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  "body": "\n## Definition\n\nLet $H \\in (0, 1)$.\nA *fractional Brownian motion* with Hurst index $H$ is a real stochastic process $(B^H_t)_{t \\ge 0}$ such that:\n\n1. $(B^H_t)$ is a [[Gaussian process|22WEM6RTW0]] with $E[B^H_t] = 0$ for all $t \\ge 0$.\n2. Its [[covariance|0KBFAWYRTE]] is\n$$E[B^H_s B^H_t] = \\tfrac{1}{2}\\bigl(s^{2H} + t^{2H} - \\abs{t - s}^{2H}\\bigr) \\quad \\text{for all } s, t \\ge 0.$$\n3. Almost surely, the path $t \\mapsto B^H_t$ is continuous.\n\n## Remarks\n\nFor $H = 1/2$ the covariance is $\\frac{1}{2}(s + t - \\abs{t - s}) = \\min(s, t)$, so $B^{1/2}$ is a [[Wiener process|552AWX3QXN]].\nClause 2 gives $B^H_0 = 0$ and $E[(B^H_t - B^H_s)^2] = \\abs{t - s}^{2H}$, so the increments are stationary and the process is [[self-similar|KS3KDRRJ58]] with index $H$.\nThe value $H = 1$ is excluded: the formula then gives covariance $st$, realised by the degenerate process $t \\xi$ with $\\xi$ standard normal, which has no randomness in its shape.\nDisjoint increments are positively correlated for $H \u003e 1/2$ and negatively correlated for $H \u003c 1/2$.\nAlmost every path is Hölder continuous of every order less than $H$, and for $H \\ne 1/2$ the process is not a [[semimartingale|MZNFPJJMY9]].\n",
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