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 model of financial markets for buying or selling). Another assumption is that asset prices have no jumps, that is there are no surprises in the market. This last assumption is removed in jump diffusion models. Consider a financial market consisting of formula_1 financial assets, where one of these assets, called a "bond" or "money market", is risk free while the remaining formula_2 assets, called "stocks", are risky. A "financial market" is defined as formula_3 that satisfies the following: Let formula_17 be a probability space, and a formula_18 be D-dimensional Brownian motion stochastic process, with the natural filtration: If formula_20 are the measure 0 (i.e.