Jump-diffusion models

Jump diffusion is a that involves jumps and diffusion. It has important applications in magnetic reconnection, coronal mass ejections, condensed matter physics, in Pattern theory and computational vision and in option pricing.

In crystals, atomic diffusion typically consists of jumps between vacant lattice sites. On time and length scales that average over many single jumps, the net motion of the jumping atoms can be described as regular diffusion.

Jump diffusion can be studied on a microscopic scale by inelastic neutron scattering and by Mößbauer spectroscopy. Closed expressions for the have been derived for several jump(-diffusion) models:

In option pricing, a jump-diffusion model is a form of mixture model, mixing a jump process and a diffusion process. Jump-diffusion models have been introduced by Robert C. Merton as an extension of jump models. Due to their computational tractability, the special case of a basic affine jump diffusion is popular for some credit risk and short-rate models.

In Pattern theory and computational vision in Medical imaging, jump-diffusion processes were first introduced by Grenander and Miller as a form of random sampling algorithm which mixes "focus" like motions, the diffusion processes, with "saccade" like motions, via jump processes. The approach modelled sciences of electron-micrographs as containing multiple shapes, each having some fixed dimensional representation, with the collection of micrographs filling out the sample space corresponding to the unions of multiple finite-dimensional spaces. Using techniques form Pattern theory, a posterior probability model was constructed over the countable union of sample space; this is therefore a hybrid system model, containing the discrete notions of object number along with the continuum notions of shape. The jump-diffusion process was constructed to have ergodic properties so that after initially flowing away from its initial condition it would generate samples from the posterior probability model.

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