Why Discrete Diffusion Data Can Resemble Jump Diffusions
Summary
The document asks whether discretely sampled data from a diffusion with large time steps can be distinguished numerically from data generated by a jump diffusion. One answer explains that a compensated Poisson process, when suitably scaled as its jump intensity grows, converges to Brownian motion. This means discrete observations consistent with Brownian motion cannot rule out a sufficiently high intensity jump process as their source.
A second answer outlines a model comparison approach: specify candidate processes, estimate their parameters from the sample, and compare their fit using goodness of fit statistics. This can suggest which model is more plausible, but it does not establish the true data generating process. The discussion gives no worked example, diagnostic procedure, or guidance on selecting a statistic; its main caution is the fundamental difficulty of distinguishing models from discrete samples.
Key ideas
- A compensated Poisson process with high intensity can converge, after scaling, to Brownian motion.
- Discrete observations that fit a diffusion do not necessarily exclude a jump process.
- Model comparison can start by fitting parameters for a set of candidate stochastic processes.
- Goodness of fit statistics can rank candidate models, but the document presents this only as an inference, not proof.
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# Does discretizing a diffusion model make it look like a jump diffusion model?
# Does discretizing a diffusion model make it look like a jump diffusion model?
Can we distinguish a sample generated from a diffusion model with large time steps from a sample generated from a jump diffusion model. Not mathematically but numerically (if we ask a computer to tell the difference) ?
## Answer by Kurt G. (score 2)
https://quant.stackexchange.com/a/75883
This not possible. The compensated Poisson process $N_t-\lambda t$ converges in the limit of large intensity $\lambda$ to a Brownian motion with variance rate $\lambda\,.$ Therefore, the pure jump process $(N_t-\lambda t)/\sqrt{\lambda}$ converges to a standard Brownian motion. The consequence is: even if you give me discrete data that were simulated with a Brownian motion I could not rule out that they were generated by $(N_t-\lambda t)/\sqrt{\lambda}$ with a large enough $\lambda\,.$
## Answer by NN2 (score 1)
https://quant.stackexchange.com/a/75882
The principle is to assume some mathematical models (for example: the sample is generated from a log-normal process, or log-normal process with jumps, or CIR process, ...) and then estimate the parameters of these models.
Then we determine how well these assumed models fit the sample data. We need to choose a (or some) goodness of fit statistics for this evaluation (R-squared for example).
And by basing on the final statistics, we can guess which model may be the one that generated the sample data, in particular, whether the sample is generated from a jump diffusion model or not.Shown 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.