General alpha-Wiener bridges

使用Wiener混沌展开模拟扩散桥

Simulating diffusion bridges using the Wiener chaos expansion

本文提出基于Wiener混沌展开的扩散桥模拟方法，速度快且无拒绝率，通过Ornstein-Uhlenbeck过程验证，并应用于三个SDE示例，适用于统计推断。

Brownian hitting distributions in space-time of bounded sets and the expected volume of Wiener sausage for Brownian bridges

The space-time distribution, $Q_A(x,dt dξ)$ say, of Brownian hitting of a bounded Borel set $A$ of the $d$-dimensional Euclidian space is studied. We derive the asymptotic form of the leading term of the time-derivative $Q_A(x, dtdξ)/dt$ for each $d =2, 3, ...$, valid uniformly with respect to the starting point $x$ of the Brownian motion, which result extends significantly the classical results for $Q_A(x, dt dξ)$ itself by Hunt ($d=2$), Joffe and Spitzer ($d= 3, 4,...$). The results are applied to find the asymptotic form of the expected volume of Wiener sausage for the Brownian bridge joini…

Weierstrass 桥

Weierstrass Bridges

本文引入分数维 Wiener-Weierstrass 桥，通过卷积构造并分析其变差，揭示两种样本路径性质，证明其非半鞅且协方差为分形，并推广至一般高斯鞅桥。

Examples of random fields that can be represented as space-domain scaled stationary Ornstein-Uhlenbeck fields

We give some examples of random fields that can be represented as space-domain scaled stationary Ornstein-Uhlenbeck fields defined on the plane. Namely, we study a tied-down Wiener bridge, tied-down scaled Wiener bridges, a Kiefer process and so called (F,G)-Wiener bridges.

分数维纳-魏尔斯特拉斯桥的样本路径性质 II

Variation and Hausdorff dimension of fractional Wiener-Weierstrass bridges

本文研究了分数维纳-魏尔斯特拉斯桥的样本路径性质，证明了在魏尔斯特拉斯机制主导时b-进变差系数具有绝对连续分布，建立了临界点的幂变差公式和Φ-变差极限，并推导了高赫斯特指数情形下样本路径图集的豪斯多夫维数。

Large deviations for Hilbert space valued Wiener processes: a sequence space approach

Ciesielski's isomorphism between the space of alpha-Hölder continuous functions and the space of bounded sequences is used to give an alternative proof of the large deviation principle for Wiener processes with values in Hilbert space.

On self-intersection local times for generalized Brownian bridges and the distance between step functions

In the paper $k$-multiple self-intersection local time for planar Gaussian integrators generated by linear operator with nontrivial kernel is studied. In this case additional singularities arise in its formal Fourier--Wiener transform. In case $k=2$ the set of singularities is the finite number of points. In case $k>2$ it contains intervals and hyperplanes. In the first and the second cases using two different approaches related on structure of set of singularities we show that "new" singularities do not imply on the convergence of integral corresponding to the formal Fourier--Wiener transform…

Karhunen-Loève expansion for a generalization of Wiener bridge

We derive a Karhunen-Loève expansion of the Gauss process $B_t - g(t)\int_0^1 g'(u)\,d B_u$, $t\in[0,1]$, where $(B_t)_{t\in[0,1]}$ is a standard Wiener process and $g:[0,1]\to R$ is a twice continuously differentiable function with $g(0) = 0$ and $\int_0^1 (g'(u))^2\,d u =1$. This process is an important limit process in the theory of goodness-of-fit tests. We formulate two special cases with the function $g(t)=\frac{\sqrt{2}}π\sin(πt)$, $t\in[0,1]$, and $g(t)=t$, $t\in[0,1]$, respectively. The latter one corresponds to the Wiener bridge over $[0,1]$ from $0$ to $0$.

On Schrödinger's equation, 3-dimensional Bessel bridges, and passage time problems

In this work we relate the density of the first-passage time of a Wiener process to a moving boundary with the three dimensional Bessel bridge process and a solution of the heat equation with a moving boundary. We provide bounds.

Representations of multidimensional linear process bridges

We derive bridges from general multidimensional linear non time-homogeneous processes using only the transition densities of the original process giving their integral representations (in terms of a standard Wiener process) and so-called anticipative representations. We derive a stochastic differential equation satisfied by the integral representation and we prove a usual conditioning property for general multidimensional linear process bridges. We specialize our results for the one-dimensional case; especially, we study one-dimensional Ornstein-Uhlenbeck bridges.

Parameter estimation for alpha-fractional bridges

Let alpha,T>0. We study the asymptotic properties of a least squares estimator for the parameter alpha of a fractional bridge defined as dX_t=-alpha*X_t/(T-t)dt+dB_t, with t in [0,T) and where B is a fractional Brownian motion of Hurst index H>1/2. Depending on the value of alpha, we prove that we may have strong consistency or not as t tends to T. When we have consistency, we obtain the rate of this convergence as well. Also, we compare our results to the (known) case where B is replaced by a standard Brownian motion W.

Explicit formulas for Laplace transforms of certain functionals of some time inhomogeneous diffusions

We consider a process $(X_t)_{t\in[0,T)}$ given by the SDE $dX_t = αb(t)X_t dt + σ(t) dB_t$, $t\in[0,T)$, with initial condition $X_0=0$, where $T\in(0,\infty]$, $α\in R$, $(B_t)_{t\in[0,T)}$ is a standard Wiener process, $b:[0,T)\to R\setminus\{0\}$ and $σ:[0,T)\to(0,\infty)$ are continuously differentiable functions. Assuming that $b$ and $σ$ satisfy a certain differential equation we derive an explicit formula for the joint Laplace transform of $\int_0^t\frac{b(s)^2}{σ(s)^2}(X_s)^2 ds$ and $(X_t)^2$ for all $t\in[0,T)$. As an application, we study asymptotic behavior of the maximum likeliho…