Modeling Correlated Jumps with Time-Varying Intensities

Summary

The document considers how to model dependence between jump events in two assets when their jump sizes follow double exponential distributions. Its answer cautions that discrete jump processes cannot simply be correlated in the same way as normally distributed variables, and proposes a different mechanism: make each asset's Poisson jump intensity stochastic and time varying.

Specifically, it suggests positive mean-reverting intensity processes, such as exponential Ornstein–Uhlenbeck processes, with correlated increments. A rise in one intensity can then coincide with a rise in the other, increasing the likelihood of related jump activity while keeping the intensities positive and limiting their tendency to drift upward without bound. This couples jump likelihoods rather than directly imposing conventional correlation on jump occurrences. The reply is conceptual and does not provide a full simulation algorithm, parameterization, validation, or empirical evidence, so implementation details remain open.

Key ideas

• The proposed dependence mechanism correlates stochastic jump intensities rather than discrete jump events directly.
• Mean-reverting intensity dynamics can keep jump rates positive and bounded in tendency.
• Correlated intensity increments can make simultaneous or related jumps more likely.
• The answer outlines a modeling direction but does not specify a complete simulation procedure.

Tags

# Simulate double exponential process with correlated jumps?


# Simulate double exponential process with correlated jumps?












So, I'm trying to simulate a correlated double exponential jump process for two assets, and I understand the pure exponential jump process ($\eta_1$ and $\eta_2$, the probability of an upward jump occurring, the size of the jump, etc, etc), but it's trying to correlate the two jump occurrences that's confusing me.

For example, correlating normally distributed jumps processes is tractable, where $n_{t}^{i}$ are distinct Poisson processes, and $K_i$ is relatively easily computed,

However, for the double exponential, the best resource I've found is here on pg. 40, but its explanation is quite frankly inscrutable. Could anyone explain to an advanced beginner how this could be simulated? Even pointing toward some successful simulation code would go a long way.

Thank you to all in advance.

## Answer by James Spencer-Lavan (score 1)

https://quant.stackexchange.com/a/38662

I don't think you can correlate discrete processes in the traditional sense.

Instead, I would make the two Poisson intensities time-varying through which a degree of "jump similarity" can be injected

Say each jump intensity is a positive mean reverting process, such as an exponential OU, where the increments are jointly distributed (I.e. with correlation)

Now when one jump intensity moves up, you can influence the other, but know the intensities are always positive and never drift off too high

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.