Fractional Brownian motion stationary and self-similar increments. It has stationary increments: For "H" > ½ the process exhibits long-range dependence, 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 for 0 < "s","t" < "T". With probability 1, the graph of "B"("t") has both Hausdorff dimension and box dimension of 2−"H". As for regular Brownian motion, one can define stochastic integrals with respect to fractional Brownian motion, usually called "fractional
Brownian bridge Brownian bridge A Brownian bridge is a continuous-time stochastic process "B"("t") whose probability distribution is the conditional probability distribution of a Wiener process "W"("t") (a mathematical model of Brownian motion) subject to the condition (when standardized) that "W"(T) = 0, so that the process is pinned at the origin at both "t=0" and "t=T". More precisely: The expected value of the bridge is zero, with variance formula_2, implying that the most uncertainty is in the middle of the bridge, with zero uncertainty at the nodes. The covariance of "B"("s") and "B"("t") is "s"(T − "t")/T if "s" < "t". The