Generalized fractional Brownian motion

Abstract

We introduce a new Gaussian process, a generalization of both fractional and subfractional Brownian motions, which could serve as a good model for a larger class of natural phenomena. We study its main stochastic properties and some increments characteristics. As an application, we deduce the properties of nonsemimartingality, Hölder continuity, nondifferentiablity, and existence of a local time.

Mixed Generalized Fractional Brownian Motion

“…Let us check the mixed-self-semilarity property of the mgfBm. This property was introduced in [29] for the mfBm and investigated to show the Hölder continuity of the mfBm. See also [12] for the sfBm case.…”

“…When a = 1 and b = 0, the mfBm is the Brownian motion and when a = 0 and b = 1, is the fBm. We refer also to [11,10,29,27] for further information on this process.…”

SPDE with generalized drift and fractional-type noise

“…Example 4. Let us denote by M H = {M H t (a, b); t ≥ 0} = {M H t ; t ≥ 0} the mixed-fractional Brownian motion (mfBm, see [13]) of parameters a, b and H such that 0 < H < 1, (a, b) ∈ R 2 \{(0, 0)}; that is the centered Gaussian process, starting from zero, with covariance…”

“…So the mfBm is clearly an extension of the fractional Brownian motion and of the Wiener process. We refer to [13] for further information on this process.…”

Variable annuities valuation under a mixed fractional Brownian motion environment with jumps considering mortality risk

“…The stock price returns are positively long-range dependent, therefore, we assume H > 1 2 . 17 Another feature present in stock returns is the presence of infrequent jumps. Therefore, to capture sudden and infrequent jumps, and long-range dependency in stock returns, we have used a combination of MFBM and Poisson jumps denoted by jump-diffusion mixed fractional Brownian motion (JDMFBM) model.…”