Jump process
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]
See also
- Poisson process, an example of a jump process
- Continuous-time Markov chain (CTMC), an example of a jump process and a generalization of the Poisson process
- Counting process, an example of a jump process and a generalization of the Poisson process in a different direction than that of CTMCs
- Interacting particle system, an example of a jump process
- Kolmogorov equations (continuous-time Markov chains)
References
- ^ 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
- Additive process
- Bessel process
- Birth–death process
- Brownian motion
- 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
- Schramm–Loewner evolution
- Semimartingale
- Sigma-martingale
- Stable process
- Superprocess
- Telegraph process
- Variance gamma process
- Wiener process
- Wiener sausage
- 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
- 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)
- 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
- Stopping time
- Stratonovich integral
- Uniform integrability
- Usual hypotheses
- Wiener space
- 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
This probability-related article is a stub. You can help Wikipedia by expanding it. |
- v
- t
- e