Semimartingale Property and Its Connections to Arbitrage

Sallieu Kabay Samura, Junjun Mao, Dengbao Yao

School of Mathematical Science, Anhui University, Hefei, China.

DOI: 10.4236/jmf.2013.32023                

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 = X₀ + M + A, where M₀ = A₀ = 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

Conflicts of Interest

The authors declare no conflicts of interest.

References

[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