Super-Diffusive Noise Source in Asset Dynamics

Given an asset with value , we revisit the Black and Scholes dynamics t S   d log d t S t t         when the driving noise t  is a non-Gaussian super-diffusive stochastic process with variance of the type 2 t t   . 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 0  S  , 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.