Bifractional Brownian motion: existence and border cases

Abstract

Bifractional Brownian motion (bfBm) is a centered Gaussian process with covarianceWe study the existence of bfBm for a given pair of parameters (h, k) and encounter some related limiting processes.MSC: primary 60G15, secondary 42A82.

Least squares estimator for non-ergodic Ornstein–Uhlenbeck processes driven by Gaussian processes

“…so B H,K has (HK − ε)−Hölder continuous paths for any ε ∈ (0, HK) thanks to Kolmogorov's continuity criterion. The bifBm B H,K can be extended for 1 < K < 2 with H ∈ (0, 1) and HK ∈ (0, 1) (see [3] and [13]).…”

Bifractional Brownian motion for H>1 and 2HK≤Talarczyk1 2020Statistics & Probability Letters5010View full textAdd to dashboardCite

“…This has not been formally proved. However, for large H the bound K ≤ 1 2H is not far short of the optimal one, since we have the following: Note that there is some discrepancy between this result and the simulations in [12] for large H. The latter suggested that the critical value of HK should be approaching 0.6 as H → ∞.…”

“…Unfortunately, Theorem 1.1 probably does not cover the whole range of parameters for which (1.1) is a covariance function. The simulations of [12] suggest that for any H > 1 there exists someK H satisfying 1 2H <K H < 1 H such that R H,K is nonnegative definite also for any 0 < K <K H . This has not been formally proved.…”

Pricing Vulnerable Options in the Bifractional Brownian Environment with Jumps

“…Es-Sebaiy and Tudor [49] gave the stochastic integral of the BFBM and illustrated that the BFBM was a valuable stochastic process to capture dynamics of financial asset prices. Lifshits and Volkova [50] considered the existence of the BFBM for a given pair of parameters. Xu [51] proposed a pricing model for European options and compound options in the BFBM environment and derived the pricing formulae by the method of variable substitution and risk-neutral principle.…”