Karhunen-Loève expansions of mean-centered Wiener processes

Karhunen-Loeve expansions of alpha-Wiener bridges

We study Karhunen-Loeve expansions of the process $(X_t^{(α)})_{t\in[0,T)}$ given by the stochastic differential equation $dX_t^{(α)} = -\fracα{T-t} X_t^{(α)} dt+ dB_t,$ $t\in[0,T),$ with an initial condition $X_0^{(α)}=0,$ where $α>0,$ $T\in(0,\infty)$ and $(B_t)_{t\geq 0}$ is a standard Wiener process. This process is called an $α$-Wiener bridge or a scaled Brownian bridge, and in the special case of $α=1$ the usual Wiener bridge. We present weighted and unweighted Karhunen-Loeve expansions of $X^{(α)}$. As applications, we calculate the Laplace transform and the distribution function of the…

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 $0$.

Karhunen-Loeve expansions of Levy processes

Karhunen-Loeve expansions (KLE) of stochastic processes are important tools in mathematics, the sciences, economics, and engineering. However, the KLE is primarily useful for those processes for which we can identify the necessary components, i.e., a set of basis functions, and the distribution of an associated set of stochastic coefficients. Our ability to derive these components explicitly is limited to a handful processes. In this paper we derive all the necessary elements to implement the KLE for a square-integrable Levy process. We show that the eigenfunctions are sine functions, identica…

Convergence Types and Rates in Generic Karhunen-Loève Expansions with Applications to Sample Path Properties

We establish a Karhunen-Lo`eve expansion for generic centered, second order stochastic processes, which does not rely on topological assumptions. We further investigate in which norms the expansion converges and derive exact average rates of convergence for these norms. For Gaussian processes we additionally prove certain sharpness results in terms of the norm. Moreover, we show that the generic Karhunen-Lo`eve expansion can in some situations be used to construct reproducing kernel Hilbert spaces (RKHSs) containing the paths of a version of the process. As applications of the general theory, …

量子卡尔门-洛埃夫展开与二次指数函数在线性量子随机系统中的应用

A quantum Karhunen-Loeve expansion and quadratic-exponential functionals for linear quantum stochastic systems

本文将经典的卡尔门-洛埃夫表示扩展到量子维纳过程，用于建模开放量子系统的外部玻色场。该展开形式在有限时间区间内以正弦函数组织，其系数由非交换的位置和动量算子组成。该方法用于计算二次指数函数，作为风险敏感控制框架中的性能标准。

Wiener integrals with respect to Yeh processes

We define Wiener integrals with respect to Yeh processes and study their properties. In particular, we obtain the martingale property of the associated stochastic processes and give a series expansion of Wiener integrals with respect to centered Yeh process. Moreover, we derive a representation of an Yeh process in terms of a random series.