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Fractional Brownian motion
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<h2 id="background-and-definition">Background and definition</h2> <p>Prior to the introduction of the fractional Brownian motion, Lévy (1953) used the <a href="/facts/Riemann%25E2%2580%2593Liouville_integral/2W9zsLdy">Riemann–Liouville fractional integral</a> to define the process </p> 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)} <p>where integration is with respect to the <a href="/facts/White_noise_measure/4FLPWdfp">white noise measure</a> <i>dB</i>(<i>s</i>). 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. </p><p>The idea instead is to use a different fractional integral of white noise to define the process: the <a href="/facts/Weyl_integral/m0QBT9jo">Weyl integral</a> </p> 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\}} <p>for <i>t</i> > 0 (and similarly for <i>t</i> < 0). The resulting process has stationary increments. </p><p>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. </p> <h2 id="properties">Properties</h2> <h3>Self-similarity</h3> <p>The process is <a href="/facts/Self-similarity/9Nvo7edY">self-similar</a>, since in terms of <a href="/facts/Probability_distribution/EpsKKVRu">probability distributions</a>: </p> B H ( a t ) ∼ | a | H B H ( t ) . {\displaystyle B_{H}(at)\sim |a|^{H}B_{H}(t).} <p>This property is due to the fact that the covariance function is homogeneous of order 2H and can be considered as a <a href="/facts/Fractal/sXj38JeN">fractal</a> property. FBm can also be defined as the unique mean-zero <a href="/facts/Gaussian_process/MrBq7kYW">Gaussian process</a>, null at the origin, with stationary and self-similar increments. </p> <h3>Stationary increments</h3> <p>It has <a href="/facts/Stationary_increments/m9A3YQW8">stationary increments</a>: </p> B H ( t ) − B H ( s ) ∼ B H ( t − s ) . {\displaystyle B_{H}(t)-B_{H}(s)\;\sim \;B_{H}(t-s).} <h3>Long-range dependence</h3> <p>For <i>H</i> > 1/2 the process exhibits <a href="/facts/Long-range_dependence/R07tX3zk">long-range dependence</a>, </p> ∑ 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 .} <h3>Regularity</h3> <p>Sample-paths are <a href="/facts/Almost_nowhere/1quRFtJE">almost</a> <a href="/facts/Nowhere_differentiable/jQqTgLk1">nowhere differentiable</a>. However, <a href="/facts/Almost_surely/fWaxzpBA">almost-all</a> trajectories are locally <a href="/facts/H%25C3%25B6lder_condition/CI0yp6kD">Hölder continuous</a> of any order strictly less than <i>H</i>: for each such trajectory, for every <i>T</i> > 0 and for every <i>ε</i> > 0 there exists a (random) constant <i>c</i> such that </p> | B H ( t ) − B H ( s ) | ≤ c | t − s | H − ε {\displaystyle |B_{H}(t)-B_{H}(s)|\leq c|t-s|^{H-\varepsilon }} <p>for 0 < <i>s</i>,<i>t</i> < <i>T</i>. </p> <h3>Dimension</h3> <p>With probability 1, the graph of <i>BH</i>(<i>t</i>) has both <a href="/facts/Hausdorff_dimension/rtX9ryrp">Hausdorff dimension</a><a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> and <a href="/facts/Box_dimension/lDLdIezz">box dimension</a><a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> of 2−<i>H</i>. </p> <h3>Integration</h3> <p>As for regular Brownian motion, one can define <a href="/facts/Ito_calculus/2uSR02fY">stochastic integrals</a> 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 <a href="/facts/Semimartingale/DUU5kWtC">semimartingales</a>. </p> <h3>Frequency-domain interpretation</h3> <p>Just as Brownian motion can be viewed as white noise filtered by ω − 1 {\displaystyle \omega ^{-1}} (i.e. integrated), fractional Brownian motion is white noise filtered by ω − H − 1 / 2 {\displaystyle \omega ^{-H-1/2}} (corresponding to <a href="/facts/Fractional_integration/gkWncv4v">fractional integration</a>). </p> <h2 id="sample-paths">Sample paths</h2> <p>Practical computer realisations of an <i>fBm</i> can be generated,<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> although they are only a finite approximation. The sample paths chosen can be thought of as showing discrete sampled points on an <i>fBm</i> process. Three realizations are shown below, each with 1000 points of an <i>fBm</i> with Hurst parameter 0.75. </p> <table><tbody><tr><td></td><td></td><td></td></tr></tbody></table> <p>Realizations of three different types of <i>fBm</i> 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. </p> <table><tbody><tr><td></td><td></td><td></td></tr></tbody></table> <h3>Method 1 of simulation</h3> <p>One can simulate sample-paths of an <i>fBm</i> using methods for generating stationary Gaussian processes with known covariance function. The simplest method relies on the <a href="/facts/Cholesky_decomposition/Q3IjkuTs">Cholesky decomposition method</a> 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). </p><p>Suppose we want to simulate the values of the <i>fBM</i> at times t 1 , … , t n {\displaystyle t_{1},\ldots ,t_{n}} using the <a href="/facts/Cholesky_decomposition/Q3IjkuTs">Cholesky decomposition method</a>. </p> <ul><li>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} .</li> <li>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 } .</li> <li>Construct a vector v {\displaystyle \,v} of <i>n</i> numbers drawn independently according to a standard Gaussian distribution,</li> <li>If we define u = Σ v {\displaystyle \,u=\Sigma v} then u {\displaystyle \,u} yields a sample path of an <i>fBm</i>.</li></ul> <p>In order to compute Σ {\displaystyle \,\Sigma } , we can use for instance the <a href="/facts/Cholesky_decomposition/Q3IjkuTs">Cholesky decomposition method</a>. An alternative method uses the <a href="/facts/Eigenvalues/8TjEoT8u">eigenvalues</a> of Γ {\displaystyle \,\Gamma } : </p> <ul><li>Since Γ {\displaystyle \,\Gamma } is <a href="/facts/Symmetric_matrix/08CGwjNl">symmetric</a>, <a href="/facts/Positive-definite_matrix/Hbr6AuPS">positive-definite</a> matrix, it follows that all <a href="/facts/Eigenvalues/8TjEoT8u">eigenvalues</a> 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} ).</li> <li>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 <a href="/facts/Kronecker_delta/gcGAy0bm">Kronecker delta</a>. 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}} .</li></ul> <p>Note that the result is real-valued because λ i > 0 {\displaystyle \lambda _{i}>0} . </p> <ul><li>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}} .</li></ul> <p>Note that since the eigenvectors are linearly independent, the matrix P {\displaystyle \,P} is invertible. </p> <ul><li>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}} .</li></ul> <h3>Method 2 of simulation</h3> <p>It is also known that <a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> 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)} <p>where <i>B</i> is a standard Brownian motion and </p> 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).} <p>Where 2 F 1 {\displaystyle _{2}F_{1}} is the <a href="/facts/Euler_hypergeometric_integral/yTEC6NBJ">Euler hypergeometric integral</a>. </p><p>Say we want to simulate an <i>fBm</i> at points 0 = t 0 < t 1 < ⋯ < t n = T {\displaystyle 0=t_{0}<t_{1}<\cdots <t_{n}=T} . </p> <ul><li>Construct a vector of <i>n</i> numbers drawn according to a standard Gaussian distribution.</li> <li>Multiply it component-wise by √<i>T</i>/<i>n</i> to obtain the increments of a Brownian motion on [0, <i>T</i>]. Denote this vector by ( δ B 1 , … , δ B n ) {\displaystyle (\delta B_{1},\ldots ,\delta B_{n})} .</li> <li>For each t j {\displaystyle t_{j}} , compute</li></ul> 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}.} <p>The integral may be efficiently computed by <a href="/facts/Gaussian_quadrature/MLmU1Dgc">Gaussian quadrature</a>. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Brownian_surface/EjsICVJS">Brownian surface</a></li> <li><a href="/facts/Autoregressive_fractionally_integrated_moving_average/EE3m3vg5">Autoregressive fractionally integrated moving average</a></li> <li><a href="/facts/Multifractal/Fly3MCTb">Multifractal</a>: The generalized framework of fractional Brownian motions.</li> <li><a href="/facts/Pink_noise/pUPduD35">Pink noise</a></li> <li><a href="/facts/Tweedie_distributions/A1PM0vQ2">Tweedie distributions</a></li></ul> <h2 id="notes">Notes</h2> <ul><li>Beran, J. (1994), <i>Statistics for Long-Memory Processes</i>, Chapman & Hall, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-412-04901-5.</li> <li>Craigmile P.F. (2003), "Simulating a class of stationary Gaussian processes using the Davies–Harte Algorithm, with application to long memory processes", <i>Journal of Times Series Analysis</i>, 24: 505–511.</li> <li>Dieker, T. (2004). <a href="http://www.columbia.edu/~ad3217/fbm/thesis.pdf"><i>Simulation of fractional Brownian motion</i></a> (PDF) (M.Sc. thesis). Retrieved 29 December 2012.</li> <li>Dietrich, C. R.; Newsam, G. N. (1997), "Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix.", <i>SIAM Journal on Scientific Computing</i>, 18 (4): 1088–1107, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1997SJSC...18.1088D">1997SJSC...18.1088D</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1137%2Fs1064827592240555">10.1137/s1064827592240555</a>.</li> <li>Falconer, Kenneth (2003), <a href="https://archive.org/details/FractalGeometry/page/n289/mode/2up"><i>Fractal Geometry Mathematical Foundations and Applications</i></a> (2 ed.), Wiley, pp. 267–271, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-470-84861-8, retrieved 23 January 2024.</li> <li>Lévy, P. (1953), <i>Random functions: General theory with special references to Laplacian random functions</i>, University of California Publications in Statistics, vol. 1, pp. 331–390.</li> <li><a href="/facts/Benoit_Mandelbrot/fa8tKP93">Mandelbrot, B.</a>; van Ness, J.W. (1968), "Fractional Brownian motions, fractional noises and applications", <i>SIAM Review</i>, 10 (4): 422–437, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1968SIAMR..10..422M">1968SIAMR..10..422M</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1137%2F1010093">10.1137/1010093</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2027184">2027184</a>.</li> <li>Orey, Steven (1970), "Gaussian sample functions and the Hausdorff dimension of level crossings", <i>Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete</i>, 15 (3): 249–256, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2FBF00534922">10.1007/BF00534922</a>, <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:121253646">121253646</a>.</li> <li>Perrin, E.; Harba, R.; Berzin-Joseph, C.; Iribarren, I.; Bonami, A. (2001). <a href="https://ieeexplore.ieee.org/document/917808">"NTH-order fractional Brownian motion and fractional Gaussian noises"</a>. <i>IEEE Transactions on Signal Processing</i>. 49 (5): 1049–1059. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2001ITSP...49.1049P">2001ITSP...49.1049P</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1109%2F78.917808">10.1109/78.917808</a>.</li> <li>Samorodnitsky G., Taqqu M.S. (1994), <i>Stable Non-Gaussian Random Processes</i>, Chapter 7: "Self-similar processes" (Chapman & Hall).</li></ul> <h2 id="further-reading">Further reading</h2> <ul><li>Sainty, P. (1992), "Construction of a complex-valued fractional Brownian motion of order <i>N</i>", <i><a href="/facts/Journal_of_Mathematical_Physics/hfGFBt5M">Journal of Mathematical Physics</a></i>, 33 (9): 3128, <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/1992JMP....33.3128S">1992JMP....33.3128S</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1063%2F1.529976">10.1063/1.529976</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Perrin et al., 2001. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Orey, 1970. <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Falconer, Kenneth (2003). Fractal Geometry Mathematical Foundations and Applications (2 ed.). Wiley. p. 268. ISBN 0-470-84861-8. Retrieved 23 January 2024. <a href="0-470-84861-8" target="_blank">0-470-84861-8</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>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. <a href="978-3-319-10063-0" target="_blank">978-3-319-10063-0</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>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. <a href="https://doi.org/10.18637%2Fjss.v005.i07" target="_blank">https://doi.org/10.18637%2Fjss.v005.i07</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Decreusefond, Laurent; Üstünel, Ali Süleyman (1999). "Stochastic analysis of the fractional Brownian motion". Potential Analysis. 10 (2): 177–214. doi:10.1023/A:1008634027843. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> </ol>