NONLINEAR TRANSFORMATIONS ON THE WIENER SPACE

• 成果类型: Article
• 署名作者: ENCHEV, O
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176989015
• 发表日期: 1993
• 页码: 2169-2188
• 关键词: calculus

摘要

We study shift transformations on a general abstract Wiener space (E, H, mu), which have the form: E contains omega --> T(phi)omega equivalent to omega - integral(0)(T) phi(t)(omega)Z(dt) is an element of E, where phi(t)(omega) is a scalar function on [0,T] x E and Z is an orthogonal H-valued measure. Under suitable conditions for the kernel phi, we construct explicitly a probability measure mu(phi) on E, which is equivalent to the standard Wiener measure mu and has the property: mu(phi){T-phi is an element of A} = mu(A), A is an element of B-E. The main result presents an analog of the well-known Cameron-Martin-Girsanov theorem for the case where the shift is allowed to anticipate. This leads to an additional integral term in the Girsanov exponent. Also, the Wiener-Ito integral in this exponent is now replaced by an extended stochastic integral.