Fractional Brownian motion and packing dimension

LetX(t) (tR ^N ) be a fractional Brownian motion of index inR ^d . For any compact setER ^N , we compute the packing dimension ofX(E).Partially supported by an NSF grant.     

The heat semigroup and Brownian motion on strip complexes

We introduce the notion of strip complex. A strip complex is a special type of complex obtained by gluing “strips” along their natural boundaries according to a given graph structure. The most familiar example is the one-dimensional complex classically associated with a graph, in which case the strips are simply copies of the unit interval (our setup actually allows for variable edge length). A leading key example is treebolic space, a geometric object studied in a number of recent articles, which arises as a horocyclic product of a metric tree with the hyperbolic plane. In this case, the graph is a regular tree, the strips are [0,1]×R, and each strip is equipped with the hyperbolic geometry of a specific strip in upper half plane. We consider natural families of Dirichlet forms on a general strip complex and show that the associated heat kernels and harmonic functions have very strong smoothness properties. We study questions such as essential self-adjointness of the underlying differential operator acting on a suitable space of smooth functions satisfying a Kirchhoff type condition at points where the strip complex bifurcates. Compatibility with projections that arise from proper group actions is also considered.     

Strong solutions of stochastic equations with singular time dependent drift

We prove existence and uniqueness of strong solutions to stochastic equations in domains with unit diffusion and singular time dependent drift b up to an explosion time. We only assume local L_q_L_p-integrability of b in ×G with d/p+2/q<1. We also prove strong Feller properties in this case. If b is the gradient in x of a nonnegative function blowing up as GxG, we prove that the conditions 2D_tK,2D_t+Ke, [0,2), imply that the explosion time is infinite and the distributions of the solution have sub Gaussian tails.The work of the first author was partially supported by NSF Grant DMS-0140405Mathematics Subject Classification (2000): 60J60, 31C25Acknowledgement Financial support by the Humboldt Foundation, the BiBoS-Research Centre and the DFG-Forschergruppe Spectral Analysis, Asymptotic Distributions and Stochastic Dynamics is gratefully acknowledged. The authors are also sincerely grateful to the referees for their helpful suggestions.     

Horizontal lift of the Brownian motion on the hyperbolic plane and the Selberg trace formula

By using an explicit representation for the horizontal lift of the Brownian motion on the Poincaré upper half-plane H², we show an expression for the heat kernel for the de Rham-Kodaira Laplacian on H². We apply the result to a study on the Selberg trace formula.     

Distribution of maximum loss of fractional Brownian motion with drift

In this paper, we find bounds on the distribution of the maximum loss of fractional Brownian motion with $H\ge 1/2$ and derive estimates on its tail probability. Asymptotically, the tail of the distribution of maximum loss over $[0,t]$ behaves like the tail of the marginal distribution at time $t$.     

Range of Brownian Motion with Drift

Let (B _δ (t)) _{t ≥ 0} be a Brownian motion starting at 0 with drift δ > 0. Define by induction S ₁=− inf _{t ≥ 0} B _δ (t), ρ₁ the last time such that B _δ (ρ₁)=−S ₁, S ₂=sup_{0≤ t ≤ρ 1} B _δ (t), ρ₂ the last time such that B _δ (ρ₂)=S ₂ and so on. Setting A _k =S _k +S _k+1; k ≥ 1, we compute the law of (A ₁,...,A _k ) and the distribution of (B _δ (t+ρ l) − B _δ (ρ _l ); 0 ≤ t ≤ ρ _l-1 − ρ _l )_{2 ≤ l ≤ k} for any k ≥ 2, conditionally on (A ₁,...,A _k ). We determine the law of the range R _δ (t) of (B _δ (s)) _{s≥ 0} at time t, and the first range time θ_δ (a) (i.e. θ_δ (a)=inf{t > 0; R _δ (t) > a}). We also investigate the asymptotic behaviour of θ _δ (a) (resp. R _δ (t)) as a → ∞ (resp. t → ∞).     

On the Wiener integral with respect to a sub-fractional Brownian motion on an interval

The domain of the Wiener integral with respect to a sub-fractional Brownian motion , , k≠0, is characterized. The set is a Hilbert space which contains the class of elementary functions as a dense subset. If , any element of is a function and if , the domain is a space of distributions.     

Smooth densities for SDEs driven by subordinated Brownian motion with Markovian switching

We consider a class of stochastic differential equations driven by subordinated Brownian motion with Markovian switching. We use Malliavin calculus to study the smoothness of the density for the solution under uniform Hörmander type condition.     

Continuity in law with respect to the Hurst index of some additive functionals of sub-fractional Brownian motion

In this article, first, we prove some properties of the sub-fractional Brownian motion introduced by Bojdecki et al. [Statist. Probab. Lett. 69(2004):405–419]. Second, we prove the continuity in law, with respect to small perturbations of the Hurst index, in some anisotropic Besov spaces, of some continuous additive functionals of the sub-fractional Brownian motion. We prove that our result can be obtained easily, by using the decomposition in law of the sub-fractional Brownian motion given by Bardina and Bascompte [Collect. Math. 61(2010):191–204] and Ruiz de Chavez and Tudor [Math. Rep. 11(2009):67–74], without using the result of Wu and Xiao [Stoch. Proc. Appl. 119(2009):1823–1844] by connecting the sub-fractional Brownian motion to its stationary Gaussian process through Lamperti’s transform. This decomposition in law leads to a better understanding and simple proof of our result.     

Numerical analysis for stochastic age-dependent population equations with fractional Brownian motion

Stochastic age-dependent population equations, one of the important classes of hybrid systems are studied. In general most equations of stochastic age-dependent population do not have explicit solutions. Thus numerical approximation schemes are invaluable tools for exploring their properties. The main purpose of this paper is to develop a numerical scheme and show the convergence of the numerical approximation solution to the analytic solution. In the last section a numerical example is given.     

On delayed averages of Brownian motion in Banach spaces

Let {W(t): t ≥ 0} be μ-Brownian motion in a real separable Banach space B, and let a_T be a nondecreasing function of T for which (i) 0 < a_T ≤ T (T ≥ 0), (ii) $\text{a}{}_{\mathrm{T}}\text{T}$ is nonincreasing. We establish a Strassen limit theorem for the net {ξ_T: T ≥ 3}, where

$\text{ξ}{}_{\mathrm{T}}\text{=}\text{W(T + ta}{}_{\mathrm{T}}\text{) ? W(T)}\text{{2a}{}_{\mathrm{T}}\text{[}\text{log}\text{(}\text{T}\text{a}{}_{\mathrm{T}}\text{) +}\text{log log}\text{T]}}\text{1}\text{2}\text{,}\text{0 ? t ? 1}$

     

Estimates on Green functions and Schrödinger-type equations for non-symmetric diffusions with measure-valued drifts

In this paper, we establish sharp two-sided estimates for the Green functions of non-symmetric diffusions with measure-valued drifts in bounded Lipschitz domains. As consequences of these estimates, we get a 3G type theorem and a conditional gauge theorem for these diffusions in bounded Lipschitz domains.Informally the Schrödinger-type operators we consider are of the form L+μ⋅∇+ν where L is a uniformly elliptic second order differential operator, μ is a vector-valued signed measure belonging to Kd,1 and ν is a signed measure belonging to Kd,2. In this paper, we establish two-sided estimates for the heat kernels of Schrödinger-type operators in bounded C1,1-domains and a scale invariant boundary Harnack principle for the positive harmonic functions with respect to Schrödinger-type operators in bounded Lipschitz domains.     

On the Wiener integral with respect to the fractional Brownian motion on an interval

We characterize the domain of the Wiener integral with respect to the fractional Brownian motion of any Hurst parameter H(0,1) on an interval [0,T]. The domain is the set of restrictions to of the distributions of with support contained in [0,T]. In the case H1/2 any element of the domain is given by a function, but in the case H>1/2 this space contains distributions that are not given by functions. The techniques used in the proofs involve distribution theory and Fourier analysis, and allow to study simultaneously both cases H<1/2 and H>1/2.     

Approximate controllability of fractional neutral stochastic evolution equations in Hilbert spaces with fractional Brownian motion

In this paper, the approximate controllability for Sobolev-type fractional neutral stochastic evolution equations with fractional stochastic nonlocal conditions and fractional Brownian motion in a Hilbert space are studied. The results are obtained by using semigroup theory, fractional calculus, stochastic integrals for fractional Brownian motion, Banach's fixed point theorem, and methods adopted directly from deterministic control problems for the main results. Finally, an example is given to illustrate the application of our result.     

Dirichlet外问题与漂移布朗运动

利用Dirichlet外问题与漂移布朗运动之间存在的密切联系,对Dirichlet外问题提出了一种新的有效的概率数值方法,这种方法运用了解的随机表达式、布朗运动、漂移布朗运动以及球面首中位置和时间的分布等．     

Spectral multipliers for Laplacians with drift on Damek–Ricci spaces

We prove a multiplier theorem for certain Laplacians with drift on Damek–Ricci spaces, which are a class of Lie groups of exponential growth. Our theorem generalizes previous results obtained by W. Hebisch, G. Mauceri and S. Meda on Lie groups of polynomial growth.     

Pension funding problem with regime‐switching geometric Brownian motion assets and liabilities

This paper extends the pension funding model in (N. Am. Actuarial J. 2003; 7 :37–51) to a regime‐switching case. The market mode is modeled by a continuous‐time stationary Markov chain. The asset value process and liability value process are modeled by Markov‐modulated geometric Brownian motions. We consider a pension funding plan in which the asset value is to be within a band that is proportional to the liability value. The pension plan sponsor is asked to provide sufficient funds to guarantee the asset value stays above the lower barrier of the band. The amount by which the asset value exceeds the upper barrier will be paid back to the sponsor. By applying differential equation approach, this paper calculates the expected present value of the payments to be made by the sponsor as well as that of the refunds to the sponsor. In addition, we study the effects of different barriers and regime switching on the results using some numerical examples. The optimal dividend problem is studied in our examples as an application of our theory. Copyright © 2009 John Wiley & Sons, Ltd.