Existence uniqueness and condition parents for stochastic functional differential equation
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This paper is devoted to existence and uniqueness of solutions for some stochastic functional differential equations with infinite delay in a fading memory phase space.
1. Introduction
Let denote the Euclidian norm in . If is a vector or a matrix, its transpose is denoted by and its trace norm is represented by . Let be the minimum (maximum) for .
Let be a complete probability space with a filtration satisfying the usual conditions; that is, it is right continuous and contains all -null sets.
denotes the family of all -measurable valued processes , such that
Assume that is an -dimensional Brownian motion which is defined on ; that is, .
Let denote the family of continuous functions defined on with norm .
Consider the -dimensional stochastic functional differential equationwhere can be regarded as a -value stochastic process, and and are Borel measurable.
The initial data of the stochastic process is defined on . That is, the initial value
1. Introduction
Let denote the Euclidian norm in . If is a vector or a matrix, its transpose is denoted by and its trace norm is represented by . Let be the minimum (maximum) for .
Let be a complete probability space with a filtration satisfying the usual conditions; that is, it is right continuous and contains all -null sets.
denotes the family of all -measurable valued processes , such that
Assume that is an -dimensional Brownian motion which is defined on ; that is, .
Let denote the family of continuous functions defined on with norm .
Consider the -dimensional stochastic functional differential equationwhere can be regarded as a -value stochastic process, and and are Borel measurable.
The initial data of the stochastic process is defined on . That is, the initial value
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