• Title, Summary, Keyword: Stochastic differential equations

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INFINITE HORIZON OPTIMAL CONTROL PROBLEMS OF BACKWARD STOCHASTIC DELAY DIFFERENTIAL EQUATIONS IN HILBERT SPACES

  • Liang, Hong;Zhou, Jianjun
    • Bulletin of the Korean Mathematical Society
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    • v.57 no.2
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    • pp.311-330
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    • 2020
  • This paper investigates infinite horizon optimal control problems driven by a class of backward stochastic delay differential equations in Hilbert spaces. We first obtain a prior estimate for the solutions of state equations, by which the existence and uniqueness results are proved. Meanwhile, necessary and sufficient conditions for optimal control problems on an infinite horizon are derived by introducing time-advanced stochastic differential equations as adjoint equations. Finally, the theoretical results are applied to a linear-quadratic control problem.

Lie Algebraic Solution of Stochastic Differential Equations

  • Kim, Yoon-Tae;Jeon, Jong-Woo
    • Proceedings of the Korean Statistical Society Conference
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    • pp.25-30
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    • 2003
  • We prove that the logarithm of the flow of stochastic differential equations is an element of the free Lie algebra generated by a finite set consisting of vector fields being coefficients of equations. As an application, we directly obtain a formula of the solution of stochastic differential equations given by Castell(1993) without appealing to an expansion for ordinary differential equations given by Strichartz (1987).

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MEAN SQUARE EXPONENTIAL DISSIPATIVITY OF SINGULARLY PERTURBED STOCHASTIC DELAY DIFFERENTIAL EQUATIONS

  • Xu, Liguang;Ma, Zhixia;Hu, Hongxiao
    • Communications of the Korean Mathematical Society
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    • v.29 no.1
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    • pp.205-212
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    • 2014
  • This paper investigates mean square exponential dissipativity of singularly perturbed stochastic delay differential equations. The L-operator delay differential inequality and stochastic analysis technique are used to establish sufficient conditions ensuring the mean square exponential dissipativity of singularly perturbed stochastic delay differential equations for sufficiently small ${\varepsilon}$ > 0. An example is presented to illustrate the efficiency of the obtained results.

BACKWARD SELF-SIMILAR STOCHASTIC PROCESSES IN STOCHASTIC DIFFERENTIAL EQUATIONS

  • Oh, Jae-Pill
    • Korean Journal of Mathematics
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    • v.6 no.2
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    • pp.259-279
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    • 1998
  • For the forward-backward semimartingale, we can define the backward semimartingale flow which is generated by the backward canonical stochastic differential equation. Therefore, we define the backward self-similar stochastic processes, and we study the backward self-similar stochastic flows through the canonical stochastic differential equations.

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MEAN-FIELD BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS ON MARKOV CHAINS

  • Lu, Wen;Ren, Yong
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.1
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    • pp.17-28
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    • 2017
  • In this paper, we deal with a class of mean-field backward stochastic differential equations (BSDEs) related to finite state, continuous time Markov chains. We obtain the existence and uniqueness theorem and a comparison theorem for solutions of one-dimensional mean-field BSDEs under Lipschitz condition.

EULER-MARUYAMA METHOD FOR SOME NONLINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS WITH JUMP-DIFFUSION

  • Ahmed, Hamdy M.
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.18 no.1
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    • pp.43-50
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    • 2014
  • In this paper we discussed Euler-Maruyama method for stochastic differential equations with jump diffusion. We give a convergence result for Euler-Maruyama where the coefficients of the stochastic differential equation are locally Lipschitz and the pth moments of the exact and numerical solution are bounded for some p > 2.

THE SOLUTIONS OF BACKWARD DOUBLY STOCHASTIC DIFFERENTIAL EQUATIONS WITH NON-LIPSCHITZ COEFFICIENTS

  • Han, Baoyan;Zhu, Bo
    • Journal of applied mathematics & informatics
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    • v.29 no.5_6
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    • pp.1143-1155
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    • 2011
  • In this paper, we shall establish a new theorem on the existence and uniqueness of the solution to a backward doubly stochastic differential equations under a weaker condition than the Lipschitz coefficient. We also show a comparison theorem for this kind of equations.

STOCHASTIC DIFFERENTIAL EQUATION MODELS FOR EXTRACELLULAR SIGNAL-REGULATED KINASE PATHWAYS

  • Choo, S.M.;Kim, Y.H.
    • Journal of applied mathematics & informatics
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    • v.31 no.3_4
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    • pp.457-467
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    • 2013
  • There exist many deterministic models for signaling pathways in systems biology. However the models do not consider the stochastic properties of the pathways, which means the models fit well with experimental data in certain situations but poorly in others. Incorporating stochasticity into deterministic models is one way to handle this problem. In this paper the way is used to produce stochastic models based on the deterministic differential equations for the published extracellular signal-regulated kinase (ERK) pathway. We consider strong convergence and stability of the numerical approximations for the stochastic models.

PERIODIC SOLUTIONS OF STOCHASTIC DELAY DIFFERENTIAL EQUATIONS AND APPLICATIONS TO LOGISTIC EQUATION AND NEURAL NETWORKS

  • Li, Dingshi;Xu, Daoyi
    • Journal of the Korean Mathematical Society
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    • v.50 no.6
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    • pp.1165-1181
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    • 2013
  • In this paper, we consider a class of periodic It$\hat{o}$ stochastic delay differential equations by using the properties of periodic Markov processes, and some sufficient conditions for the existence of periodic solution of the delay equations are given. These existence theorems improve the results obtained by It$\hat{o}$ et al. [6], Bainov et al. [1] and Xu et al. [15]. As applications, we study the existence of periodic solution of periodic stochastic logistic equation and periodic stochastic neural networks with infinite delays, respectively. The theorem for the existence of periodic solution of periodic stochastic logistic equation improve the result obtained by Jiang et al. [7].