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

In physics

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 autocorrelation function have been derived for several jump(-diffusion) models:

Singwi, Sjölander 1960:^[1] alternation between oscillatory motion and directed motion
Chudley, Elliott 1961:^[2] jumps on a lattice
Sears 1966,^[3] 1967:^[4] jump diffusion of rotational degrees of freedom
Hall, Ross 1981:^[5] jump diffusion within a restricted volume

In economics and finance

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.^[6] Due to their computational tractability, the special case of a basic affine jump diffusion is popular for some credit risk and short-rate models.^{[citation needed]}

In pattern theory, computer vision, and medical imaging

In pattern theory and computational vision in medical imaging, jump-diffusion processes were first introduced by Grenander and Miller^[7] as a form of random sampling algorithm that 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 from 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.

References

^ Singwi, K.; Sjölander, A. (1960). "Resonance Absorption of Nuclear Gamma Rays and the Dynamics of Atomic Motions". Physical Review. 120 (4): 1093. doi:10.1103/PhysRev.120.1093.
^ Chudley, C. T.; Elliott, R. J. (1961). "Neutron Scattering from a Liquid on a Jump Diffusion Model". Proceedings of the Physical Society. 77 (2): 353. doi:10.1088/0370-1328/77/2/319.
^ Sears, V. F. (1966). "Theory of Cold Neutron Scattering by Homonuclear Diatomic Liquids: I. Free Rotation". Canadian Journal of Physics. 44 (6): 1279–1297. doi:10.1139/p66-108.
^ Sears, V. F. (1967). "Cold Neutron Scattering by Molecular Liquids: Iii. Methane". Canadian Journal of Physics. 45 (2): 237–254. doi:10.1139/p67-025.
^ Hall, P. L.; Ross, D. K. (1981). "Incoherent neutron scattering functions for random jump diffusion in bounded and infinite media". Molecular Physics. 42 (3): 673. doi:10.1080/00268978100100521.
^ Merton, R. C. (1976). "Option pricing when underlying stock returns are discontinuous". Journal of Financial Economics. 3 (1–2): 125–144. doi:10.1016/0304-405X(76)90022-2. hdl:1721.1/1899.
^ Grenander, U.; Miller, M.I. (1994). "Representations of Knowledge in Complex Systems". Journal of the Royal Statistical Society, Series B. 56 (4): 549–603. JSTOR 2346184.

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References