On the pathwise uniqueness Of solutions of stochastic integral equations driven by martingales

1983 ◽  
pp. 145-152
Author(s):  
I. I. Galčuk
Keyword(s):  
Integral Equations ◽  
Stochastic Integral ◽  
Pathwise Uniqueness ◽  
Uniqueness Of Solutions ◽  
Stochastic Integral Equations

scholarly journals Set-valued and fuzzy stochastic integral equations driven by semimartingales under Osgood condition

2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Marek T. Malinowski
Keyword(s):  
Integral Equations ◽  
Stochastic Integral ◽  
Local Martingale ◽  
Type Condition ◽  
Finite Variation ◽  
Uniqueness Of Solutions ◽  
Stochastic Integral Equations ◽  
Bounded Convex ◽  
Continuous Dependence Of Solutions ◽  
Osgood Condition

AbstractWe analyze the set-valued stochastic integral equations driven by continuous semimartingales and prove the existence and uniqueness of solutions to such equations in the framework of the hyperspace of nonempty, bounded, convex and closed subsets of the Hilbert space L2 (consisting of square integrable random vectors). The coefficients of the equations are assumed to satisfy the Osgood type condition that is a generalization of the Lipschitz condition. Continuous dependence of solutions with respect to data of the equation is also presented. We consider equations driven by semimartingale Z and equations driven by processes A;M from decomposition of Z, where A is a process of finite variation and M is a local martingale. These equations are not equivalent. Finally, we show that the analysis of the set-valued stochastic integral equations can be extended to a case of fuzzy stochastic integral equations driven by semimartingales under Osgood type condition. To obtain our results we use the set-valued and fuzzy Maruyama type approximations and Bihari’s inequality.

scholarly journals Existence of Solutions of Nonlinear Stochastic Volterra Fredholm Integral Equations of Mixed Type

2010 ◽  
Vol 2010 ◽  
pp. 1-16 ◽  
Author(s):  
K. Balachandran ◽  
J.-H. Kim
Keyword(s):  
Integral Equations ◽  
Existence And Uniqueness ◽  
Mixed Type ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Stochastic Integral ◽  
Sufficient Conditions ◽  
Fredholm Integral Equations ◽  
Equations Of Mixed Type ◽  
Stochastic Integral Equations

We establish sufficient conditions for the existence and uniqueness of random solutions of nonlinear Volterra-Fredholm stochastic integral equations of mixed type by using admissibility theory and fixed point theorems. The results obtained in this paper generalize the results of several papers.

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