A joint Laplace transform for pre-exit diffusion of occupation times

For a < r < b, the approach of Li and Zhou (2014) is adopted to find joint Laplace transforms of occupation times over intervals (a, r) and (r, b) for a time homogeneous diffusion process before it first exits from either a or b. The results are expressed in terms of solutions to the differential equations associated with the diffusions generator. Applying these results, we obtain more explicit expressions on the joint Laplace transforms of occupation times for Brownian motion with drift, Brownian motion with alternating drift and skew Brownian motion, respectively.     

Asymptotic Behavior of Weighted Power Variations of Fractional Brownian Motion in Brownian Time

We study the asymptotic behavior of weighted power variations of fractional Brownian motion in Brownian time \(Z_t:= X_{Y_t},t \geqslant 0\), where X is a fractional Brownian motion and Y is an independent Brownian motion.     

A Brownian Motion on the Diffeomorphism Group of the Circle

Let Diff(S ¹) be the group of orientation preserving C ^?∞? diffeomorphisms of S ¹. In 1999, P. Malliavin and then in 2002, S. Fang constructed a canonical Brownian motion associated with the H ^3/2 metric on the Lie algebra diff(S ¹). The canonical Brownian motion they constructed lives in the group Homeo(S ¹) of Hölderian homeomorphisms of S ¹, which is larger than the group Diff(S ¹). In this paper, we present another way to construct a Brownian motion that lives in the group Diff(S ¹), rather than in the larger group Homeo(S ¹).     

Harnack Inequality and Regularity for a Product of Symmetric Stable Process and Brownian Motion

In this paper, we consider a product of a symmetric stable process in ? ^d and a one-dimensional Brownian motion in ?^?+?. Then we define a class of harmonic functions with respect to this product process. We show that bounded non-negative harmonic functions in the upper-half space satisfy Harnack inequality and prove that they are locally Hölder continuous. We also argue a result on Littlewood–Paley functions which are obtained by the α-harmonic extension of an L ^p (? ^d ) function.     

On nonparametric change point estimator based on empirical characteristic functions

We propose a nonparametric change point estimator in the distributions of a sequence of independent observations in terms of the test statistics given by Huˇskov′a and Meintanis(2006) that are based on weighted empirical characteristic functions. The weight function ω(t; a) under consideration includes the two weight functions from Huˇskov′a and Meintanis(2006) plus the weight function used by Matteson and James(2014),where a is a tuning parameter. Under the local alternative hypothesis, we establish the consistency, convergence rate, and asymptotic distribution of this change point estimator which is the maxima of a two-side Brownian motion with a drift. Since the performance of the change point estimator depends on a in use, we thus propose an algorithm for choosing an appropriate value of a, denoted by a_s which is also justified. Our simulation study shows that the change point estimate obtained by using a_s has a satisfactory performance. We also apply our method to a real dataset.     

The Joint Distribution of Running Maximum of a Slepian Process

Consider the Slepian process S defined by S(t) = B(t +?1) ? B(t),t ∈ [0, 1] with B(t), t ∈ ? a standard Brownian motion. In this contribution we analyze the properties between the maximum \(m_{s}=\max \limits _{0\leq u\leq s}S(u)\) and the maximum \(m_{t}=\max \limits _{0\leq u\leq t}S(u)\) for 0 ≤ s < t ≤?1 fixed. Explicit integral expressions are obtained for the joint distribution function between m _s and m _t and the distribution function of the partial maximum m _s . Further, we apply our results for the determination of the moments of m _s .     

Estimates for the Torsion Function and Sobolev Constants

Let Ω be an open set in Euclidean space, and let u : Ω → ?^?+? be the expected lifetime of Brownian motion in Ω. It is shown that if u?∈?L ^p (Ω) for some p?∈?[1, ?∞?) then (i) u?∈?L ^q (Ω) for all q?∈?[p,?∞?], and (ii) \({trace}\left(e^{t\Delta_{\Omega}}\right)<\infty\) for all t?>?0, where ??Δ_Ω is the Dirichlet Laplacian acting in L ²(Ω). Pointwise bounds are obtained for u in terms of the first Dirichlet eigenfunction for Ω, assuming that the spectrum of ??Δ_Ω is discrete. It is shown that if Ω is open, bounded and connected in the plane and \(\partial\Omega\) has an interior wedge with opening angle α at vertex v then the first Dirichlet eigenfunction and u are comparable near v if and only if α?≥?π/2. Two sided estimates are obtained for the Sobolev constant

$ C_p(\Omega):= \inf\left\{\Vert \nabla u \Vert_2^2: u \in C_0^{\infty}(\Omega),\ \Vert u\Vert_p = 1\right\}, $

where 0?p?Ω satisfies a strong Hardy inequality, and the distance to the boundary function δ?∈?L ^2p/(2???p)(Ω).

     

Torsional Rigidity for Regions with a Brownian Boundary

Let ?? ^m be the m-dimensional unit torus, m ∈ ?. The torsional rigidity of an open set Ω ? ?? ^m is the integral with respect to Lebesgue measure over all starting points x ∈ Ω of the expected lifetime in Ω of a Brownian motion starting at x. In this paper we consider Ω = ?? ^m \β[0, t], the complement of the path ß[0, t] of an independent Brownian motion up to time t. We compute the leading order asymptotic behaviour of the expectation of the torsional rigidity in the limit as t → ∞. For m = 2 the main contribution comes from the components in ??²\β0, t] whose inradius is comparable to the largest inradius, while for m = 3 most of ??³\β[0, t] contributes. A similar result holds for m ≥ 4 after the Brownian path is replaced by a shrinking Wiener sausage W _r(t)[0, t] of radius r(t) = o(t ^-1/(m-2)), provided the shrinking is slow enough to ensure that the torsional rigidity tends to zero. Asymptotic properties of the capacity of ß[0, t] in ?³ and W ₁[0, t] in ? ^m , m ≥ 4, play a central role throughout the paper. Our results contribute to a better understanding of the geometry of the complement of Brownian motion on ?? ^m , which has received a lot of attention in the literature in past years.     

Regularity of Intersection Local Times of Fractional Brownian Motions

Let \(B^{\alpha_{i}}\) be an (N _i ,d)-fractional Brownian motion with Hurst index α _i (i=1,2), and let \(B^{\alpha_{1}}\) and \(B^{\alpha_{2}}\) be independent. We prove that, if \(\frac{N_{1}}{\alpha_{1}}+\frac{N_{2}}{\alpha_{2}}>d\), then the intersection local times of \(B^{\alpha_{1}}\) and \(B^{\alpha_{2}}\) exist, and have a continuous version. We also establish Hölder conditions for the intersection local times and determine the Hausdorff and packing dimensions of the sets of intersection times and intersection points.One of the main motivations of this paper is from the results of Nualart and Ortiz-Latorre (J. Theor. Probab. 20:759–767, 2007), where the existence of the intersection local times of two independent (1,d)-fractional Brownian motions with the same Hurst index was studied by using a different method. Our results show that anisotropy brings subtle differences into the analytic properties of the intersection local times as well as rich geometric structures into the sets of intersection times and intersection points.     

The Exact Hausdorff Measure of the Zero Set of Fractional Brownian Motion

Let {X(t), t∈? ^N } be a fractional Brownian motion in ? ^d of index H. If L(0,I) is the local time of X at 0 on the interval I?? ^N , then there exists a positive finite constant c(=c(N,d,H)) such that

$m_\phi\bigl(X^{-1}(0)\cap I\bigr)=cL(0,I),$

where \(\phi(t)=t^{N-dH}(\log\log\frac{1}{t})^{dH/N}\), and m _φ (E) is the Hausdorff φ-measure of E. This refines a previous result of Xiao (Probab. Theory Relat. Fields 109: 126–197, 1997) on the relationship between the local time and the Hausdorff measure of zero set for d-dimensional fractional Brownian motion on ? ^N .

     

On Generalised Piterbarg Constants

We investigate generalised Piterbarg constants

$$\mathcal{P}_{\alpha, \delta}^{h}=\lim\limits_{T \rightarrow \infty} \mathbb{E}\left\{ \sup\limits_{t\in \delta \mathbb{Z} \cap [0,T]} e^{\sqrt{2}B_{\alpha}(t)-|t|^{\alpha}- h(t)}\right\} $$

determined in terms of a fractional Brownian motion B _α with Hurst index α/2∈(0,1], the non-negative constant δ and a continuous function h. We show that these constants, similarly to generalised Pickands constants, appear naturally in the tail asymptotic behaviour of supremum of Gaussian processes. Further, we derive several bounds for \(\mathcal {P}_{\alpha , \delta }^{h}\) and in special cases explicit formulas are obtained.

     

Convex hull of Brownian motion ind-dimensions

We supposeK(w) to be the boundary of the closed convex hull of a sample path ofZ _t(w), 0 ≦t ≦ 1 of Brownian motion ind-dimensions. A combinatorial result of Baxter and Borndorff Neilson on the convex hull of a random walk, and a limiting process utilizing results of P. Levy on the continuity properties ofZ _t(w) are used to show that the curvature ofK(w) is concentrated on a metrically small set.     

Parisian ruin over a finite-time horizon

For a risk process R_u(t) = u + ct- X(t), t≥0, where u≥0 is the initial capital, c 0 is the premium rate and X(t), t≥0 is an aggregate claim process, we investigate the probability of the Parisian ruin P_S(u, T_u) = P{inf (t∈[0,S]_(s∈[t,t+T_u])) sup R_u(s) 0}, S, T_u 0.For X being a general Gaussian process we derive approximations of P_S(u, T_u) as u →∞. As a by-product, we obtain the tail asymptotic behaviour of the infimum of a standard Brownian motion with drift over a finite-time interval.     

Fekete-Szegö problem for close-to-convex functions with respect to a certain convex function dependent on a real parameter

Given α ∈ [0, 1], let h _α (z):= z/(1 - αz), z ∈ D:= {z ∈ D: |z| < 1}. An analytic standardly normalized function f in D is called close-to-convex with respect to h _α if there exists δ ∈ (-π/2, π/2) such that Re{e^iδ zf′(z)/h _α (z)} > 0, z ∈ D. For the class ? (h _α ) of all close-to-convex functions with respect to h _α , the Fekete-Szegö problem is studied.     

An Analytic Expression for the Distribution of the Generalized Shiryaev–Roberts Diffusion

We consider the quickest change-point detection problem where the aim is to detect the onset of a pre-specified drift in “live”-monitored standard Brownian motion; the change-point is assumed unknown (nonrandom). The topic of interest is the distribution of the Generalized Shryaev–Roberts (GSR) detection statistic set up to “sense” the presence of the drift. Specifically, we derive a closed-form formula for the transition probability density function (pdf) of the time-homogeneous Markov diffusion process generated by the GSR statistic when the Brownian motion under surveillance is “drift-free”, i.e., in the pre-change regime; the GSR statistic’s (deterministic) nonnegative headstart is assumed arbitrarily given. The transition pdf formula is found analytically, through direct solution of the respective Kolmogorov forward equation via the Fourier spectral method to achieve separation of the spacial and temporal variables. The obtained result generalizes the well-known formula for the (pre-change) stationary distribution of the GSR statistic: the latter’s stationary distribution is the temporal limit of the distribution sought in this work. To conclude, we exploit the obtained formula numerically and briefly study the pre-change behavior of the GSR statistic versus three factors: (a) drift-shift magnitude, (b) time, and (c) the GSR statistic’s headstart.     

Stochastic Heat Equation with Multiplicative Fractional-Colored Noise

We consider the stochastic heat equation with multiplicative noise \(u_{t}=\frac{1}{2}\Delta u+u\dot{W}\) in ?₊×? ^d , whose solution is interpreted in the mild sense. The noise \(\dot{W}\) is fractional in time (with Hurst index H≥1/2), and colored in space (with spatial covariance kernel f). When H>1/2, the equation generalizes the Itô-sense equation for H=1/2. We prove that if f is the Riesz kernel of order α, or the Bessel kernel of order α<d, then the sufficient condition for the existence of the solution is d≤2+α (if H>1/2), respectively d<2+α (if H=1/2), whereas if f is the heat kernel or the Poisson kernel, then the equation has a solution for any d. We give a representation of the kth order moment of the solution in terms of an exponential moment of the “convoluted weighted” intersection local time of k independent d-dimensional Brownian motions.     

Ultrafilter semigroups generated by direct sums

Let λ be an infinite cardinal and for every ordinal α<λ, let A _α be a set with a distinguished element 0 _α ∈A _α . The direct sum of sets A _α , α<λ, is the subset \(X=\bigoplus_{\alpha<\lambda}A_{\alpha}\) of the Cartesian product ∏_α<λ A _α consisting of all x with finite supp?(x)={α<λ:x(α)≠0 _α }. Endow X with a topology by taking as a neighborhood base at x∈X the subsets of the form {y∈X:y(α)=x(α) for all α<γ} where γ<λ. Let Ult?(X) denote the set of all nonprincipal ultrafilters on X converging to 0∈X. There is a natural partial semigroup operation on X which induces a semigroup operation on Ult?(X). We show that if direct sums X and Y are homeomorphic, then the semigroups Ult?(X) and Ult?(Y) are isomorphic.     

Brownian Motion with Singular Time-Dependent Drift

In this paper, we study weak solutions for the following type of stochastic differential equation

Open image in new window

where \(b: [0,\infty ) \times \mathbb {R}^{d}\rightarrow \mathbb {R}^{d}\) is a measurable drift, \(W=(W_{t})_{t \ge 0}\) is a d-dimensional Brownian motion and \((s,x)\in [0,\infty ) \times \mathbb {R}^{d}\) is the starting point. A solution \(X=(X_t)_{t \ge s}\) for the above SDE is called a Brownian motion with time-dependent drift b starting from (s, x). Under the assumption that |b| belongs to the forward-Kato class \(\mathcal {F}\mathcal {K}_{d-1}^{\alpha }\) for some \(\alpha \in (0,1/2)\), we prove that the above SDE has a unique weak solution for every starting point \((s,x)\in [0,\infty ) \times \mathbb {R}^{d}\).

     

Polycyclic groups with automorphisms of order four

n this paper, we study the structure of polycyclic groups admitting an automorphism of order four on the basis of Neumann’s result, and prove that if α is an automorphism of order four of a polycyclic group G and the map φ: G → G defined by g^φ = [g,α] is surjective, then G contains a characteristic subgroup H of finite index such that the second derived subgroup H″ is included in the centre of H and C_H(α²) is abelian, both C_G(α²) and G/[G, α²] are abelian-by-finite. These results extend recent and classical results in the literature.     

Skew inverse power series rings over a ring with projective socle

A ring R is called a right PS-ring if its socle, Soc(R _R ), is projective. Nicholson and Watters have shown that if R is a right PS-ring, then so are the polynomial ring R[x] and power series ring R[[x]]. In this paper, it is proved that, under suitable conditions, if R has a (flat) projective socle, then so does the skew inverse power series ring R[[x ^?1; α, δ]] and the skew polynomial ring R[x; α, δ], where R is an associative ring equipped with an automorphism α and an α-derivation δ. Our results extend and unify many existing results. Examples to illustrate and delimit the theory are provided.