Author(s)

Nicholas S. Gonchar

Bogolyubov Institute for Theoretical Physics of NAS, Kiev, Ukraine.

In the paper, a class of discrete evolutions of risk assets having the memory is considered. For such evolutions the description of all martingale measures is presented. It is proved that every martingale measure is an integral on the set of extreme points relative to some measure on it. For such a set of evolutions of risk assets, the contraction of the set of martingale measures on the filtration is described and the representation for it is found. The inequality for the integrals from a nonnegative random value relative to the contraction of the set of martingale measure on the filtration which is dominated by one is obtained. Using these inequalities a new proof of the optional decomposition theorem for super-martingales is presented. The description of all local regular super-martingales relative to the regular set of measures is presented. The applications of the results obtained to mathematical finance are presented. In the case, as evolution of a risk asset is given by the discrete geometric Brownian motion, the financial market is incomplete and a new formula for the fair price of super-hedge is founded.

Random Process, Regular Set of Measures, Optional Doob Decomposition, Local Regular Super-Martingale, Martingale, Discrete Geometric Brownian Motion

Gonchar, N. (2019) Description of Incomplete Financial Markets for Time Evolution of Risk Assets. Advances in Pure Mathematics, 9, 567-610. doi: 10.4236/apm.2019.96029.