• Journal of the Korean Statistical Society
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• 제31권3호
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• pp.329-342
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• 2002

We deal with asymptotic properties of Maximum Likelihood Estimator(MLE) for the parameters appearing in a Hilbert space-valued Stochastic Differential Equation(SDE) and a Stochastic Partial Differential Equation(SPDE). In paractice, the available data are only the finite dimensional projections to the solution of the equation. Using these data we obtain MLE and consider the asymptotic properties as the dimension of projections increases. In particular we explore a relationship between the conditions for the solution and asymptotic properties of MLE.

• 충청수학회지
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• 제24권1호
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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.

• 대한수학회지
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• 제42권1호
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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.

• Journal of the Korean Statistical Society
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• 제25권2호
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• pp.161-173
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• 1996

When we deal with a Hilbert space-valued Stochastic Differential Equation (SDE) (or Stochastic Partial Differential Equation (SPDE)), depending on some unknown parameters, the solution usually has a Fourier series expansion. In this situation we consider the maximum likelihood method for the statistical estimation problem and derive the asymptotic properties (consistency and normality) of the Maximum Likelihood Estimator (MLE).

• 충청수학회지
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• 제29권2호
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• pp.337-346
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• 2016

Within white noise approach, we study the existence and uniqueness of the solution of an initial value problem for generalized white noise functionals, and then as a corollary we discuss the linear stochastic differential equation associated with a convolution of white noise functionals.

• 대한수학회보
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• 제40권4호
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• pp.677-683
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• 2003

In multi-allelic model $X\;=\;(x_1,\;x_2,\;\cdots\;,\;x_d),\;M_f(t)\;=\;f(p(t))\;-\;{\int_0}^t\;Lf(p(t))ds$ is a P-martingale for diffusion operator L under the certain conditions. In this note, we examine the stochastic differential equation for model X and find the properties using stochastic differential equation.

• 대한수학회보
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• 제37권3호
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• pp.619-631
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• 2000

We study a class of catalytic super Brownian motion X in 1-dimension. We show under some conditions of catalyst, the process X is absolutely continuous and we get a stochastic partial differential equation for X.

• Korean Journal of Mathematics
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• 제23권3호
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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.

• Korean Journal of Mathematics
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• 제6권2호
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• pp.259-279
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• 1998

For the forward-backward semimartingale, we can define the backward semimartingale flow which is generated by the backward canonical stochastic differential equation. Therefore, we define the backward self-similar stochastic processes, and we study the backward self-similar stochastic flows through the canonical stochastic differential equations.

• Journal of applied mathematics & informatics
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• 제34권5_6호
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• pp.421-434
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• 2016

The main aim of this paper is to discuss the difference between the Euler-Maruyama's approximate solutions and the accurate solution to stochastic differential delay equation. To make the theory more understandable, we impose the non-uniform Lipschitz condition and weakened linear growth condition. Furthermore, we give the pth moment continuous of the approximate solution for the delay equation.