First passage time distribution via Monte Carlo simulation

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?

1 Answer

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 $$p_t=\mathbb{P}(\tau=t)$$.

• thank you very much for the answer. The first approach gives me a different perspective to think about it. Thank you!! I would be much interested in the second approach for my specific problem though. Would you be so kind to explain a bit more about the formula (what is $p$ and where do you get that formula) or point me in the right direction with references? Thanks! – gioxc88 Jan 22 '19 at 12:58