State-Dependent Jump Intensity in a Compound Poisson Process
Summary
The discussion clarifies the pure jump component used in an affine jump-diffusion model. It is essentially a compound Poisson process: jumps accumulate according to a counting process whose intensity varies with the state of another process, X. Over an interval, the count has a Poisson distribution with parameter equal to the integral of that state-dependent intensity across the interval.
Each jump has a size drawn from a specified distribution, with the stated independence condition applying to the prior path of X. This differs from treating the count at time t as a Poisson variable with parameter given by the instantaneous intensity at t. The answer connects the construction to finite-activity jump-diffusion terminology, but the exchange is a brief conceptual clarification rather than a full derivation or a guide to estimating the model parameters.
Key ideas
- The jump process is a compound Poisson process with state-dependent intensity.
- The expected jump count over an interval depends on the integrated intensity across that interval.
- Jump sizes follow a specified distribution and are independent of the prior path of the state process under the stated setup.
- Using the instantaneous intensity at a single time as the Poisson parameter for the cumulative count is not the described construction.
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# Pure jump process in Duffie, Pan and Singleton's paper
# Pure jump process in Duffie, Pan and Singleton's paper
In page 1349 or Section 2.1 of "Duffie, D., Pan, J., & Singleton, K. (2000). Transform Analysis and Asset Pricing for Affine Jump-Diffusions. Econometrica, 68(6), 1343-1376" the pure jump process Z is defined as follow:
> conditional on the path of X, the jump times of Z are the jump times of a Poisson process with time-varying intensity $\{\lambda(X_s):0\le{s}\le{t}\}$, and that the size of the jump of Z at a jump time T is independent of $\{X_s:0\le{s}<T\}$ and has the probability distribution $\nu$.
Since I can't find a formal definition of Z in the same paper, I'm writing to ask if it is talking about a compound Poisson Process. Saying, $(Z_t)_{t\ge{0}}$ is a stochastic process such that: $Z_t=\sum_{i=1}^{N_t}{Y_i}$, where $Y_1,...,Y_{N_t}$ are random variables with distribution $\nu$ and $N_t$ follows a $Pois(\lambda(X_t))$.
## Answer by Achrbot (score 1, accepted)
https://quant.stackexchange.com/a/76035
Essentially yes - $Z_t$ is a compound Poisson process, except that the underlying counting process $N_t$ has intensity $\lambda(X_t)$. I.e $$ N_t - N_s \sim Pois\bigg( \int_s^t \lambda(X_u) \mathrm{d}u\bigg). $$
In Lewis (2001), he refers to the 'jump diffusion-case', when the underlying process is a Levy process with finite Levy measure - Which is consistent with the above.
Lewis, Alan L., A Simple Option Formula for General Jump-Diffusion and Other Exponential Levy Processes (September 2001). Available at SSRN: https://ssrn.com/abstract=282110 or http://dx.doi.org/10.2139/ssrn.282110Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.