Abstract
In the paper, the martingales and super-martingales relative to a convex set of equivalent measures are systematically studied. The notion of local regular super-martingale relative to a convex set of equivalent measures is introduced and the necessary and sufficient conditions of the local regularity of it in the discrete case are founded. The description of all local regular super-martingales relative to a convex set of equivalent measures is presented. The notion of the complete set of equivalent measures is introduced. We prove that every bounded in some sense super-martingale relative to the complete set of equivalent measures is local regular. A new definition of the fair price of contingent claim in an incomplete market is given and the formula for the fair price of Standard Option of European type is found. The proved Theorems are the generalization of the famous Doob decomposition for super-martingale onto the case of super-martingales relative to a convex set of equivalent measures.
Highlights
In the paper, a new method of investigation of martingales and super-martingales relative to a convex set of equivalent measures is developed
If the nonnegative random value ξ is such that sup E Pξ < ∞, fm = ess sup E P {ξ | m}, m is a m=0 super-martingale relative to the convex set of equivalent measures M
{ } fmE P {ξ | m}, m is a local regular super-martingale relative to the convex set of equivalent measures M
Summary
A new method of investigation of martingales and super-martingales relative to a convex set of equivalent measures is developed. Let {Ω, } be a measurable space with a filtration m on it and let ξ be a nonnegative integrable random value with respect to a set of equivalent measures P1, , Pk. The necessary and sufficient conditions of the local regularity of the super-martingale f m. ∞ m=0 is a nonnegative uniformly integrable super-martingale relative to a convex set of equivalent measures M, the necessary and sufficient conditions for it to be a local regular one is belonging it to the set K. We introduce the notion of complete set of equivalent measures and prove that non negative super-martingales are local regular ones with respect to this set of measures For this purpose we are needed the auxiliary statement.
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