Derivatives of sup-functionals of fractional Brownian motion evaluated at H=1/2

Abstract

We consider a family of sup-functionals of (drifted) fractional Brownian motion with Hurst parameter H ∈ (0, 1). This family includes, but is not limited to: expected value of the supremum, expected workload, Wills functional, and Piterbarg-Pickands constant. Explicit formulas for the derivatives of these functionals as functions of Hurst parameter evaluated at H = 1 2 are established. In order to derive these formulas, we develop the concept of derivatives of fractional α-stable fields introduced by Stoev & Taqqu ( 2004) and propose Paley-Wiener-Zygmund representation of fractional Brownian motion.1 T H H (T ) is known as Pickands constant; see e.g. [31,32].

Lower bound for the expected supremum of fractional Brownian motion using coupling

“…We remark that the representation (4) was recently rediscovered in [14]. Following [5], we use the Paley-Wiener-Zygmund (PWZ) representation of processes…”

“…Finally, we introduce a certain functional of Brownian motion, which plays an important role in this manuscript. It is noted that its special case (H = 1 2 ) appeared in [5,Eq. (11)].…”

“…In this work we present a new theoretical lower bound for M H (T, a) for general T > 0, a ∈ R (including the case T = ∞, a > 0). Our approach is loosely based on recent work [5], where the authors consider a coupling between H-fBms with different values of H ∈ (0, 1) on the Mandelbrot & van Ness field [18]. The idea of considering such a coupling dates back at least to [20,3], who introduced a so-called multi-fractional Brownian motion.…”

“…In Section 3 we show our main results; our lower bound M H (T, a) is presented in Corollary 2. Additionally, in Corollary 4 we present an explicit formula for the derivative ∂ ∂H M H (T, a)| H=1/2 , which was given in terms of definite integral in [5]. The main results are compared to numerical simulations in Section 4, where the results are also discussed.…”

“…Extensive simulation experiments indicate that our lower bound outperforms the Monte Carlo estimates based on very dense grids for H ∈ (0, 1 2 ). Additionally, we derive the Paley-Wiener-Zygmund representation of a Linear Fractional Brownian motion and give an explicit expression for the derivative of the expected supremum at H = 1 2 in the sense of recent work by Bisewski, Dȩbicki & Rolski (2021).…”