Jump process WikiZero

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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, 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
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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
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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
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Jump process

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