• Title/Summary/Keyword: non-Lipschitz condition

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AN EXISTENCE OF THE SOLUTION TO NEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS UNDER SPECIAL CONDITIONS

  • KIM, YOUNG-HO
    • Journal of applied mathematics & informatics
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    • v.37 no.1_2
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    • pp.53-63
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    • 2019
  • In this paper, we show the existence of solution of the neutral stochastic functional differential equations under non-Lipschitz condition, a weakened linear growth condition and a contractive condition. Furthermore, in order to obtain the existence of solution to the equation we used the Picard sequence.

A NOTE ON THE APPROXIMATE SOLUTIONS TO STOCHASTIC DIFFERENTIAL DELAY EQUATION

  • KIM, YOUNG-HO;PARK, CHAN-HO;BAE, MUN-JIN
    • Journal of applied mathematics & informatics
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    • v.34 no.5_6
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    • pp.421-434
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    • 2016
  • The main aim of this paper is to discuss the difference between the Euler-Maruyama's approximate solutions and the accurate solution to stochastic differential delay equation. To make the theory more understandable, we impose the non-uniform Lipschitz condition and weakened linear growth condition. Furthermore, we give the pth moment continuous of the approximate solution for the delay equation.

AN ESTIMATE OF THE SOLUTIONS FOR STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATIONS

  • Kim, Young-Ho
    • Journal of applied mathematics & informatics
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    • v.29 no.5_6
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    • pp.1549-1556
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    • 2011
  • In this paper, we give an estimate on the difference between $x^n(t)$ and x(t) and it clearly shows that one can use the Picard iteration procedure to the approximate solutions to stochastic functional differential equations with infinite delay at phase space BC(($-{\infty}$, 0] : $R^d$) which denotes the family of bounded continuous $R^d$-valued functions ${\varphi}$ defined on ($-{\infty}$, 0] with norm ${\parallel}{\varphi}{\parallel}={\sup}_{-{\infty}<{\theta}{\leq}0}{\mid}{\varphi}({\theta}){\mid}$ under non-Lipschitz condition being considered as a special case and a weakened linear growth condition.

CONTINUOUS DEPENDENCE PROPERTIES ON SOLUTIONS OF BACKWARD STOCHASTIC DIFFERENTIAL EQUATION

  • Fan, Sheng-Jun;Wu, Zhu-Wu;Zhu, Kai-Yong
    • Journal of applied mathematics & informatics
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    • v.24 no.1_2
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    • pp.427-435
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    • 2007
  • The existence theorem and continuous dependence property in $"L^2"$ sense for solutions of backward stochastic differential equation (shortly BSDE) with Lipschitz coefficients were respectively established by Pardoux-Peng and Peng in [1,2], Mao and Cao generalized the Pardoux-Peng's existence and uniqueness theorem to BSDE with non-Lipschitz coefficients in [3,4]. The present paper generalizes the Peng's continuous dependence property in $"L^2"$ sense to BSDE with Mao and Cao's conditions. Furthermore, this paper investigates the continuous dependence property in "almost surely" sense for BSDE with Mao and Cao's conditions, based on the comparison with the classical mathematical expectation.