The Study Of Two Classes Of Special Martingales And Their Applications

This paper mainly studies the theories of set-valued martingales and strictlocal martingales, we discuss the applications of the two class of specialmartingales.First of all, we briefly introduce the development of set-valuedmartingales and strict local martingales. In the study of set-valued martingales andits application, we presents the definitions and the necessary results aboutcontinuous set-valued martingales, set-valued square integrable martingales, localset-valued martingales and set-valued semi-martingales, based on this we analysisthe connection of these results. We discuss stochastic integral with respect to set-valued square integrable martingales, illustrate why we should consider this kind ofstochastic integral, give the defnition of stochastic integral of a stochastic processwith respect to a set-valued square integrable martingale and the representationtheorem of this kind of integrals and prove that the stochastic integral is a set-valued sub-martingale. In the study of strict local martingales and its application,we introduce what are strict local martingales, discuss the differences and relationsof martingales, local martingales and strict local martingale, present the defaultfunction which plays an important role in judging whether a local martingale is astrict local martingale or not. We study a BSDE whose terminal condition is anintegrable function of a local martingale. When the local martingale is a strict localmartingale, the BSDE admits at least two different solutions. These two differentBSDE solutions generate different viscosity solutions to the associated quasi-linearpartial differential equation. In the last part, we study the application of strict localmartingales in financial modeling, mainly discuss the the economic plausibility ofstrict local martingales in financial modeling.