• 대한수학회지
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• 제59권6호
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• pp.1083-1101
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• 2022

In this paper, we study backward stochastic differential equations (BSDEs shortly) with jumps that have Lipschitz generator in a general filtration supporting a Brownian motion and an independent Poisson random measure. Under just integrability on the data we show that such equations admit a unique solution which belongs to class 𝔻.

• Journal of Applied and Pure Mathematics
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• 제4권5_6호
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• pp.299-315
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• 2022

This manuscript deals with the exact (complete) controllability of semilinear stochastic differential equations with infinite delay and Poisson jumps utilizing some basic and readily verified conditions. The results are obtained by using fixed-point approach and by using advance phase space definition for infinite delay part. We have used the axiomatic definition of the phase space in terms of stochastic process to consider the time delay of the system. An infinite delay along with the Poisson jump is the new investigation for the given stochastic system. An example is given to illustrate the effectiveness of the results.

• Journal of Applied and Pure Mathematics
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• 제4권3_4호
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• pp.107-122
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• 2022

This manuscript aims to investigate the existence, uniqueness, and stability of non-local random impulsive neutral stochastic differential time delay equations (NRINSDEs) with Poisson jumps. First, we prove the existence of mild solutions to this equation using the Banach fixed point theorem. Next, we demonstrate the stability via continuous dependence initial value. Our study extends the work of Wang, and Wu [16] where the time delay is addressed by the prescribed phase space 𝓑 (defined in Section 3). To illustrate the theory, we also provide an example of our methods. Using our results, one could investigate the controllability of random impulsive neutral stochastic differential equations with finite/infinite states. Moreover, one could extend this study to analyze the controllability of fractional-order of NRINSDEs with Poisson jumps as well.

• 대한수학회보
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• 제55권2호
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• pp.479-497
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• 2018

This paper deals with an optimal control on semilinear stochastic functional differential equations with Poisson jumps in a Hilbert space. The existence of an optimal control is derived by the solution of proposed system which satisfies weakly sequentially compactness. Also the stochastic maximum principle for the optimal control is established by using spike variation technique of optimal control with a convex control domain in Hilbert space. Finally, an application of retarded type stochastic Burgers equation is given to illustrate the theory.

• 대한수학회지
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• 제42권2호
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• pp.269-289
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• 2005

An infinite system of stochastic differential equations (SDE)driven by Brownian motions and compensated Poisson random measures for the locations and weights of a collection of particles is considered. This is an analogue of the work by Kurtz and Xiong where compensated Poisson random measures are replaced by white noise. The particles interact through their weighted measure V, which is shown to be a solution of a stochastic differential equation. Also a limit theorem for system of SDE is proved when the corresponding Poisson random measures in SDE converge to white noise.

• Journal of the Korean Statistical Society
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• 제26권1호
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• pp.147-154
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• 1997

An infinite dam with a compound Poisson input having exponential jumps is considered. As an output policy, we adopt the $P_{\lambda}$$^{M}$ Policy. After assigning costs to the dam we obtain the long-rum average cost per unit time of operating the dam and find the optimal values of .lambda. and M which minimize the long-run average cost.t.

• Journal of applied mathematics & informatics
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• 제29권5_6호
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• pp.1311-1325
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• 2011

In this paper the Milstein method is proposed to approximate the solution of a linear stochastic differential equation with Poisson-driven jumps. The strong Milstein method and the weak Milstein method are shown to capture the mean square stability of the system. Furthermore using some technique, our result shows that these two kinds of Milstein methods can well reproduce the stochastically asymptotical stability of the system for all sufficiently small time-steps. Some numerical experiments are given to demonstrate the conclusions.

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

• Communications for Statistical Applications and Methods
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• 제11권3호
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• pp.495-501
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• 2004

HJM representation of the term structure of interest rates sometimes produces the negative interest rates with positive probability. This paper shows that the condition of positive interest rates can be derived from the jump diffusion process, if a proper positive martingale process with the compensated jump process is chosen. As in Flesaker and Hughston, the condition is incorporated into the bond price process.

• 한국통계학회:학술대회논문집
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• 한국통계학회 2004년도 학술발표논문집
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• pp.123-128
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• 2004

A finite dam under $P_{\lambda,;,T}^M-policy$ is considered, where the input of water is formed by a Wiener process subject to random jumps arriving according to a Poisson process. Explicit expression is deduced for the stationary distribution of the level of water. And the long-run average cost per unit time is obtained after assigning costs to the changes of release rate, a reward to each unit of output, and a penalty which is a function of the level of water in the reservoir.