Carciola, Alessandro and Pascucci, Andrea and Polidoro, Sergio (2009): Harnack inequality and noarbitrage bounds for selffinancing portfolios.

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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 noarbitrage bounds for the value of selffinancing portfolios in terms of the initial wealth.
Item Type:  MPRA Paper 

Original Title:  Harnack inequality and noarbitrage bounds for selffinancing portfolios 
English Title:  Harnack inequality and noarbitrage bounds for selffinancing portfolios 
Language:  English 
Keywords:  Harnack inequality; noarbitrage principle; selffinancing 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 
Item ID:  15665 
Depositing User:  Andrea Pascucci 
Date Deposited:  12. Jun 2009 03:40 
Last Modified:  16. Feb 2013 00:43 
References:  [1] A. A. Agrachev and Y. L. Sachkov, Control Theory from the Geometric Viewpoint, Springer, 2004. [2] G. Auchmuty and D. Bao, Harnacktype 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. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/15665 