Jump process

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Stochastic process with discrete movements

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A jump process is a type of stochastic process that has discrete movements, called jumps, with random arrival times, rather than continuous movement. It is typically modelled as a simple or compound Poisson process.^[1]

In finance, various stochastic models are used to model the price movements of financial instruments; for example the Black-Scholes model for pricing options assumes that the underlying instrument follows a traditional diffusion process, with continuous, random movements at all scales, no matter how small. John Carrington Cox and Stephen Ross^[2]^{: 145-166} proposed that prices actually follow a 'jump process'.

Robert C. Merton extended this approach to a hybrid model known as jump diffusion, which states that the prices have large jumps interspersed with small continuous movements.^[3]

References

[edit]

^ Tankov, P. (2003). Financial modelling with jump processes (Vol. 2). CRC press.
^ Cox, J. C.; Ross, S. A. (1976). "The valuation of options for alternative stochastic processes". Journal of Financial Economics. 3 (1-2): 145-166. CiteSeerX 10.1.1.540.5486. doi:10.1016/0304-405X(76)90023-4.
^ Merton, R. C. (1976). "Option pricing when underlying stock returns are discontinuous". Journal of Financial Economics. 3 (1-2): 125-144. CiteSeerX 10.1.1.588.7328. doi:10.1016/0304-405X(76)90022-2. hdl:1721.1/1899.

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 Airy process Bessel process Birth-death process pure birth Brownian motion Bridge Dyson Excursion Fractional Geometric Meander Cauchy process Contact process Continuous-time random walk Cox process Diffusion process Empirical process Feller process Fleming-Viot process Gamma process Geometric process Hawkes process Hunt process Interacting particle systems Ito diffusion Ito process Jump diffusion Jump process Levy process Local time Markov additive process McKean-Vlasov process Ornstein-Uhlenbeck process Poisson process Compound Non-homogeneous Quasimartingale 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 Determinantal 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 Vasicek Wilkie
Actuarial models	Buhlmann Cramer-Lundberg Risk process Sparre-Anderson
Queueing models	Bulk Fluid Generalized queueing network M/G/1 M/M/1 M/M/c
Properties	Cadlag 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, Levy)
Inequalities	Burkholder-Davis-Gundy Doob's martingale Doob's upcrossing Kunita-Watanabe Marcinkiewicz-Zygmund
Tools	Cameron-Martin theorem Convergence of random variables Doleans-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 Ito integral Ito's lemma Kolmogorov continuity theorem Kolmogorov extension theorem Kosambi-Karhunen-Loeve theorem Levy-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
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