University of Khartoum

Stability Theorems for Stochastic Differential Equations (S.D.E.'s) with Memory (Part 1)

Stability Theorems for Stochastic Differential Equations (S.D.E.'s) with Memory (Part 1)

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Title: Stability Theorems for Stochastic Differential Equations (S.D.E.'s) with Memory (Part 1)
Author: Ahmed, Tagelsir A.; Casteren, J. A. Van
Abstract: Here stochastic differential equations with memory means delay stochastic differential equations. In the present work we have formulated an example of the main delay stochastic differential equation,see [2] and [11] and [9]. The example which we have considered is of the following form: 𝑑 𝑥1 𝑡 𝑥2(𝑡) 𝑥 𝑡(𝑡) = 𝑥2 𝑡 βˆ’𝑥1 𝑡 + 𝑒βˆ’𝑠 𝑥𝑡 1 𝑠 βˆ’ 𝑥1(𝑡) 𝑑𝑠 ∞ 0 βˆ’ 𝑥 𝑡 ˊ βˆ’ 𝑥2(𝑡) 𝑑𝑡 + 0 𝛼 𝑥 𝑡 0 𝑑𝑊𝑡 where the ordered triple (𝑥1 𝑡 , 𝑥2 𝑡 , 𝑥 𝑡 ) can be considered as representing position, velocity and history ofposition respectively. We will call the space containing this triple "the history space 𝑋". In section two of thiswork we have proved a stability theorem for a diffusion of a S.D.E. in 𝑅𝑛 . With a suitable choice of Lyapunovfunctional we have proved that the motion will finally come to rest at the origin. In section three we haveextended the space 𝑅𝑛 of section two to a history space, i.e. to a space with three components; position,velocity and history of position. Also we formulate our S.D.E. on this history space 𝑋 and also we found thegenerator of the diffusion. The initiation of the present work was suggested by Prof. Maassen,J.D.M.,KatholikUniversity of Nijmegen,The Netherlands.
URI: http://khartoumspace.uofk.edu/123456789/24754
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