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.

has been cited by the following article:

• TITLE: Super-Diffusive Noise Source in Asset Dynamics

  AUTHORS: Max-Olivier Hongler

  KEYWORDS: Black-Scholes Dynamics; Non-Gaussian Volatility; Optimal Stopping; Adaptive Optimal Control; Exact Solutions

  JOURNAL NAME: Journal of Mathematical Finance, Vol.3 No.1, February 26, 2013

  ABSTRACT: Given an asset with value St, we revisit the Black and Scholes dynamicswhen the driving noise ξtis 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 asset’s 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.