Yoon Tae Kim

S-shaped software reliability growth models derived from stochastic differential equations

This paper presents a software reliability growth model based on Itô type Stochastic Differential Equations (SDE). As the size of a software system becomes larger, the number of faults remaining in the system during the testing phase can be considered to be a continuous stochastic process. In practice, if the per-fault detection rate is subject to certain random effects, we may consider the use of a SDE to describe the average behavior of the software fault detection process during the testing phase. As a result, we derive several software reliability measures by utilizing the mean value function which is the expected value of the SDE. We also derive the maximum likelihood estimators of the unknown parameters for the model. Futhermore, we compare our model with other software reliability growth models in terms of several reliability measures and goodness-of-fit for the same data set.

An Ito formula for a fractional Brownian sheet with arbitrary Hurst parameters

By using the white noise theory for a fractional Brownian sheet, we derive an Ito formula for the fractional Brownian sheet with arbitrary Hurst parameters H 1 , H 2 ∈ (0,1).

An Itô Formula of Generalized Functionals and Local Time for Fractional Brownian Sheet

Abstract By using the white noise theory for a fractional Brownian sheet, we derive an Itô formula for the generalized functionals for the fractional Brownian sheet with arbitrary Hurst parameters H 1, H 2 ∈ (0,1). As an application, we give the integral representations for two versions of local times of a fractional Brownian sheet, respectively.

A Wong–Zakai Type Approximation for Multiple Wiener–Stratonovich Integrals

Abstract We present an extension of the Wong-Zakai type approximation theorem for a multiple stochastic integral. Using a piecewise linear approximation w (n) of a Wiener process w, we prove that the multiple integral process where f is a given symmetric function in the space 𝒞([0, T] m ), converge to the multiple Stratonovich integral of f in the uniform L 2-sense.

Stratonovich Calculus with Respect to Fractional Brownian Sheet

Abstract We introduce two types of Stratonovich stochastic integrals for two-parameter process. The relationship of Stratonovich integrals to Skorohod integrals will be investigated. By using this relationship, we prove that a differentiation formula for fractional Brownian sheet in Stratonovich form can be expressed as the sum of Stratonovich integrals of two types introduced in this article.

A Wong–Zakai type approximation for two-parameter processes

We present an extension of the Wong–Zakai approximation theorem for a stochastic differential equation on the plane driven by a two-parameter Wiener process. For an approximation of the two-parameter Wiener process, we use a two-parameter version of the one-parameter piecewise linear approximation. By our approximation to the two-parameter Wiener process we show that the solution of an ordinary differential equation converges, in the uniform L 2-sense, to that of a stochastic differential equation obtained by using Stratonovich integral. *This research was supported (in part) by KOSEF through Statistical Research Center for Complex Systems at Seoul National University.

Optimal Berry-Esseen bound for statistical estimations and its application to SPDE

We consider asymptotically normal statistics of the form F n / G n , where F n and G n are functionals of Gaussian fields. For these statistics, we establish an optimal Berry-Esseen bound for the Central Limit Theorem (CLT) of the sequence F n / G n is ź ( n ) in the following sense: there exist constants 0 < c < C < ∞ such that c ź d Kol ( F n / G n , Z ) / ź ( n ) ź C , where d Kol ( F n , Z ) = sup z ź R ź Pr ( F n ź z ) - Pr ( Z ź z ) ź . As an example, we find an optimal Berry-Esseen bound for the CLT of the maximum likelihood estimators for parameters occurring in parabolic stochastic partial differential equations.

Central Limit Theorem of the Cross Variation Related to Fractional Brownian Sheet

By using Malliavin calculus, we study a central limit theorem of the cross variation related to fractional Brownian sheet with Hurst parameter H = (H1,H2) such that 1/4 < H1 < 1/2 and 1/4 < H2 < 1/2.

Mean distance of Brownian motion on a Riemannian manifold

Consider the mean distance of Brownian motion on Riemannian manifolds. We obtain the first three terms of the asymptotic expansion of the mean distance by means of stochastic differential equation for Brownian motion on Riemannian manifold. This method proves to be much simpler for further expansion than the methods developed by Liao and Zheng (Ann. Probab. 23(1) (1995) 173). Our expansion gives the same characterizations as the mean exit time from a small geodesic ball with regard to Euclidean space and the rank 1 symmetric spaces.

A geometric approach to singularity for Hilbert space-valued SDEs

We find a geometric condition for singularity of measures which depend on the parameters appearing in Hilbert space-valued stochastic differential equations (SDEs) (or stochastic partial differential equations (SPDEs)).