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Girsanov Transformations for Non-Symmetric Diffusions

Published online by Cambridge University Press:  20 November 2018

Chuan-Zhong Chen  and
Wei Sun
Chuan-Zhong Chen
Affiliation:
Department of Mathematics, Hainan Normal University, Haikou, 571158, China, ccz0082@yahoo.com.cn
Wei Sun
Affiliation:
Department of Mathematics and Statistics, Concordia University, Montreal, H3G 1M8, Canada, wsun@mathstat.concordia.ca
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Abstract

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Let $X$ be a diffusion process, which is assumed to be associated with a (non-symmetric) strongly local Dirichlet form $(\varepsilon ,\mathcal{D}(\varepsilon ))$ on ${{L}^{2}}(E;m)$ . For $u\,\in \,\mathcal{D}{{(\varepsilon )}_{e}}$ , the extended Dirichlet space, we investigate some properties of the Girsanov transformed process $Y$ of $X$ . First, let $\hat{X}$ be the dual process of $X$ and $\hat{Y}$ the Girsanov transformed process of $\hat{X} $ . We give a necessary and sufficient condition for $(Y,\hat{Y})$ to be in duality with respect to the measure ${{e}^{2u}}m$ . We also construct a counterexample, which shows that this condition may not be satisfied and hence $(Y,\hat{Y})$ may not be dual processes. Then we present a sufficient condition under which $Y$ is associated with a semi-Dirichlet form. Moreover, we give an explicit representation of the semi-Dirichlet form.

Keywords

60J4531C2560J57Diffusionnon-symmetric Dirichlet formGirsanov transformationh-transformationperturbation of Dirichlet formgeneralized Feynman-Kac semigroup

Information

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 3 , 01 June 2009 , pp. 534 - 547
Copyright
Copyright © Canadian Mathematical Society 2009

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