Diffusion process

Inprobability theoryandstatistics,diffusion processesare a class of continuous-timeMarkov processwithalmost surely continuoussample paths. Diffusion process is stochastic in nature and hence is used to model many real-life stochastic systems.Brownian motion,reflected Brownian motionandOrnstein–Uhlenbeck processesare examples of diffusion processes. It is used heavily instatistical physics,statistical analysis,information theory,data science,neural networks,financeandmarketing.

A sample path of a diffusion process models the trajectory of a particle embedded in a flowing fluid and subjected to random displacements due to collisions with other particles, which is calledBrownian motion.The position of the particle is then random; itsprobability density functionas afunction of space and timeis governed by aconvection–diffusion equation.

Mathematical definition[edit]

Adiffusion processis aMarkov processwithcontinuous sample pathsfor which theKolmogorov forward equationis theFokker–Planck equation.^[1]

References[edit]

^"9. Diffusion processes"(pdf).RetrievedOctober 10,2011.

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[1] "9. Diffusion processes"(pdf).RetrievedOctober 10,2011.

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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
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Actuarial models	Bühlmann Cramér–Lundberg Risk process Sparre–Anderson
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Properties	Càdlàg paths Continuous Continuous paths Ergodic Exchangeable Feller-continuous Gauss–Markov Markov Mi xing Piecewise-deterministic Predictable Progressively measurable Self-similar Stationary Time-reversible
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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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Mathematical definition[edit]

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