Is the Driving Force of a Continuous Process a Brownian Motion or Fractional Brownian Motion?

Is the Driving Force of a Continuous Process a Brownian Motion or Fractional Brownian Motion?

Itô’s semimartingale driven by a Brownian motion is typically used in modeling the asset prices, interest rates and exchange rates, and so on. However, the assumption of Brownian motion as a driving force of the underlying asset price processes is rarely contested in practice. This naturally raises the question of whether this assumption is really appropriate. In the paper we propose a statisti...

Lacunary Fractional Brownian Motion

In this paper, a new class of Gaussian field is introduced called Lacunary Fractional Brownian Motion. Surprisingly we show that usually their tangent fields are not unique at every point. We also investigate the smoothness of the sample paths of Lacunary Fractional Brownian Motion using wavelet analysis.

Tempered fractional Brownian motion

On time-dependent neutral stochastic evolution equations with a fractional Brownian motion and infinite delays

In this paper, we consider a class of time-dependent neutral stochastic evolution equations with the infinite delay and a fractional Brownian motion in a Hilbert space. We establish the existence and uniqueness of mild solutions for these equations under non-Lipschitz conditions with Lipschitz conditions being considered as a special case. An example is provided to illustrate the theory

Continuous time process and Brownian motion

Consider a complete probability space (Ω,F ,P;F) equipped with the Þltration F = {Ft; 0 ≤ t <∞}. A stochastic process is a collection of random variables X = {Xt; 0 ≤ t <∞} where, for every t, Xt : Ω → Rd is a random variable. We assume the space Rd is equipped with the usual Borel σ-algebra B(Rd). Every Þxed ω ∈ Ω corresponds to a sample path (or, trajectory), that is, the function t 7→ Xt(ω) ...