On transformations of Wiener space

Anatoli V. Skorokhod

International Journal of Stochastic Analysis, 1994, vol. 7, 1-8

Abstract:

We consider transformations of the form ( T a x ) t = x t + ∫ 0 t a ( s , x ) d s on the space C of all continuous functions x = x t : [ 0 , 1 ] → ℝ , x 0 = 0 , where a ( s , x ) is a measurable function [ 0 , 1 ] × C → ℝ which is 𝒞 ˜ s -measurable for a fixed s and 𝒞 ˜ s is the σ -algebra generated by { x u , u ≤ t } . It is supposed that T a maps the Wiener measure μ 0 on ( C , 𝒞 ˜ s ) into a measure μ a which is equivalent with respect to μ 0 . We study some conditions of invertibility of such transformations. We also consider stochastic differential equations of the form d y ( t ) = d w ( t ) + a ( t , y ( t ) ) d t , y ( 0 ) = 0 where w ( t ) is a Wiener process. We prove that this equation has a unique strong solution if and only if it has a unique weak solution.

Date: 1994
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnijsa:584613

DOI: 10.1155/S1048953394000249

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