Question

Give an efficient method for simulating a nonhomogeneous Poisson process with intensity function $$ \lambda(t)=b+\frac{1}{t+a}, \quad t \geqslant 0 $$

Short answer

To efficiently simulate a nonhomogeneous Poisson process with intensity function \(\lambda(t) = b + \frac{1}{t+a}\), follow these steps: 1. Calculate the maximum intensity value: \(\lambda_{max} = \lambda(0) = b + \frac{1}{a}\). 2. Simulate a homogeneous Poisson process with intensity \(\lambda_{max}\) using standard methods, such as the inter-arrival times having an exponential distribution with mean \(1/\lambda_{max}\). 3. Perform the thinning process by accepting or rejecting each point \(t_k\) in the homogeneous process, based on a randomly generated number \(u_k\), using the rule: \(u_k \leq \frac{\lambda(t_k)}{\lambda_{max}}\). 4. Collect the accepted points from the thinning process, which will represent the simulated nonhomogeneous Poisson process.

Step-by-step solution

Identify the maximum intensity function value

First, we need to find the maximum value of the given intensity function over the interval of interest. This will be used to scale our homogeneous Poisson process to ensure we accept/reject points correctly. In our case, we have: $$ \lambda(t) = b + \frac{1}{t+a}, \quad t \geqslant 0 $$ For this exercise, we will assume that \(b>0\) and \(a>0\). The maximum value of the intensity function can be found by taking the first derivative and setting it to zero to find the critical points. Then, we will evaluate the intensity function at the critical points and boundaries to determine the maximum value.

First derivative

To find the maximum value of the intensity function, we need to find the first derivative of the function and set it to zero to find the critical points: $$ \frac{d}{dt} \bigg( b + \frac{1}{t+a} \bigg) = -\frac{1}{(t+a)^2} $$ Since the first derivative is always negative, the intensity function is strictly decreasing. Therefore, its maximum value occurs at the lower-bound of the interval (\(t=0\)).

Calculate maximum intensity value

Evaluate the intensity function at the lower bound of the interval to find the maximum value: $$ \lambda_{max} = \lambda(0) = b + \frac{1}{a} $$

Simulate a homogeneous Poisson process

Next, we simulate a homogeneous Poisson process with intensity \(\lambda_{max}\). This can be done using standard methods, such as the inter-arrival times having an exponential distribution with mean \(1/\lambda_{max}\). Generate points \(t_1, t_2, \dots\) in the homogeneous Poisson process until the last point exceeds the time horizon of interest.

Thinning the homogeneous Poisson process

Now, we will thin the homogeneous Poisson process by accepting or rejecting each point \(t_k\) in the process. To do this, generate a random number \(u_k\) between 0 and 1 for each point \(t_k\). If: $$ u_k \leq \frac{\lambda(t_k)}{\lambda_{max}} $$ we accept the point \(t_k\) in the nonhomogeneous Poisson process. Otherwise, we reject it. Repeat this process for all generated points in the homogeneous Poisson process.

Combine accepted points

Finally, collect all the accepted points from Step 5. These accepted points represent the occurrences of events in our simulated nonhomogeneous Poisson process. The generated nonhomogeneous Poisson process with intensity function \(\lambda(t) = b + \frac{1}{t+a}\) is now successfully simulated using the thinning method.