• Communications of the Korean Mathematical Society
• /
• v.29 no.1
• /
• pp.173-185
• /
• 2014

In the present paper, we consider an anomalous diffusion problem in two dimensional space involving Caputo time and Riesz-Feller fractional derivatives and then solve it by using a series involving bilateral eigen-functions. Also, we obtain a numerical approximation formula of this problem and discuss some of its particular cases.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.27 no.3
• /
• pp.180-193
• /
• 2023

Approximating the implied volatilities and estimating the model parameters are important topics in quantitative finance. This study proposes an approximation formula for short-maturity near-the-money implied volatilities in stochastic volatility models. A general second-order nonlinear PDE for implied volatility is derived in terms of time-to-maturity and log-moneyness from the Feyman-Kac formula. Using regularity conditions and the Taylor expansion, an approximation formula for implied volatility is obtained for short-maturity nearthe-money call options in two stochastic volatility models: Heston model and SABR model. In addition, we proposed a novel numerical method to estimate model parameters. This method reduces the number of model parameters that should be estimated. Generating sample data on log-moneyness, time-to-maturity, and implied volatility, we estimate the model parameters fitting the sample data in the above two models. Our method provides parameter estimates that are close to true values.

• Industrial Engineering and Management Systems
• /
• v.7 no.2
• /
• pp.143-149
• /
• 2008

There have been some approximation analysis methods for a GI/G/1 queueing system. As one of them, an approximation technique for the steady-state probability in the GI/G/1 queueing system based on the iteration numerical calculation has been proposed. As another one, an approximation formula of the average queue length in the GI/G/1 queueing system by using the diffusion approximation or the heuristics extended diffusion approximation has been developed. In this article, an approximation technique in order to analyze the GI/G/1 queueing system is considered and then the formulae of both the steady-state probability and the average queue length in the GI/G/1 queueing system are proposed. Through some numerical examples by the proposed technique, the existing approximation methods, and the Monte Carlo simulation, the effectiveness of the proposed approximation technique is verified.

• Journal of the Korean Mathematical Society
• /
• v.46 no.6
• /
• pp.1267-1276
• /
• 2009

This paper presents a method for approximation of the standard normal distribution by using hyperbolic tangent based functions. The presented approximate formula for the cumulative distribution depends on one numerical coefficient only, and its accuracy is admissible. Furthermore, in some particular cases, closed forms of inverse formulas are derived. Numerical results of the present method are compared with those of an existing method.

• Journal of the Chungcheong Mathematical Society
• /
• v.29 no.1
• /
• pp.95-102
• /
• 2016

Multi-server queueing systems with retrials are widely used to model problems in a call center. We present an explicit formula for an approximation of the queue length distribution in a multi-server retrial queue, by using the Lerch transcendent. Accuracy of our approximation is shown in the numerical examples.

• Honam Mathematical Journal
• /
• v.41 no.2
• /
• pp.403-420
• /
• 2019

The Helmholtz equation represents acoustic or electromagnetic scattering phenomena. The Method of Lines are known to have many advantages in simulation of forward and inverse scattering problems due to the usage of angle rays and Bessel functions. However, the method does not account for the jump phenomena on obstacle boundary and the approximation includes many high order Bessel functions. The high order Bessel functions have extreme blow-up or die-out features in resonance region obstacle boundary. Therefore, in particular, when we consider shape reconstruction problems, the method is suffered from severe instabilities due to the logical confliction and the severe singularities of high order Bessel functions. In this paper, two approximation formulas for the Helmholtz equation are introduced. The formulas are new and powerful. The derivation is based on Method of Lines, Huygen's principle, boundary jump relations, Addition Formula, and the orthogonality of the trigonometric functions. The formulas reduce the approximation dimension significantly so that only lower order Bessel functions are required. They overcome the severe instability near the obstacle boundary and reduce the computational time significantly. The convergence is exponential. The formulas adopt the scattering jump phenomena on the boundary, and separate the boundary information from the measured scattered fields. Thus, the sensitivities of the scattered fields caused by the boundary changes can be analyzed easily. Several numerical experiments are performed. The results show the superiority of the proposed formulas in accuracy, efficiency, and stability.

• The Korean Journal of Applied Statistics
• /
• v.24 no.2
• /
• pp.347-357
• /
• 2011

In this paper, we propose an approximation to the overshoot in M/$E_n$/1 queues. Overshoot means the size of excess over the threshold when the workload process of an M/$E_n$/1 queue exceeds a prespecified threshold. The distribution, $1^{st}$ and $2^{nd}$ moments of overshoot have an important role in solving some kind of optimization problems. For the approximation to the overshoot, we propose a formula that is a convex sum of the service time distribution and an exponential distribution. We also do a numerical study to check how exactly the proposed formula approximates the overshoot.

• The Journal of Korean Institute of Communications and Information Sciences
• /
• v.25 no.10A
• /
• pp.1566-1575
• /
• 2000

This paper studies mismatched scalar quantization of a generalized gamma source by a quantizer that is optimally (in the mean square error sense) designed for another generalized gamma source. Specifically, it considers variance-mismatched quantization which occurs when the variance of the source to be quantized differs from tat of the designed-for source. The main result is the two distortion formulas derived from Bennett's integral. The first formula is an approximation expression that uses the outermost threshold of an optimum scalar quantizer, and the second formula, in turn, uses an approximation formula for this outermost threshold. Numerical results are obtained for Laplacian sources, which are example of a generalized gamma source, and comparisons are made between actual mismatched distortions and the two formulas. These numerical results show that the two formulas become more accurate, as the number of quantization points gets larger and the ratio of the source variance to that of the designed-for source gets bigger. For example, the formulas are within 2~4% of the actual distortion for approximately 64 quantization points or more. In conclusion, the proposed approximation formulas are considered to have contribution as closed formulas and for their accuracy.

• Journal of the Korean Operations Research and Management Science Society
• /
• v.19 no.3
• /
• pp.93-109
• /
• 1994

Even though sojourn time distributions are essential information in analyzing queueing networks, there are few methods to compute them accurately in non-product form queueing networks. In this study, we model the location process of a typical customer as a GMPH semi-Markov chain and develop computationally useful formula for the transition function and the first-passage time distribution in the GMPH semi-Markov chain. We use the formula to develop an effcient method for approximating sojourn time distributions in the non-product form queueing networks under quite general situation. We demonstrate its validity through numerical examples.

• The Pure and Applied Mathematics
• /
• v.12 no.4 s.30
• /
• pp.275-287
• /
• 2005

Geometric invariants are basic tools for geometric processing and computer vision. In this paper, we give a linear approximation for the differentiation of the binormal vector field of a space curve by using the forward and backward differences of discrete binormal vectors. Two kind of discrete torsion, say, back-ward torsion $T_b$ and forward torsion $T_f$ can be defined by the dot product of the (backward and forward) discrete differentiation of binormal vectors that are linear approximations of torsion. Using Frenet formula and Taylor series expansion, we give error estimations for the discrete torsions. We also give numerical tests for a curve. Notably the average of $T_b$ and $T_f$ looks more stable in errors.