Jump diffusion is a stochastic process that involves jumps and diffusion. It has important applications in magnetic reconnection, coronal mass ejections, condensed matter physics 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 autocorrelation function have been derived for several jump(-diffusion) models:
- Singwi, Sjölander 1960: alternation between oscillatory motion and directed motion
- Chudley, Elliott 1961: jumps on a lattice
- Sears 1966, 1967: jump diffusion of rotational degrees of freedom
- Hall, Ross 1981: 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. Due to their computational tractability, the special case of a basic affine jump diffusion is popular for some credit risk and short-rate models.
- 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: 125–144. doi:10.1016/0304-405X(76)90022-2. hdl:1721.1/1899.