of the time where my simulated $X$ first hits the barrier. This is one realization of my random variable $\tau$

My question is:

If inside the path $X_t$ never hits the barrier $l$ what should the realization of $\tau$ be for that sample path?

Should I completely discard the information for these sample paths?

Answer 1

Rather than fixing the simulation time horizon, you can run simulations until you hit the barrier $\ell$. This does not induce the bias the first approach induces. However, if $\tau$ has a non-zero probability to take values beyond the time $T$ allocated to the experiment, the outcome of this first approach has to be handled as a censored experiment, with the likelihood associated being $$\prod_{t=1}^{T-1} p_t^{n_t} \mathbb{P}(\tau\ge T)^{n_T}$$ where $n_t$ is the number of exit times equal to $t$ and $p_t=\mathbb{P}(\tau=t)$.