Harnack inequality and no-arbitrage bounds for self-financing portfolios

Carciola, Alessandro and Pascucci, Andrea and Polidoro, Sergio (2009): Harnack inequality and no-arbitrage bounds for self-financing portfolios.

Abstract

We give a direct proof of the Harnack inequality for a class of Kolmogorov operators associated with a linear SDE and we find the explicit expression of the optimal Harnack constant. We discuss some possible implication of the Harnack inequality in finance: specifically we infer no-arbitrage bounds for the value of self-financing portfolios in terms of the initial wealth.

• Item Type: MPRA Paper
• Language: English
• Keywords: Harnack inequality; no-arbitrage principle; self-financing portfolio; Kolmogorov equation; linear stochastic equation
• Subjects: G - Financial Economics > G1 - General Financial Markets; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C65 - Miscellaneous Mathematical Tools; C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods

References

[1] A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, Springer, 2004.

[2] G. Auchmuty and D. Bao, Harnack-type inequalities for evolution equations, Proc. Amer. Math. Soc., 122 (1994), pp. 117{129.

[3] E. Barucci, S. Polidoro, and V. Vespri, Some results on partial differential equations and Asian options, Math. Models Methods Appl. Sci., 11 (2001), pp. 475{497.

[4] U. Boscain and S. Polidoro, Gaussian estimates for hypoelliptic operators via optimal control, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 18 (2007), pp. 333{342.

[5] M. Bossy, J. Jabir, and D. Talay, On conditional McKean lagrangian stochastic models, Rapport de recherche n.6761 - INRIA, (2008).

[6] H. D. Cao and S. T. Yau, Gradient estimates, Harnack inequalities and estimates for heat kernels of the sum of squares of vector ¯elds, Math. Z., 211 (1992), pp. 485{504.

[7] N. Garofalo and E. Lanconelli, Level sets of the fundamental solution and Harnack inequality for degenerate equations of Kolmogorov type, Trans. Amer. Math. Soc., 321 (1990), pp. 775{792.

[8] L. HÄormander, Hypoelliptic second order di®erential equations, Acta Math., 119 (1967), pp. 147{171.

[9] V. Jurdjevic, Geometric control theory, Cambridge Univ. Press, 1997.

[10] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer, 1987.

[11] A. Kolmogorov, ZufÄallige Bewegungen. (Zur Theorie der Brown- schen Bewegung.)., Ann. of Math., II. Ser., 35 (1934), pp. 116{117.

[12] L. P. Kupcov, The fundamental solutions of a certain class of elliptic- parabolic second order equations, Di®erential' nye Uravneija, 8 (1972), pp. 1649{1660, 1716.

[13] E. Lanconelli and S. Polidoro, On a class of hypoelliptic evolution operators, Rend. Sem. Mat. Univ. Politec. Torino, 52 (1994), pp. 29{63.

[14] E. B. Lee and L. Markus, Foundations of optimal control theory, John Wiley & Sons Inc., New York, 1967.

[15] P. Li and S. T. Yau, On the parabolic kernel of SchrÄodinger operator, Acta Math., 156 (1986), pp. 153{201.

[16] A. Pascucci and S. Polidoro, On the Harnack inequality for a class of hypoelliptic evolution equations, Trans. Amer. Math. Soc., 356 (2004), pp. 4383{4394.