Let X = {X(t), t ∈ RN} be a multiparameter fractional Brownian motion of index α (0 < α < 1) in R. We prove that if N < αd , then there exist positive finite constants K1 and K2 such that with probability 1, K1 ≤ φ-p(X([0, 1] )) ≤ φ-p(GrX([0, 1] )) ≤ K2 where φ(s) = s/(log log 1/s), φ-p(E) is the φ-packing measure of E, X([0, 1] ) is the image and GrX([0, 1] ) = {(t,X(t)); t ∈ [0, 1]N} is the g...

Given an integer m, a probability measure ν on [0, 1], a process X and a real function g, we define the m-order ν-integral having as integrator X and as integrand g(X). In the case of the fractional Brownian motion B , for any locally bounded function g, the corresponding integral vanishes for all odd indices m > 1 2H and any symmetric ν. One consequence is an Itô-Stratonovich type expansion fo...

This paper deals with the dynamics of stochastic delay lattice systems fractional discrete Laplacian driven by nonlinear noise. The existence a invariant measure for nonlocal is established under certain conditions. idea uniform estimates on tails solutions, technique diadic division and Arzela-Ascoli theorem are employed to show tightness family probability distributions solutions.

Instantaneous volatility of logarithmic return in the lognormal fractional SABR model is driven by exponentiation a correlated Brownian motion. Due to mixed nature driving and motions, probability density for such less studied literature. We show this paper bridge representation joint Fourier space. Evaluating along properly chosen deterministic path yields small time asymptotic expansion leadi...

This is sometimes denoted simply “X−1(B) ⊂ F.” Since the probability measure P is only defined on sets F ∈ F, a random variable must satisfy this condition if we are to be able to find the probability Pr[X ∈ B] for each Borel set B, or even if we want to find the distribution function (DF) FX(b) ≡ Pr[X ≤ b] for each rational number b. Note that set-inverses are rather well-behaved functions fro...