Existence, Uniqueness, and Asymptotic Behavior of Mild Solutions to Stochastic Functional Differential Equations in Hilbert Spaces

By Takeshi Taniguchi, Kai Liu, A. Truman. 2002

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Abstract In this paper we shall consider the existence, uniqueness, and asymptotic behavior of mild solutions to stochastic partial functional differential equations with finite delay r >0:  dX ( t )=[− AX ( t )+ f ( t ,  X t )]  dt + g ( t ,  X t )  dW ( t ), where we assume that − A is a closed, densely defined linear operator and the generator of a certain analytic semigroup. f : (−∞, +∞)× C α → H , g : (−∞, +∞)× C α → L 0 2 ( K ,  H ) are two locally Lipschitz continuous functions, where C α = C ([− r , 0],  D ( A α )), L 0 2 ( K ,  H ) are two proper infinite dimensional spaces, 0 α W ( t ) is a given K -valued Wiener process and both H and K are separable Hilbert spaces.

Subfields: Applied Mathematics, Control and Systems Engineering, Finance

View Paper: https://www.semanticscholar.org/paper/7eea06b829eec876f20071296a44713dd3f69f06