Girsanov Transformations For Non-symmetric Markov Processes

Dirichlet Forms originated from the classical potential theory,and it is associated with Markov processes. Therefor we can transform some problems about stochastic processes into analytical problems.The Girsanov transformation is always the element and the most important tool in stochastic analysis.There have been a lot of beautiful results about the Girsanov transformation for symmetric Markov processes. So we will talk about the Girsanov thansformation for the non-symmetric Markov processes associated with Dirichlet Forms.Let X be a Markov process, which is assumed to be associated with a(non-symmetric) Dirichlet form (ε,D(ε)) on L2(E; m). In the second chapter, for u ∈D(ε)e, the extended Dirichlet space, we give necessary and sufficient conditions for the Girsanov transform to induce a positive local martingale by using the Doleans-Dade formula in stochastic differential equations. In the third chapter, we assume the Girsanov transform is a positive local martingale and hence to determine the Girsanov transformed process Y of X. At first we define a bilinear form. Moreover we present a sufficient condition under which Y is associated with a semi-Dirichlet form and give an explicit representation of the semi-Dirichlet which is the defined form.