• Journal of the Chungcheong Mathematical Society
• /
• v.24 no.1
• /
• pp.45-57
• /
• 2011

We study the stochastic integral of an operator valued process against with a Banach space valued regular process. We establish the existence and uniqueness of solution of the stochastic differential equation for a Banach space valued regular process under the certain conditions. As an application of it, we study a noncommutative stochastic differential equation.

• Korean Journal of Mathematics
• /
• v.8 no.1
• /
• pp.1-17
• /
• 2000

The existence of density of process, which is given by canonical stochastic differential equation, can be proved by the Picard's method([5]) also.

• Korean Journal of Mathematics
• /
• v.5 no.2
• /
• pp.199-209
• /
• 1997

The existence and the smoothness of densities of random variables, which are generated by the canonical stochastic differential equation, can be proved by the Malliavin - Bismut method.

• Korean Journal of Mathematics
• /
• v.5 no.1
• /
• pp.91-106
• /
• 1997

The existence and the smoothness of density of random variables which are solutions of the canonical stochastic differential equation can be proved by simpler conditions in the Malliavin-Bismut method.

• The Pure and Applied Mathematics
• /
• v.14 no.4
• /
• pp.335-354
• /
• 2007

In this paper, we define an analogue of generalized Wiener measure and investigate its basic properties. We define (${\hat}It{o}$ type) stochastic integrals with respect to the generalized Wiener process and prove the ${\hat}It{o}$ formula. The existence and uniqueness of the solution of stochastic differential equation associated with the generalized Wiener process is proved. Finally, we generalize the linear filtering theory of Kalman-Bucy to the case of a generalized Wiener process.

• Wind and Structures
• /
• v.5 no.5
• /
• pp.423-440
• /
• 2002

The effects of the nonlinear (quadratic) term in wind pressure have been analyzed in many papers with reference to linear structural models. The present paper addresses the problem of the response of nonlinear structures to stochastic nonlinear wind pressure. Adopting a single-degree-of-freedom structural model with polynomial nonlinearity, the solution is obtained by means of the moment equation approach in the context of It$\hat{o}$'s stochastic differential calculus. To do so, wind turbulence is idealized as the output of a linear filter excited by a Gaussian white noise. Response statistical moments are computed for both the equivalent linear system and the actual nonlinear one. In the second case, since the moment equations form an infinite hierarchy, a suitable iterative procedure is used to close it. The numerical analyses regard a Duffing oscillator, and the results compare well with Monte Carlo simulation.

• East Asian mathematical journal
• /
• v.29 no.1
• /
• pp.13-22
• /
• 2013

As a fuzzy counterpart of stochastic calculus, fuzzy calculus including Liu integral and Liu formula were introduced. In order to deal with the problems with several fuzzy dynamic factors, Liu process, Liu integral and Liu formula are extended to the case of multi-dimensional in this paper.

• Journal of applied mathematics & informatics
• /
• v.18 no.1_2
• /
• pp.145-158
• /
• 2005

We formulate a stochastic control problem on proportional reinsurance that includes impulse and regular control strategies. For the first time we combine impulse control with regular control, and derive the expected total discount pay-out (return function) from present to bankruptcy. By relying on both stochastic calculus and the classical theory of impulse and regular controls, we state a set of sufficient conditions for its solution in terms of optimal return function. Moreover, we also derive its explicit form and corresponding impulse and regular control strategies.

• Communications for Statistical Applications and Methods
• /
• v.14 no.2
• /
• pp.459-479
• /
• 2007

Theories related to financial market has received big attention from the statistics community. However, not many courses on the topic are provided in statistics departments. Because the financial theories are entangled with many complicated mathematical and physical theories as well as ambiguously stated financial terminologies. Based on our experience on the topic, we try to explain the rather complicated terminologies and theories with easy-to-understand words. This paper will briefly cover the topics of basic terminologies of derivatives, Black-Scholes pricing idea, and related basic mathematical terminologies.

• Journal of the Korean Mathematical Society
• /
• v.38 no.2
• /
• pp.297-309
• /
• 2001

White noise distribution theory over the complex Gaussian space is established on the basis of the recently developed white noise operator theory. Unitarity condition for a white noise operator is discussed by means of the operator symbol and complex Gaussian integration. Concerning the overcompleteness of the exponential vectors, a coherent sate representation of a white noise function is uniquely specified from the diagonal coherent state representation of the associated multiplication operator.