Share This Article:

Super-Diffusive Noise Source in Asset Dynamics

Full-Text HTML XML Download Download as PDF (Size:161KB) PP. 53-58
DOI: 10.4236/jmf.2013.31004    3,227 Downloads   5,159 Views  


Given an asset with value St, we revisit the Black and Scholes dynamics  when the driving noise ξt is a non-Gaussian super-diffusive stochastic process with variance of the type . This super-diffusive quadratic variance behavior, synthesizes a ballistic component which would occur in strongly fluctuating environments. When , the assets can, with high probability, be driven towards the bankruptcy . This extra dynamic feature significantly affects the management of an optimal portfolio. In this context, we focus on basic decisions like: 1) determine the optimal level to sell the asset; 2) determine how to balance a portfolio which incorporates such a high volatility asset; and 3) when facing incertitudes on the assets growth rate μ, construct an optimal adaptive portfolio control. In all mentioned cases and despite the presence of this highly non-Gaussian noise source, we are able to deliver simple exact and fully explicit optimal control rules.


Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

M. Hongler, "Super-Diffusive Noise Source in Asset Dynamics," Journal of Mathematical Finance, Vol. 3 No. 1, 2013, pp. 53-58. doi: 10.4236/jmf.2013.31004.


[1] O. E. Barndorff-Nielsen and N. Shepard, “Non-Gaussian Ornstein-Uhlenbeck-Based Models and Some of Their Uses in Financial Economics,” Journal of the Royal Statistical Society, Vol. 63, No. 2, 2001, pp. 167-241.
[2] R. C. Dalang and M.-O. Hongler, “The Right Time to Sell a Stock Whose Price Is Driven by Markovian Noise,” Annals of Applied Probability, Vol. 14, No. 4, 2004, pp. 2176-2201. doi:10.1214/105051604000000747
[3] M.-O. Hongler, R. Filliger and P. Blanchard, “Soluble Models for a Dynamics Driven by Super-Diffuisve Noise,” Physica A: Statistical Mechanics and Its Applications, Vol. 370, No. 2, 2006, pp. 301-315.
[4] M.-O. Hongler, R. Filliger, P. Blanchard and J. Rodriguez, “On Stochastic Processes Driven by Ballistic Noise Sources,” In: A. Adhikari, M. R. Adhikari and Y. P. Chaubey, Eds., Contemporary Topics in Mathematics and Statistics with Applications, Asian Books Private Ltd., New Delhi, 2003.
[5] I. Benjamini and S. Lee, “Conditional Diffusions Which Are Brownian Bridges,” Journal of Theoretical Probability, Vol. 10, No. 3, 1997, pp. 733-736. doi:10.1023/A:1022657828923
[6] L. C. G. Rogers and J. W. Pitman, “Markov Functions,” Annals of Probability, Vol. 9, No. 4, 1981, pp. 573-582. doi:10.1214/aop/1176994363
[7] B. Oksendal, “Stochastic Differential Equations—An Introduction with Applications,” Springer, 1998.
[8] I. Karatzas, “Adaptive Control of a Diffusion to a Goal and a Parabolic Monge-Ampère Equation,” The Asian Journal of Mathematics, Vol. 1, No. 2, 1997, pp. 295313.
[9] M. Kulldorff, “Optimal Control of Favorable Games with a Time Limit,” SIAM Journal on Control and Optimization, Vol. 31, No. 1, 1993, pp. 52-69. doi:10.1137/0331005
[10] R. Filliger and M.-O. Hongler, “Explicit Gittins’ Indices for a Class of Super-Diffusive Processes,” Journal of Applied Probability, Vol. 44, No. 2, 2007, pp. 554-559. doi:10.1239/jap/1183667421

comments powered by Disqus

Copyright © 2018 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.