Semimartingale Property and Its Connections to Arbitrage

ABSTRACT

In this paper, we prove the celebrated Bichteler-Dellaccherie Theorem which states that the class of stochastic processes *X* allowing for a useful integration theory consists precisely of those processes which can be written in the form *X* = *X*_{0} + *M* + *A*, where *M*_{0} = *A*_{0} = 0, *M* is a local martingale, and *A* is of finite variation process. We obtain this decomposition rather direct form an elementary discrete-time Doob-Meyer decomposition. By moving to convex combination we obtain a direct continuous time decomposition, which then yield the desired decomposition. We also obtain a characterization of semi-martingales in terms of a variant no free lunch with vanishing risk.

KEYWORDS

Bichteler-Dellaccherie Theorem; Doob-Meyer Decomposition; Semi-Martingales; Arbitrage; Komlos Lemma

Bichteler-Dellaccherie Theorem; Doob-Meyer Decomposition; Semi-Martingales; Arbitrage; Komlos Lemma

Cite this paper

S. Samura, J. Mao and D. Yao, "Semimartingale Property and Its Connections to Arbitrage,"*Journal of Mathematical Finance*, Vol. 3 No. 2, 2013, pp. 237-241. doi: 10.4236/jmf.2013.32023.

S. Samura, J. Mao and D. Yao, "Semimartingale Property and Its Connections to Arbitrage,"

References

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[7] C. Kardaras and E. Platen, “On the Semimartingale Property of Discounted Asset-Price Processes,” Stochastic Processes and Their Applications, Vol. 121, 2011, pp. 2678-2691.

[8] F. Delbaen and W. Schachermayer, “Arbitrage and Free Lunch with Bounded Risk for Unbounded Continuous Processes,” Mathematical Finance, Vol. 4, No. 4, 1994, pp. 343-348. doi:10.1111/j.1467-9965.1994.tb00063.x

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[1] F. Delbaen and W. Schachermayer, “A General Version of the Fundamental Theorem of Asset Pricing,” Mathematische Annalen, Vol. 300 No. 3, 1994, pp.463-520. doi:10.1007/BF01450498

[2] P. E. Protter, “Stochastic Integration and Differential Equation,” 2nd Edition, Springer-Verlag, Berlin, 2004.

[3] M. Beiglblock, W. Schachermayer and B. Veliyev, “A Direct Proof of the Bichteler-Dellacherie Theorem and Connections to Arbitrage,” 2010.

[4] M. Loewenstein and G. A. Willard, “Local Martingales, Arbitrage, and Viability Free Snacks and Cheap Thrills,” Economic Theory, Vol. 16, No. 1, 2000, pp.135-161. doi:10.1007/s001990050330

[5] E. Platen, “Arbitrage in Continueous Complete Markets,” Advances in Applied Probability, Vol. 34, No. 3, 2002, pp.540-558. doi:10.1239/aap/1033662165

[6] R. Fernholz, I. Karatzas and C. Kardaras, “Diversity and Relative Arbitrage in Equity Markets,” Finance and Stochastics, Vol. 9, No. 1, 2005, pp.1-27. doi:10.1007/s00780-004-0129-4

[7] C. Kardaras and E. Platen, “On the Semimartingale Property of Discounted Asset-Price Processes,” Stochastic Processes and Their Applications, Vol. 121, 2011, pp. 2678-2691.

[8] F. Delbaen and W. Schachermayer, “Arbitrage and Free Lunch with Bounded Risk for Unbounded Continuous Processes,” Mathematical Finance, Vol. 4, No. 4, 1994, pp. 343-348. doi:10.1111/j.1467-9965.1994.tb00063.x

[9] D. Williams, “Probability with Martingales,” Cambridge University Press, Cambridge, 1991. doi:10.1017/CBO9780511813658

[10] I. Karatzas and S. E. Shrve, “Brownian Motion and Stochastic Calculus of Graduate Text in Mathematics,” 2nd Edition, Springer-Verlag, Berlin, 1991.

[11] J. Komlos, “A generalization of problem of Steinhaus,” Acta Mathematica Academiae Scientiarum Hungarica, Vol. 18, No. 1-2, 1967, pp. 217-229. doi:10.1007/BF02020976