• Korean Journal of Mathematics
• /
• 제23권3호
• /
• pp.491-502
• /
• 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.

• 대한수학회보
• /
• 제47권1호
• /
• pp.39-51
• /
• 2010

In this paper we consider the uniform central limit theorem for a martingale-difference array of a function-indexed stochastic process under the uniformly integrable entropy condition. We prove a maximal inequality for martingale-difference arrays of process indexed by a class of measurable functions by a method as Ziegler [19] did for triangular arrays of row wise independent process. The main tools are the Freedman inequality for the martingale-difference and a sub-Gaussian inequality based on the restricted chaining. The results of present paper generalizes those of Ziegler [19] and other results of independent problems. The results also generalizes those of Bae and Choi [3] to martingale-difference array of a function-indexed stochastic process. Finally, an application to classes of functions changing with n is given.

• Journal of the Korean Statistical Society
• /
• 제24권1호
• /
• pp.219-224
• /
• 1995

A central limit theorem with random indices is obtained for stattionary martingale difference sequences.

• Journal of applied mathematics & informatics
• /
• 제31권1_2호
• /
• pp.241-246
• /
• 2013

Choi [1] identified and characterized the limiting diffusion of this diploid model by defining discrete generator for the rescaled Markov chain. In this note, we define the operator of projection $S_t$ on limiting diffusion and new measure $dQ=S_tdP$. We show the martingale property on this operator and measure. Also we conclude that the martingale problem for diffusion operator of projection is well-posed.

• 대한수학회보
• /
• 제41권2호
• /
• pp.337-345
• /
• 2004

In allelic model X = ($\chi_1\chi$_2ㆍㆍㆍ, \chi_d$), M_f(t) = f(p(t)) - ${{\int^t}_0}\;Lf(p(t))ds$ is a P-martingale for diffusion operator L under the certain conditions. In this note, we can show existence and uniqueness of solution for stochastic differential equation and martingale problem associated with mean vector. Also, we examine that if the operator related to this martingale problem is connected with Markov processes under certain circumstance, then this operator must satisfy the maximum principle.

• Journal of applied mathematics & informatics
• /
• 제27권3_4호
• /
• pp.757-762
• /
• 2009

In allelic model $X\;=\;(x_1,\;x_2,\;{\cdots},\;x_d)$, $$M_f(t)\;=\;f(p(t))\;-\;{\int}^t_0\;Lf(p(t))ds$$ is a P-martingale for diffusion operator L under the certain conditions. We can also obtain a new diffusion operator $L^*$ for diffusion coefficient and we prove that unique solution for $L^*$-martingale problem exists. In this note, we define new symmetric preserving transformation. Uniqueness for martingale problem and symmetric property will be proved.

• 대한수학회논문집
• /
• 제18권4호
• /
• pp.713-723
• /
• 2003

We study a limit problem of reflected solutions of parabolic stochastic partial differential equations driven by martingale measures. The existence of solutions is found in an extension of the work with respect to white noise by Donati-Martin and Pardoux [4]. We show that if a certain sequence of driving martingale measures converges, the corresponding solutions also converge in the Skorohod topology.

• 호남수학학술지
• /
• 제36권1호
• /
• pp.33-41
• /
• 2014

Ko(2013, JIA 2013:473) discussed complete convergence for weighted sum of martingale difference field when all indices have the same powers in the normalization. In this paper we generalize this law to the case where different indices have different powers in the normalization.

• Journal of applied mathematics & informatics
• /
• 제26권1_2호
• /
• pp.417-425
• /
• 2008

For stationary m-dimensional martingale difference sequences we prove the random functional central limit theorems and propose an almost sure consistent estimator for the limiting covariance matrix.

• 대한수학회지
• /
• 제33권4호
• /
• pp.909-928
• /
• 1996

In this paper, an empirical law of the iterated logarithm is investigated in the context of a stationary and ergodic martingale differences whose values taking in a measurable space.