Abstract
In a recent paper, [1], Phillipe Biane introduced martingales M k associated with the different jump ‘sizes’ of a time homogeneous, finite Markov chain and developed homogeneous chaos expansions. It has long been known that the Kolmogorov equation for the probability densities of a Markov chain gives rise to a canonical martingale M. The modest contributions of this note, are that working with a non-homogeneous chain, we relate Biane’s martingales M k to M, calculate the quadratic variation of M and thereby that of the M k. In addition, square field identities are obtained for each jump size.
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References
P. Biane, Chaotic representation for finite Markov chains. Stochastics and Stock. Reports 30 (1990), 61-68.
R.J. Elliott, Smoothing for a finite state Markov process. Springer Lecture Notes in Control and Info. Sciences, Vol. 69, (1985), 199-206.
R.J. Elliott and M. Kohlmann, Integration by parts, homogeneous chaos expansions and smooth densities. Ann. of Prob. 17 (1989), 194-207.
R.S. Liptser and A.N. Shiryayev, Statistics of Random Processes, Vol. 1, Springer Verlag, Berlin, Heidelberg, New York, 1977.
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Elliott, R.J. (1991). Martingales Associated with Finite Markov Chains. In: Çinlar, E., Fitzsimmons, P.J., Williams, R.J. (eds) Seminar on Stochastic Processes, 1990. Progress in Probability, vol 24. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4684-0562-0_6
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DOI: https://doi.org/10.1007/978-1-4684-0562-0_6
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-0-8176-3488-9
Online ISBN: 978-1-4684-0562-0
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