Time inhomogeneous jump-diffusion model with exponential frequency function

[Robin Schneider]

Dear Quantnet,

I want to implement a real options model that models the movement of the underlying based on a Brownian diffusion process and compound Poisson jump process with a time-inhomogeneous (deterministic) frequency function lambda(t). More precisely, I want to find the solution for a standard jump-diffusion SDE with drift, while the frequency function lambda(t) follows a simple exponential function (for example lambda = 2^t), which means that the frequency of jumps increases exponentially with time (doubles every year).

The Merton model is an example that provides a closed-form solution to a similar type of model, however, with a constant frequency. Does there already exist a model/approach that might be able to provide an analytical solution a SDE as described above?
Unfortunately, I don't have a strong mathematical background, so I would be really happy for any hint that could help me in finding a solution to this problem.

Is it even possible to solve this analytically?
I suppose Monte Carlo Simulation can help to approximate a solution, however, I would prefer to find an analytical solution.

Also I would very much appreciate any ideas about relevant literature that might help to understand / solve this issue.

Thanks and best regards!

 

[Daniel Duffy]

The book by Cont and Tankov is probably a good place to start.


I would prefer to find an analytical solution.

In general, 'analytical' solutions don't exist/only on paper.

Monte Carlo can
Another robust solution is PDE/FDM.