Necessary and Sufficient Conditions for a Class Positive Local Martingale


Let X be a Markov process, which is assumed to be associated with a (non-symmetric) Dirichlet form (E,D(E)) on L2 (E;m). For , the extended Dirichlet space, we give necessary and sufficient conditions for a multiplicative functional to be a positive local martingale.

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Chen, C. and Yang, S. (2014) Necessary and Sufficient Conditions for a Class Positive Local Martingale. Advances in Pure Mathematics, 4, 545-549. doi: 10.4236/apm.2014.410063.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Ma, Z.M. and Rockner, M. (1992) Introduction to Theory of (Non-Symmetric) Dirichlet Forms. Springer-Verlag, Berlin.
[2] Fukushima, M., Oshima, Y. and Takeda, M. (1994) Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter Berlin, New York.
[3] Chen, C.-Z. and Sun, W. (2009) Girsanov Transformations for Non-symmetric Diffusions. Canadian Journal of Mathematics, 61, 534-547.
[4] Chen, Z.-Q. and Zhang, T.-S. (2002) Girsanov and Feynman-Kac Type Transformations for Symmetric Markov Processes. Annales de l’Institut Henri Poincare (B) Probability and Statistics, 38, 475-450.
[5] Chen, C.-Z., Ma, Z.-M. and Sun, W. (2007) On Girsanov and Generalized Feynman-Kac Transformations for Symmetric Markov Process. World Scientific, 10, 141-163.
[6] Oshima. Y. (2013) Semi-Dirichlet Forms and Markov Processes. Walter de Gruyter, Berlin.
[7] He, S.W., Wang, J.G. and Yan, J.A. (1992) Semimartingale Theory and Stochastic Calculus. Science Press, Beijing.
[8] Kallsen, J. and Shiryaev, A.N. (2002) The Cumulant Process and Esscher’s Change of Measure. Finance Stochast, 6, 397-428.

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