• Journal of the Korean Statistical Society
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• v.33 no.4
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• pp.459-470
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• 2004

We study the approximation of the solution of linear stochastic evolution equations driven by infinite-dimensional fractional Brownian motion with Hurst parameter H > 1/2 through discretization of space and time. The rate of convergence of an approximation for Euler scheme is established.

• East Asian mathematical journal
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• v.14 no.2
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• pp.281-297
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• 1998

Here we establish limit theorems on one-side increments of two-parameter fractional $L\'{e}vy$ Brownian motion, via estimating upper bounds large deviation probability inequalities on the suprema of the fractional $L\'{e}vy$ Brownian motion.

• Journal of the Korean Statistical Society
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• v.34 no.4
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• pp.281-295
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• 2005

We investigate the rate of convergence of the distribution of the maximum likelihood estimator (MLE) of an unknown parameter in the drift coefficient of a stochastic process described by a linear stochastic differential equation driven by a fractional Brownian motion (fBm). As a special case, we obtain the rate of convergence for the case of the fractional Ornstein- Uhlenbeck type process studied recently by Kleptsyna and Le Breton (2002).

• Proceedings of the Korea Institute of Convergence Signal Processing
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• 2000.08a
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• pp.21-24
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• 2000

Fractional Brownian motion(fBm)은 long-term persistence 특성을 가진 자연 현상, 1/f 잡음, 깊이가 낮은 해저에서의 배경음향잡음 등을 모델링하는데 많이 사용된다. 이 fBm은 nonstationary 유색잡음이다. 이러한 유색잡음 환경 하에서 신호를 검출하기 위한 한 방법은 Fredholm 적분방정식의 해를 구하는 것이다. 이 방정식을 이산화 하면 잡음의 공분산 행렬의 역행렬이 포함되어 계산량이 많다 본 논문에서는 fBm 잡음의 공분산 행렬을 웨이브렛 변환하여 얻어지는 행렬, 즉 fBm의 멀티스케일 성분들의 공분산행렬은 밴드화된 블록들로 근사화할 수 있다는 성질을 이용하여 적은 계산량으로 신호를 검출하는 알고리즘을 제안한다.

• The Pure and Applied Mathematics
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• v.28 no.2
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• pp.119-142
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• 2021

In this paper, we first introduce a new L_p analytic Fourier-Feynman transform with respect to subordinate Brownian motion (AFFTSB), which extends the Fourier-Feynman transform in the Wiener space. We next examine several relationships involving the L_p-AFFTSB, the convolution product, and the gradient operator for several types of functionals.

• Bulletin of the Korean Mathematical Society
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• v.55 no.4
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• pp.1241-1261
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• 2018

We derive discrete time model of the geometric fractional Brownian motion. It provides numerical pricing scheme of financial derivatives when the market is driven by geometric fractional Brownian motion. With the convergence analysis, we guarantee the convergence of Monte Carlo simulations. The strong convergence rate of our scheme has order H which is Hurst parameter. To obtain our model we need to convert Wick product term of stochastic differential equation into Wick free discrete equation through Malliavin calculus but ours does not include Malliavin derivative term. Finally, we include several numerical experiments for the option pricing.

• Journal of the Korean Statistical Society
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• v.33 no.2
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• pp.245-254
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• 2004

This paper derives a distribution function of the terminal value and running maximum of two-dimensional Brownian motion {X($\tau$) = (X$_1$($\tau$), X$_2$ ($\tau$))', $\tau$ 〉0}. One random variable of the joint distribution is the terminal time value, X$_1$ (T). The other random variable is the maximum of the Brownian motion {X$_2$($\tau$), $\tau$〉} between time s and time t. With this distribution function, this paper also derives an explicit pricing formula for an outside barrier option whose monitoring period starts at an arbitrary date and ends at another arbitrary date before maturity.

• Journal of Advanced Marine Engineering and Technology
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• v.34 no.7
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• pp.981-990
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• 2010

The main objective of this study is to investigate the fluid flow and the heat transfer characteristics of nanofluids using multi-phase thermal LBM and to realize theenhancement of heat transfer characteristics considered in the Brownian motion. In multi-phase, fluid component($H_2O$) is driven by Boussinesq approximation, and nanoparticles component by the external force gravity and buoyancy. The effect of Brownian motion as a random movement is modified to the internal velocity of nanoparticles(Cu). Simultaneously, the particles of both the phases assume the local equilibrium temperature after each collision. It has been observed that when simulating $H_2O$-Cu nanoparticles, the heat transfer is the highest, at the particle volume fraction 0.5% of the particle diameter 10 nm. The average Nusselt number is increased approximately by 33% at the particle volume fraction 0.5% of the particle diameter 10 nm when compared with pure water.

• Communications of the Korean Mathematical Society
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• v.17 no.1
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• pp.81-93
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• 2002

We consider the Hausdorff measure for Brownian motion(BM) in ι$_2$. Several path properties of BM in ι$_2$ are used to show the upper bound of Hausdorff measure. We also show the lower bound of it applying a law of iterated logarithm for the occupation time of BM in ι$_2$.

• Communications of the Korean Mathematical Society
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• v.14 no.3
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• pp.569-579
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• 1999

We consider a random measure defined by the occupation time of Brownian motion in $l_2$. If it is normalized ${\lambda}^2$log then we show that its cluster set as ${\lambda}{longrightarrow}\infty$ can be represented by Ι-function on $\sigma$-finite measure in $l_2$.