A Bayesian sequential test for the drift of a fractional Brownian motion

We consider a fractional Brownian motion with linear drift such that its unknown drift coefficient has a prior normal distribution and construct a sequential test for the hypothesis that the drift is positive versus the alternative that it is negative. We show that the problem of constructing the test reduces to an optimal stopping problem for a standard Brownian motion obtained by a transformation of the fractional Brownian motion. The solution is described as the first exit time from some set, and it is shown that its boundaries satisfy a certain integral equation, which is solved numerically.

Some asymptotic formulars for a Brownian motion whose dimension is variable with a regular variation from a parabolic domain to research the variation of biological species

Consider a Brownian motion with a regular variation starting at an interior point of a domain D in R d+1 ,d ≥ 1, let τ D denote the first time the Brownian motion exits from D. Estimates with exact constants for the asymptotics of logP(τ D > T) are given for T → ∞, depending on the shape of the domain D and the order of the regular variation. Furthermore, the asymptotically equivalence are obtained. The problem is motivated by the early results of Lifshits and Shi, Li in the first exit time and Karamata in the regular variation. The methods of proof are based on their results and the calculus of variations.

Upper and lower bounds of the first exit time of Brownian motion from The Minimum and Maximun Parabolic Domains with variable dimension to research the variation of biological species

Consider a Brownian motion with variable dimension starting at an interior point of the minimum or maximum parabolic domains Dmax t and Dmin t in Rd(t)+2, d(t) ≥ 1 is an increasing integral function as t →∞,d(t) →∞, and let τDmax t and τDmin t denote the first time the Brownian motion exits from Dmax t and Dmin t , respectively. Upper and lower bounds with exact constants for the asymptotics of logP(τDmax t > t) and logP(τDmin t > t) are given as t → ∞, depending on the shape of the domain Dmax t and Dmin t . The methods of proof are based on Gordon’s inequality and early works of Li, Lifshits and Shi in the single general parabolic domain case.

On first exit times and their means for Brownian bridges

For a Brownian bridge from 0 to y, we prove that the mean of the first exit time from the interval $\left( -h,h \right),h>0$ , behaves as ${\mathrm{O}}(h^2)$ when $h \downarrow 0$ . Similar behaviour is also seen to hold for the three-dimensional Bessel bridge. For the Brownian bridge and three-dimensional Bessel bridge, this mean of the first exit time has a puzzling representation in terms of the Kolmogorov distribution. The result regarding the Brownian bridge is applied to provide a detailed proof of an estimate needed by Walsh to determine the convergence of the binomial tree scheme for European options.