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search appearance donate create account log in personal tools donate create account log in contents move to sidebar hide top 1 background and definition 2 properties toggle properties subsection 2 1 self similarity 2 2 stationary increments 2 3 long range dependence 2 4 regularity 2 5 dimension 2 6 integration 2 7 frequency domain interpretation 3 sample paths toggle sample paths subsection 3 1 method 1 of simulation 3 2 method 2 of simulation 4 see also 5 notes 6 references 7 further reading toggle the table of contents fractional brownian motion 6 languages deutsch français 日本語 português türkçe українська edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia probability theory concept in probability theory fractional brownian motion fbm also called a fractal brownian motion is a generalization of brownian motion unlike classical brownian motion the increments of fbm need not be independent fbm is a continuous time gaussian process b h t textstyle b_ h t on 0 t textstyle 0 t that starts at zero has expectation zero for all t displaystyle t in 0 t textstyle 0 t and has the following covariance function e b h t b h s 1 2 t 2 h s 2 h t s 2 h displaystyle e b_ h t b_ h s tfrac 1 2 t 2h s 2h t s 2h where h is a real number in 0 1 called the hurst index or hurst parameter associated with the fractional brownian motion the hurst exponent describes the raggedness of the resultant motion with a higher value leading to a smoother motion it was introduced by mandelbrot van ness 1968 the value of h determines what kind of process the fbm is if h 1 2 then the process is in fact a brownian motion or wiener process if h 1 2 then the increments of the process are positively correlated if h 1 2 then the increments of the process are negatively correlated fractional brownian motion has stationary increments x t b h s t b h s the value is the same for any s the increment process x t is known as fractional gaussian noise there is also a generalization of fractional brownian motion n th order fractional brownian motion abbreviated as n fbm 1 n fbm is a gaussian self similar non stationary process whose increments of order n are stationary for n 1 n fbm is classical fbm like the brownian motion that it generalizes fractional brownian motion is named after 19th century biologist robert brown fractional gaussian noise is named after mathematician carl friedrich gauss background and definition edit prior to the introduction of the fractional brownian motion lévy 1953 used the riemann liouville fractional integral to define the process b h t 1 γ h 1 2 0 t t s h 1 2 d b s displaystyle tilde b _ h t frac 1 gamma h 1 2 int _ 0 t t s h 1 2 db s where integration is with respect to the white noise measure db s this integral turns out to be ill suited as a definition of fractional brownian motion because of its over emphasis of the origin mandelbrot van ness 1968 p 424 it does not have stationary increments the idea instead is to use a different fractional integral of white noise to define the process the weyl integral b h t b h 0 1 γ h 1 2 0 t s h 1 2 s h 1 2 d b s 0 t t s h 1 2 d b s displaystyle b_ h t b_ h 0 frac 1 gamma h 1 2 left int _ infty 0 left t s h 1 2 s h 1 2 right db s int _ 0 t t s h 1 2 db s right for t 0 and similarly for t 0 the resulting process has stationary increments the main difference between fractional brownian motion and regular brownian motion is that while the increments in brownian motion are independent increments for fractional brownian motion are not if h 1 2 then there is positive autocorrelation if there is an increasing pattern in the previous steps then it is likely that the current step will be increasing as well if h 1 2 the autocorrelation is negative properties edit self similarity edit the process is self similar since in terms of probability distributions b h a t a h b h t displaystyle b_ h at sim a h b_ h t this property is due to the fact that the covariance function is homogeneous of order 2h and can be considered as a fractal property fbm can also be defined as the unique mean zero gaussian process null at the origin with stationary and self similar increments stationary increments edit it has stationary increments b h t b h s b h t s displaystyle b_ h t b_ h s sim b_ h t s long range dependence edit for h 1 2 the process exhibits long range dependence n 1 e b h 1 b h n 1 b h n displaystyle sum _ n 1 infty e b_ h 1 b_ h n 1 b_ h n infty regularity edit sample paths are almost nowhere differentiable however almost all trajectories are locally hölder continuous of any order strictly less than h for each such trajectory for every t 0 and for every ε 0 there exists a random constant c such that b h t b h s c t s h ε displaystyle b_ h t b_ h s leq c t s h varepsilon for 0 s t t dimension edit with probability 1 the graph of b h t has both hausdorff dimension 2 and box dimension 3 of 2 h integration edit as for regular brownian motion one can define stochastic integrals with respect to fractional brownian motion usually called fractional stochastic integrals in general though unlike integrals with respect to regular brownian motion fractional stochastic integrals are not semimartingales frequency domain interpretation edit just as brownian motion can be viewed as white noise filtered by ω 2 displaystyle omega 2 i e integrated fractional brownian motion is white noise filtered by ω h 1 2 displaystyle omega h 1 2 corresponding to fractional integration sample paths edit practical computer realisations of an fbm can be generated 4 5 although they are only a finite approximation the sample paths chosen can be thought of as showing discrete sampled points on an fbm process three realizations are shown below each with 1000 points of an fbm with hurst parameter 0 75 h 0 75 realisation 1 h 0 75 realisation 2 h 0 75 realisation 3 realizations of three different types of fbm are shown below each showing 1000 points the first with hurst parameter 0 15 the second with hurst parameter 0 55 and the third with hurst parameter 0 95 the higher the hurst parameter is the smoother the curve will be h 0 15 h 0 55 h 0 95 method 1 of simulation edit one can simulate sample paths of an fbm using methods for generating stationary gaussian processes with known covariance function the simplest method relies on the cholesky decomposition method of the covariance matrix explained below which on a grid of size n displaystyle n has complexity of order o n 3 displaystyle o n 3 a more complex but computationally faster method is the circulant embedding method of dietrich newsam 1997 suppose we want to simulate the values of the fbm at times t 1 t n displaystyle t_ 1 ldots t_ n using the cholesky decomposition method form the matrix γ r t i t j i j 1 n displaystyle gamma bigl r t_ i t_ j i j 1 ldots n bigr where r t s s 2 h t 2 h t s 2 h 2 displaystyle r t s s 2h t 2h t s 2h 2 compute σ displaystyle sigma the square root matrix of γ displaystyle gamma i e σ 2 γ displaystyle sigma 2 gamma loosely speaking σ displaystyle sigma is the standard deviation matrix associated to the variance covariance matrix γ displaystyle gamma construct a vector v displaystyle v of n numbers drawn independently according to a standard gaussian distribution if we define u σ v displaystyle u sigma v then u displaystyle u yields a sample path of an fbm in order to compute σ displaystyle sigma we can use for instance the cholesky decomposition method an alternative method uses the eigenvalues of γ displaystyle gamma since γ displaystyle gamma is symmetric positive definite matrix it follows that all eigenvalues λ i displaystyle lambda _ i of γ displaystyle gamma satisfy λ i 0 displaystyle lambda _ i 0 i 1 n displaystyle i 1 dots n let λ displaystyle lambda be the diagonal matrix of the eigenvalues i e λ i j λ i δ i j displaystyle lambda _ ij lambda _ i delta _ ij where δ i j displaystyle delta _ ij is the kronecker delta we define λ 1 2 displaystyle lambda 1 2 as the diagonal matrix with entries λ i 1 2 displaystyle lambda _ i 1 2 i e λ i j 1 2 λ i 1 2 δ i j displaystyle lambda _ ij 1 2 lambda _ i 1 2 delta _ ij note that the result is real valued because λ i 0 displaystyle lambda _ i 0 let v i displaystyle v_ i an eigenvector associated to the eigenvalue λ i displaystyle lambda _ i define p displaystyle p as the matrix whose i displaystyle i th column is the eigenvector v i displaystyle v_ i note that since the eigenvectors are linearly independent the matrix p displaystyle p is invertible it follows then that σ p λ 1 2 p 1 displaystyle sigma p lambda 1 2 p 1 because γ p λ p 1 displaystyle gamma p lambda p 1 method 2 of simulation edit it is also known that 6 b h t 0 t k h t s d b s displaystyle b_ h t int _ 0 t k_ h t s db s where b is a standard brownian motion and k h t s t s h 1 2 γ h 1 2 2 f 1 h 1 2 1 2 h h 1 2 1 t s displaystyle k_ h t s frac t s h frac 1 2 gamma h frac 1 2 _ 2 f_ 1 left h frac 1 2 frac 1 2 h h frac 1 2 1 frac t s right where 2 f 1 displaystyle _ 2 f_ 1 is the euler hypergeometric integral say we want to simulate an fbm at points 0 t 0 t 1 t n t displaystyle 0 t_ 0 t_ 1 cdots t_ n t construct a vector of n numbers drawn according to a standard gaussian distribution multiply it component wise by t n to obtain the increments of a brownian motion on 0 t denote this vector by δ b 1 δ b n displaystyle delta b_ 1 ldots delta b_ n for each t j displaystyle t_ j compute b h t j n t i 0 j 1 t i t i 1 k h t j s d s δ b i displaystyle b_ h t_ j frac n t sum _ i 0 j 1 int _ t_ i t_ i 1 k_ h t_ j s ds delta b_ i the integral may be efficiently computed by gaussian quadrature see also edit brownian surface autoregressive fractionally integrated moving average multifractal the generalized framework of fractional brownian motions pink noise tweedie distributions notes edit perrin et al 2001 orey 1970 falconer kenneth 2003 fractal geometry mathematical foundations and applications 2 ed wiley p 268 isbn 0 470 84861 8 retrieved 23 january 2024 kroese dirk p botev zdravko i 2015 spatial process simulation in schmidt v ed stochastic geometry spatial statistics and random fields lecture notes in mathematics vol 2120 berlin springer verlag pp 369 404 arxiv 1308 0399 doi 10 1007 978 3 319 10064 7_12 isbn 978 3 319 10063 0 coeurjolly jean françois 2000 simulation and identification of the fractional brownian motion a bibliographical and comparative study journal of statistical software 5 7 doi 10 18637 jss v005 i07 decreusefond laurent üstünel ali süleyman 1999 stochastic analysis of the fractional brownian motion potential analysis 10 2 177 214 doi 10 1023 a 1008634027843 references edit beran j 1994 statistics for long memory processes chapman hall isbn 0 412 04901 5 craigmile p f 2003 simulating a class of stationary gaussian processes using the davies harte algorithm with application to long memory processes journal of times series analysis 24 505 511 dieker t 2004 simulation of fractional brownian motion pdf m sc thesis retrieved 29 december 2012 dietrich c r newsam g n 1997 fast and exact simulation of stationary gaussian processes through circulant embedding of the covariance matrix siam journal on scientific computing 18 4 1088 1107 bibcode 1997sjsc 18 1088d doi 10 1137 s1064827592240555 falconer kenneth 2003 fractal geometry mathematical foundations and applications 2 ed wiley pp 267 271 isbn 0 470 84861 8 retrieved 23 january 2024 lévy p 1953 random functions general theory with special references to laplacian random functions university of california publications in statistics vol 1 pp 331 390 mandelbrot b van ness j w 1968 fractional brownian motions fractional noises and applications siam review 10 4 422 437 bibcode 1968siamr 10 422m doi 10 1137 1010093 jstor 2027184 orey steven 1970 gaussian sample functions and the hausdorff dimension of level crossings zeitschrift für wahrscheinlichkeitstheorie und verwandte gebiete 15 3 249 256 doi 10 1007 bf00534922 s2cid 121253646 perrin e harba r berzin joseph c iribarren i bonami a 2001 nth order fractional brownian motion and fractional gaussian noises ieee transactions on signal processing 49 5 1049 1059 bibcode 2001itsp 49 1049p doi 10 1109 78 917808 samorodnitsky g taqqu m s 1994 stable non gaussian random processes chapter 7 self similar processes chapman hall further reading edit sainty p 1992 construction of a complex valued fractional brownian motion of order n journal of mathematical physics 33 9 3128 bibcode 1992jmp 33 3128s doi 10 1063 1 529976 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 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 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 vašíček wilkie ac...