Abstract.
We present new exact expressions for a class of moments of the geometric Brownian motion in terms of determinants, obtained using a recurrence relation and combinatorial arguments for the case of a Itô's Wiener process. We then apply the obtained exact formulas to computing averages of the solution of the logistic stochastic differential equation via a series expansion, and compare the results to the solution obtained via Monte Carlo.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C. Gardiner, Stochastic Methods (Springer-Verlag, Berlin, 2009)
M. Yor, Exponential Functionals of Brownian Motion and Related Processes (Springer-Verlag, Berlin, 2001)
J.P. Bouchaud, M. Potters, Theory of Financial Risks (Cambridge University Press, Cambridge, 2000)
D.J. Wilkinson, Stochastic Modelling for Systems Biology (CRC Press, New York, 2012)
H. Matsumoto, M. Yor, Prob. Surv. 2, 312 (2005)
H. Matsumoto, M. Yor, Prob. Surv. 2, 348 (2005)
D. Dufresne, Adv. Appl. Prob. 33, 223 (2001)
C. Monthus, A. Comtet, J. Phys. A 4, 635 (1994)
A. Comtet, C. Monthus, J. Phys. A 29, 1331 (1996)
M. Yor, Adv. Appl. Prob. 24, 509 (1992)
I.V. Girsanov, Theory Probab. Appl. 5, 285 (1960)
E. Moro, J. Vicente, L.G. Moyano, A. Gerig, J.D. Farmer, G. Vaglica, F. Lillo, R.N. Mantegna, Phys. Rev. E 80, 066102 (2009)
E. Zarinelli, M. Treccani, J.D. Farmer, F. Lillo, Mark. Microstruct. Liq. 1, 1 (2015)
F. Caravelli, L. Sindoni, F. Caccioli, C. Ududec, arXiv:1510.05123
B.J.C. Baxter, R. Brummelhuis, J. Comput. Appl. Math. 236, 424 (2011)
C. De Boor, Surv. Approx. Theory 1, 46 (2005)
M. Salmhofer, Renormalization - An Introduction (Springer-Verlag, Berlin, 1999)
D. Delpini, Modeling and Analysis of Financial Time Series beyond Geometric Brownian Motion, PhD Thesis, Università degli Studi di Pavia (2010)
G. Bormetti, D. Delpini, Phys. Rev. E 81, 032102 (2010)
D. Delpini, G. Bormetti, Phys. Rev. E 83, 041111 (2011)
D. Delpini, G. Bormetti, Quant. Financ. 15, 15971608 (2015)
R. Vein, P. Dale, Determinants and Their Applications in Mathematical Physics (Springer-Verlag, Berlin, 1999)
M. Janjić, J. Integer Seq. 15, 12.3.5 (2012)
P.E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations (Springer Science and Business Media, 2013)
R.C. Merton, J. Econ. Theory 3, 373 (1971)
J.D. Farmer, L. Gillemot, F. Lillo, S. Mike, A. Sen, Quant. Financ. 4, 383 (2004)
S. Ciliberti, J. Bouchaud, M. Potters, Wilmott J. 1, 87 (2009)
O. Peters, Quant. Financ. 11, 1593 (2011)
E.O. Thorp, in Handbook of Asset and Liability Management: Theory and Methodology (Elsevier, 2006)
J. Ben Geloun, F. Caravelli, arXiv:1512.02278
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Caravelli, F., Mansour, T., Sindoni, L. et al. On moments of the integrated exponential Brownian motion. Eur. Phys. J. Plus 131, 245 (2016). https://doi.org/10.1140/epjp/i2016-16245-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/i2016-16245-9