• Korean Journal of Mathematics
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• v.23 no.3
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• pp.491-502
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• 2015

A martingale is a mathematical model for a fair wager and the modern theory of martingales plays a very important and useful role in the study of the stochastic fields. This paper is devoted to investigate a martingale and a non-martingale on the several stochastic integral or differential equations. Specially, we show that whether the stochastic integral equation involving a standard Wiener process with the associated filtration is or not a martingale.

• Journal of the Korean Mathematical Society
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• v.34 no.4
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• pp.1019-1028
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• 1997

The integral solution for a deterministic evolution equation was introduced by Benilan. Similarly, in this paper, we define the integral solution for a stochastic evolution equation with a state-dependent diffusion term and prove that there exists a unique integral solution of the stochastic evolution euation under some conditions for the coefficients. Moreover we prove that this solution is a unique strong solution.

• Journal of the Chungcheong Mathematical Society
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• v.24 no.1
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• pp.45-57
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• 2011

We study the stochastic integral of an operator valued process against with a Banach space valued regular process. We establish the existence and uniqueness of solution of the stochastic differential equation for a Banach space valued regular process under the certain conditions. As an application of it, we study a noncommutative stochastic differential equation.

• The Pure and Applied Mathematics
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• v.4 no.1
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• pp.47-60
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• 1997

In the theory of the conditional Wiener integral, the integrand is a functional of the standard Wiener process. In this paper we consider a conditional function space integral for functionals of more general stochastic process and the generalized Kac-Feynman integral equation. We first show that the existence of a partial differential equation. We then show that the generalized Kac-Feynman integral equation is equivalent to the partial differential equation.

• Journal of the Korean Mathematical Society
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• v.38 no.3
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• pp.657-667
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• 2001

We propose a statistical interpolation approximate solution for a nonlinear stochastic integral equation of a stock price process. The proposed method has the order O(h$^2$) of local error under the weaker conditions of $\mu$ and $\sigma$ than those of Milstein' scheme.

• Journal of the Korean Mathematical Society
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• v.42 no.1
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• pp.153-169
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• 2005

A fuzzy stochastic differential equation contains a fuzzy valued diffusion term which is defined by stochastic integral of a fuzzy process with respect to 1-dimensional Brownian motion. We prove the existence and uniqueness of the solution for fuzzy stochastic differential equation under suitable Lipschitz condition. To do this we prove and use the maximal inequality for fuzzy stochastic integrals. The results are illustrated by an example.

• The Pure and Applied Mathematics
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• v.14 no.4
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• pp.335-354
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• 2007

In this paper, we define an analogue of generalized Wiener measure and investigate its basic properties. We define (${\hat}It{o}$ type) stochastic integrals with respect to the generalized Wiener process and prove the ${\hat}It{o}$ formula. The existence and uniqueness of the solution of stochastic differential equation associated with the generalized Wiener process is proved. Finally, we generalize the linear filtering theory of Kalman-Bucy to the case of a generalized Wiener process.

• Journal of the Earthquake Engineering Society of Korea
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• v.9 no.6 s.46
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• pp.53-58
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• 2005

Semi-analytical methodology to examine the dynamic responses of a bridge is developed via the joint probability density function. The evolution of joint probability density function is evaluated by the semi-analytical procedure developed. The joint probability function of the bridge responses can be obtained by solving the path-integral solution of the Fokker-Planet equation corresponding to the stochastic differential equations of the system. The response characteristics are observed from the joint probability density function and the boundary of the envelope of the probability density function can provide the maxima ol the bridge responses.

• Korean Journal of Mathematics
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• v.25 no.1
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• pp.45-60
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• 2017

This paper focuses on estimation of an non-integral quadratic function (NIQF) and integral quadratic function (IQF) of a random signal in dynamic system described by a linear stochastic differential equation. The quadratic form of an unobservable signal indicates useful information of a signal for control. The optimal (in mean square sense) and suboptimal estimates of NIQF and IQF represent a function of the Kalman estimate and its error covariance. The proposed estimation algorithms have a closed-form estimation procedure. The obtained estimates are studied in detail, including derivation of the exact formulas and differential equations for mean square errors. The results we demonstrate on practical example of a power of signal, and comparison analysis between optimal and suboptimal estimators is presented.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.12 no.2
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• pp.251-264
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• 1999

본 연구에서는 추계론적 동적시스템의 응답거동을 예측할 수 있는 반해석적 절차를 개발하였으며, 이를 이용하여 구분적선형시스템의 동적거동특성을 확률적 영역에서 분석하였다. 반 해석적 절차는 시스템의 추계론적 미분방정식에 상응하는 Fokker-Planck 방정식을 path-integral solotion을 이용하여 풂으로써 구할 수 있다. 결합확률밀도함수의 시간에 따른 전개과정을 통하여 시스템의 동적 응답거동 특성의 예측과 분석을 하고 시스템의 거동에 미치는 외부노이즈의 영향 또한 조사하였다. 반 해석적 방법은 위상면 상에서 결합확률밀도 함수를 통하여 응답거동의 예측은 물론 거동특성에 대하여 적절한 정보를 제공하는 것을 밝혔다. 혼돈거동의 특성은 외부노이즈가 존재하는 상황에서도 시스템의 응답 안에 잔재하는 것을 밝혔다.