Free-Knot Spline Approximation of Stochastic Processes

The optimal free knot spline approximation of stochastic differential equations with additive noise

In this paper we analyse the pathwise approximation of stochastic differential equations by polynomial splines with free knots. The pathwise distance between the solution and its approximation is measured globally on the unit interval in the $L_{\infty}$-norm, and we study the expectation of this distance. For equations with additive noise we obtain sharp lower and upper bounds for the minimal error in the class of arbitrary spline approximation methods, which use $k$ free knots. The optimal order is achieved by an approximation method $\hat{X}_{k}^{\dagger}$, which combines an Euler scheme on…

Spline approximation of a random process with singularity

Let a continuous random process $X$ defined on $[0,1]$ be $(m+β)$-smooth, $0\le m, 0<β\le 1$, in quadratic mean for all $t>0$ and have an isolated singularity point at $t=0$. In addition, let $X$ be locally like a $m$-fold integrated $β$-fractional Brownian motion for all non-singular points. We consider approximation of $X$ by piecewise Hermite interpolation splines with $n$ free knots (i.e., a sampling design, a mesh). The approximation performance is measured by mean errors (e.g., integrated or maximal quadratic mean errors). We construct a sequence of sampling designs with asymptotic appro…

B样条在Schwartz半范数下逼近高斯密度与Hermite函数

B-splines approximate the Gaussian density and Hermite functions in Schwartz seminorms

本文证明满足Berry-Esseen定理节点条件的B样条在任意Schwartz半范数下收敛于高斯密度，并由此导出Hermite与Laguerre多项式之间的渐近公式。

Estimation of Hurst Parameter of Fractional Brownian Motion Using CMARS Method

In this study, we develop a new theory of estimating Hurst parame- ter using conic multivariate adaptive regression splines (CMARS) method. We concentrate on the strong solution of stochastic differentional equations (SDEs) driven by fractional Brownian motion (fBm). The superiority of our approach to the others is, it not only estimates the Hurst parameter but also finds spline parameters of the stochastic process in an adaptive way. We examine the performance of our estimations using simulated test data. Keywords: Stochastic differential equations, fractional Brownian motion, Hurst parameter…

Shape Splines and Stochastic Shape Evolutions: A Second Order Point of View

This article presents a new mathematical framework to perform statistical analysis on time-indexed sequences of 2D or 3D shapes. At the core of this statistical analysis is the task of time interpolation of such data. Current models in use can be compared to linear interpolation for one dimensional data. We develop a spline interpolation method which is directly related to cubic splines on a Riemannian manifold. Our strategy consists of introducing a control variable on the Hamiltonian equations of the geodesics. Motivated by statistical modeling of spatiotemporal data, we also design a stocha…

Free knot linear interpolation and the Milstein scheme for stochastic differential equations

The main purpose of this paper is to give a solution to a long-standing unsolved problem concerning the pathwise strong approximation of stochastic differential equations with respect to the global error in the $L_{\infty}$-norm. Typically, one has average supnorm error of order $\left(\ln k/k\right)^{1/2}$ for standard approximations of SDEs with $k$ discretization points, like piecewise interpolated Ito-Taylor schemes. On the other hand there is a lower bound, which indicates that the order $1/\sqrt{k}$ is best possible for spline approximation of SDEs with $k$ free knots. The present paper …