Jump diffusion

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

A jump-diffusion model is a form of mixture model, mixing a jump process and a diffusion process. In finance, jump-diffusion models were first introduced by Robert C. Merton.^[6] Such models have a range of financial applications from option pricing, to credit risk, to time series forecasting.^[7]

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^[8] 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.
^ Christensen, H. L. (2012). "Forecasting high-frequency futures returns using online Langevin dynamics". IEEE Journal of Selected Topics in Signal Processing. 6 (4): 366–380. doi:10.1109/JSTSP.2012.2191532. hdl:10.1109/JSTSP.2012.2191532.
^ 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.

[1] 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.

[2] 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.

[3] 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.

[4] 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.

[5] 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.

[6] 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.

[7] Christensen, H. L. (2012). "Forecasting high-frequency futures returns using online Langevin dynamics". IEEE Journal of Selected Topics in Signal Processing. 6 (4): 366–380. doi:10.1109/JSTSP.2012.2191532. hdl:10.1109/JSTSP.2012.2191532.

[8] 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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v t e Volatility
Modelling volatility	Implied volatility Volatility smile Volatility clustering Local volatility Stochastic volatility Jump-diffusion models ARCH and GARCH
Trading volatility	Volatility arbitrage Straddle Volatility swap IVX VIX

v t e Stochastic processes
Discrete time	Bernoulli process Branching process Chinese restaurant process Galton–Watson process Independent and identically distributed random variables Markov chain Moran process Random walk Loop-erased Self-avoiding Biased Maximal entropy
Continuous time	Additive process Bessel process Birth–death process pure birth Brownian motion Bridge Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Dyson Brownian motion Empirical process Feller process Fleming–Viot process Gamma process Geometric process Hawkes process Hunt process Interacting particle systems Itô diffusion Itô process Jump diffusion Jump process Lévy process Local time Markov additive process McKean–Vlasov process Ornstein–Uhlenbeck process Poisson process Compound Non-homogeneous Schramm–Loewner evolution Semimartingale Sigma-martingale Stable process Superprocess Telegraph process Variance gamma process Wiener process Wiener sausage
Both	Branching process Gaussian process Hidden Markov model (HMM) Markov process Martingale Differences Local Sub- Super- Random dynamical system Regenerative process Renewal process Stochastic chains with memory of variable length White noise
Fields and other	Dirichlet process Gaussian random field Gibbs measure Hopfield model Ising model Potts model Boolean network Markov random field Percolation Pitman–Yor process Point process Cox Poisson Random field Random graph
Time series models	Autoregressive conditional heteroskedasticity (ARCH) model Autoregressive integrated moving average (ARIMA) model Autoregressive (AR) model Autoregressive–moving-average (ARMA) model Generalized autoregressive conditional heteroskedasticity (GARCH) model Moving-average (MA) model
Financial models	Binomial options pricing model Black–Derman–Toy Black–Karasinski Black–Scholes Chan–Karolyi–Longstaff–Sanders (CKLS) Chen Constant elasticity of variance (CEV) Cox–Ingersoll–Ross (CIR) Garman–Kohlhagen Heath–Jarrow–Morton (HJM) Heston Ho–Lee Hull–White Korn-Kreer-Lenssen LIBOR market Rendleman–Bartter SABR volatility Vašíček Wilkie
Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson
Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c
Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mixing Piecewise-deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible
Limit theorems	Central limit theorem Donsker's theorem Doob's martingale convergence theorems Ergodic theorem Fisher–Tippett–Gnedenko theorem Large deviation principle Law of large numbers (weak/strong) Law of the iterated logarithm Maximal ergodic theorem Sanov's theorem Zero–one laws (Blumenthal, Borel–Cantelli, Engelbert–Schmidt, Hewitt–Savage, Kolmogorov, Lévy)
Inequalities	Burkholder–Davis–Gundy Doob's martingale Doob's upcrossing Kunita–Watanabe Marcinkiewicz–Zygmund
Tools	Cameron–Martin formula Convergence of random variables Doléans-Dade exponential Doob decomposition theorem Doob–Meyer decomposition theorem Doob's optional stopping theorem Dynkin's formula Feynman–Kac formula Filtration Girsanov theorem Infinitesimal generator Itô integral Itô's lemma Karhunen–Loève theorem Kolmogorov continuity theorem Kolmogorov extension theorem Lévy–Prokhorov metric Malliavin calculus Martingale representation theorem Optional stopping theorem Prokhorov's theorem Quadratic variation Reflection principle Skorokhod integral Skorokhod's representation theorem Skorokhod space Snell envelope Stochastic differential equation Tanaka Stopping time Stratonovich integral Uniform integrability Usual hypotheses Wiener space Classical Abstract
Disciplines	Actuarial mathematics Control theory Econometrics Ergodic theory Extreme value theory (EVT) Large deviations theory Mathematical finance Mathematical statistics Probability theory Queueing theory Renewal theory Ruin theory Signal processing Statistics Stochastic analysis Time series analysis Machine learning
List of topics Category

In physics

In economics and finance

In pattern theory, computer vision, and medical imaging

See also

References