• Journal of applied mathematics & informatics
• /
• 제30권3_4호
• /
• pp.541-553
• /
• 2012

In this paper, we study some limit properties of the harmonic mean of random transition probability for a second-order nonhomogeneous Markov chain on the generalized gambling system indexed by a tree by constructing a nonnegative martingale. As corollary, we obtain the property of the harmonic mean and the arithmetic mean of random transition probability for a second-order nonhomogeneous Markov chain indexed by a double root tree.

• 대한수학회지
• /
• 제38권5호
• /
• pp.1031-1046
• /
• 2001

The paper studies the evolution of the financial markets and pays the basic attention to the role of financial innovations (derivative securities) in this process. A characterization of both complete and incomplete markets is given through an identification of the sets of contingent claims and terminal wealths of self-financing portfolios. the dynamics of the financial system is described as a movement of incomplete markets to a complete one when the volume of financial innovations is growing up and the spread tends to zero (the Merton financial innovation spiral). Namely in this context the paper deals with the problem of pricing risks in both field: finance and insurance.

• 대한수학회지
• /
• 제35권3호
• /
• pp.713-725
• /
• 1998

Let X and Y be Ito processes with dX$_{s}$ = $\phi$$_{s}$dB$_{s}$ ＋ $\psi$$_{s}$ds and dY$_{s}$ = (equation omitted)dB$_{s}$ ＋ ξ$_{s}$ds. Burkholder obtained a sharp bound on the distribution of the maximal function of Y under the assumption that │Y$_{0}$│$\leq$│X$_{0}$│,│ζ│$\leq$│$\phi$│, │ξ│$\leq$│$\psi$│ and that X is a nonnegative local submartingale. In this paper we consider a wider class of Ito processes, replace the assumption │ξ│$\leq$│$\psi$│ by a more general one │ξ│$\leq$$\alpha$ │$\psi$│ , where a $\geq$ 0 is a constant, and get a weak-type inequality between X and the maximal function of Y. This inequality, being sharp for all a $\geq$ 0, extends the work by Burkholder.der.urkholder.der.

• Journal of the Korean Statistical Society
• /
• 제36권2호
• /
• pp.229-235
• /
• 2007

We consider a ruin model whose surplus process is formed by a compound Poisson process. If the level of surplus reaches V > 0, it is assumed that a certain amount of surplus is invested. In this paper, we apply the optional sampling theorem to the surplus process and obtain the expectation of period T, time from origin to the point where the level of surplus reaches either 0 or V. We also derive the total and average amount of surplus during T by establishing a backward differential equation.

• Communications for Statistical Applications and Methods
• /
• 제23권5호
• /
• pp.423-432
• /
• 2016

In this paper, we stochastically analyze the continuous time surplus process in a risk model which involves a continuous type investment. It is assumed that the investment of the surplus to other business is continuously made at a constant rate, while the surplus process stays over a given sufficient level. We obtain the stationary distribution of the surplus level and/or its moment generating function by forming martingales from the surplus process and applying the optional sampling theorem to the martingales and/or by establishing and solving an integro-differential equation for the distribution function of the surplus level.

• Journal of the Korean Statistical Society
• /
• 제24권1호
• /
• pp.197-218
• /
• 1995

We first consider the random censorship model of survival analysis. Efron (1981) introduced two equivalent bootstrap methods for censored data. We propose a new bootstrap scheme, called Method 3, that acts conditionally on the censoring pattern when making inference about aspects of the unknown life-time distribution F. This article contains (a) a motivation for this refined bootstrap scheme ; (b) a proof that the bootstrapped Kaplan-Meier estimatro fo F formed by Method 3 has the same limiting distribution as the one by Efron's approach ; (c) description of and report on simulation studies assessing the small-sample performance of the Method 3 ; (d) an illustration on some Danish data. We also consider the model in which the survival times are censered by death times due to other caused and also by known fixed constants, and propose an appropriate bootstrap method for that model. This bootstrap method is a readily modified version of the Method 3.

• 한국통계학회:학술대회논문집
• /
• 한국통계학회 2002년도 추계 학술발표회 논문집
• /
• pp.265-272
• /
• 2002

An Integer-valued autoregressive integrated (INARI) model is introduced to eliminate stochastic trend and seasonality from time series of count data. This INARI extends the previous integer-valued ARMA model. We show that it is stationary and ergodic to establish asymptotic normality for conditional least squares estimator. Optimal estimating equations are used to reflect categorical and serial correlations arising from panel count data and variations arising from three random processes for obtaining observation into estimation. Under regularity conditions for martingale sequence, we show asymptotic normality for estimators from the estimating equations. Using cancer mortality data provided by the U.S. National Center for Health Statistics (NCHS), we apply our results to estimate the probability of cells classified by 4 causes of death and 6 age groups and to forecast death count of each cell. We also investigate impact of three random processes on estimation.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• 제19권1호
• /
• pp.23-45
• /
• 2015

We consider counterparty risk in CDS rates. To do so, we use a multivariate jump diffusion process for obligors' default intensity, where jumps (i.e. magnitude of contribution of primary events to default intensities) occur simultaneously and their sizes are dependent. For these simultaneous jumps and their sizes, a homogeneous Poisson process. We apply copula-dependent default intensities of multivariate Cox process to derive the joint Laplace transform that provides us with joint survival/default probability and other relevant joint probabilities. For that purpose, the piecewise deterministic Markov process (PDMP) theory developed in [7] and the martingale methodology in [6] are used. We compute survival/default probability using three copulas, which are Farlie-Gumbel-Morgenstern (FGM), Gaussian and Student-t copulas, with exponential marginal distributions. We then apply the results to calculate CDS rates assuming deterministic rate of interest and recovery rate. We also conduct sensitivity analysis for the CDS rates by changing the relevant parameters and provide their figures.

• 대한수학회지
• /
• 제38권2호
• /
• pp.227-274
• /
• 2001

In this paper we intended to discuss the following topics: (1) Notation, generalities, Markov processes. The close relationship between (generators of) Markov processes and the martingale problem is exhibited. A link between the Korovkin property and generators of Feller semigroups is established. (2) Feynman-Kac semigroups: 0-order regular perturbations, pinned Markov measures. A basic representation via distributions of Markov processes is depicted. (3) Dirichlet semigroups: 0-order singular perturbations, harmonic functions, multiplicative functionals. Here a representation theorem of solutions to the heat equation is depicted in terms of the distributions of the underlying Markov process and a suitable stopping time. (4) Sets of finite capacity, wave operators, and related results. In this section a number of results are presented concerning the completeness of scattering systems (and its spectral consequences). (5) Some (abstract) problems related to Neumann semigroups: 1st order perturbations. In this section some rather abstract problems are presented, which lie on the borderline between first order perturbations together with their boundary limits (Neumann type boundary conditions and) and reflected Markov processes.

• 대한수학회지
• /
• 제49권3호
• /
• pp.659-670
• /
• 2012

For any $K$ > $2/{\pi}$ we determine the optimal constant $L(K)$ for which the following holds. If $u$, $tilde{u}$ are conjugate harmonic functions on the unit disc with $\tilde{u}(0)=0$, then $$ {\int}_{-\pi}^{\pi}{\mid}\tilde{u}(e^{i\phi}){\mid}\frac{d{\phi}}{2{\pi}}{\leq}K{\int}_{-\pi}^{\pi}{\mid}u(e^{i{\phi}}){\mid}{\log}^+{\mid}u(e^{i{\phi}}){\mid}\frac{d{\phi}}{2{\pi}}+L(K).$$ We also establish a related estimate for orthogonal harmonic functions given on Euclidean domains as well as an extension concerning orthogonal martingales under differential subordination.