• Journal of applied mathematics & informatics
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• v.19 no.1_2
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• pp.19-32
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• 2005

It is shown that the problem of minimizing (maximizing) a quadratic cost functional (quadratic gain functional) given the dynamics dx = (fx + gu)dt + hdb(t, a) where b(t, a) is a fractional Brownian motion of order a, 0 < 2a < 1, can be solved completely (and meaningfully!) by using the dynamical equations of the moments of x(t). The key is to use fractional Taylor's series to obtain a relation between differential and differential of fractional order.

• Korean Journal of Mathematics
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• v.22 no.1
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• pp.71-84
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• 2014

By using the multidimensional normal approximation of functionals of Gaussian fields, we prove that functionals of Gaussian fields, as functions of t, converge weakly to a standard Brownian motion. As an application, we consider the convergence of the Stratonovich-type Riemann sums, as a function of t, of fractional Brownian motion with Hurst parameter H = 1/4.

• Communications for Statistical Applications and Methods
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• v.16 no.4
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• pp.639-645
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• 2009

Fractional Brwonian motion(fBm) has properties of behaving tails and exhibiting long memory while remaining Gaussian. In particular, it is well known that interest rates show some long memories and non-Markovian. We present no aribitrage condition for HJM model under the multi-factor fBm reflecting the long range dependence in the interest rate model.

• Communications for Statistical Applications and Methods
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• v.15 no.1
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• pp.137-145
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• 2008

We present formulas for two types of cap pricing under fBm-HJM model reflecting the empirical long range dependence in the interest rate model. In particular, we propose a new approach to pricing the cap with the default risk.

• Korean Journal of Mathematics
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• v.17 no.4
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• pp.399-409
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• 2009

We study the weak convergence of various models to Fractional Brownian motion. First, we consider arima process and ON/OFF source model which allows for long packet trains and long inter-train distances. Finally, we figure out power spectrum density as a Fourier transform of autocorrelation function of arima model and binary signal model.

• Korean Journal of Mathematics
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• v.11 no.1
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• pp.35-43
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• 2003

We study the weak convergence to Fractional Brownian motion and some examples with applications to traffic modeling. Finally, we get loss probability for queue-length distribution related to self-similar process.

• Korean Journal of Mathematics
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• v.15 no.1
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• pp.71-78
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• 2007

We consider arrival process and ON/OFF source model which allows for long packet trains and long inter-train distances. We prove the weak convergence of theses processes to Fractional Brownian motion. Finally, we figure out the coefficients of $B_H(t)$ and time $t$ when ON/OFF periods have the Pareto distribution.

• Bulletin of the Korean Mathematical Society
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• v.57 no.3
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• pp.547-568
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• 2020

In this paper, we introduce the linear self-attracting diffusion driven by a weighted fractional Brownian motion with weighting exponent a > -1 and Hurst index |b| < a + 1, 0 < b < 1, which is analogous to the linear fractional self-attracting diffusion. For the 1-dimensional process we study its convergence and the corresponding weighted local time. As a related problem, we also obtain the renormalized intersection local time exists in L² if max{a₁ + b₁, a₂ + b₂} < 0.

• Journal of applied mathematics & informatics
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• v.33 no.1_2
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• pp.203-210
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• 2015

We consider fractional Brownian motion and FARIMA process with Gaussian innovations and show that the suitably scaled distributions of the FARIMA processes converge to fractional Brownian motion in the sense of finite dimensional distributions. We figure out ACF function and estimate the self-similarity parameter H of FARIMA(0, d, 0) by using R/S method. Finally, we display power spectrum density of FARIMA process.

• The Korean Journal of Financial Management
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• v.25 no.1
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• pp.85-108
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• 2008

In this paper, the fBm interest rate theory is investigated by using Wick integral. The well-known Affine, Quadratic and HJM are derived from fBm framework, respectively. We obtain new theoretical results, and zero coupon bond pricing formula from newly obtained probability measure.