Sampling Jump Times in Milstein Jump-Diffusion Schemes

Summary

The document clarifies how jump-related terms in a Milstein discretization are interpreted. They correspond to iterated integrals that combine Wiener-process increments with jumps. To evaluate these terms, a simulation must know the Wiener process at the jump times, rather than relying only on a Poisson count for each fixed time step.

Two approaches are described: sample the number of jumps within each interval and generate the needed Brownian values, or use a jump-adapted scheme that explicitly locates jump times and samples the process there. The cited schemes apply to jump-diffusions with finite jump activity; the response cautions that they do not directly cover infinite-activity processes such as alpha-stable models. It points readers toward a specialized numerical methods text for further details. No numerical comparison with Euler is provided, so the document does not establish which scheme is more accurate or efficient for a particular application.

Key ideas

• The Milstein jump terms are iterated integrals combining Brownian motion and jump increments.
• Simulation requires Brownian process values at the jump times.
• One approach samples jumps and corresponding Brownian values within each time step.
• Jump-adapted schemes explicitly locate jump times and are described for finite-activity jump-diffusions.
• The discussion does not establish a general accuracy or efficiency comparison with Euler discretization.

Tags

# Milstein Scheme for Jump-Diffusion models


# Milstein Scheme for Jump-Diffusion models












Hey in this report (Approximation of Jump Diffusions in Finance and Economics by Bruti-Liberati and Platen) is described the Milstein formula (3.5) for simulation SDE with jump component. How it is calculated? In this formula we also have to compute value of a Wiener process in a jump time, how to do it? I think that simualtion of a Poisson process by increment will be insufficient in this situation. Or maybe the Euler scheme is preferable and the Milstein scheme is not used in this case?

## Answer by Tom (score 1)

https://quant.stackexchange.com/a/74804

Those terms represent iterated integrals of the type $$ \int_{t_n}^{t_{n+1}} \int_{t_n}^{s} dW(z) dJ(s) $$ and $$ \int_{t_n}^{t_{n+1}} \int_{t_n}^{s} dJ(z) dW(s) $$ Which seem to be the third and fourth lines in that formula, respectively.

To quote that same report:

> Furthermore, one needs to sample the Wiener process W at the jump times τi, for i ∈ {1, . . . , NT }

Which either means you have to sample $N$ for every $t_n$ and then sample $\Delta W$ an $N$ number of times for every $t_n$ or use jump adapted schemes that calculate when a jump occurs and sample the process there. These work on jump-diffusions (which is the type of SDE written on that link), so not on processes with infinite activity (like an alpha-stable process). For more info on these, you can see chapter 8 of the book at Numerical Solution of Stochastic Differential Equations with Jumps in Finance, which is by the same authors.

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.