Martingale Measures For A Class of Right‐Continuous Processes

Abstract

The subject of the present paper is the following. Suppose that W is a class of adapted, right‐continuous processes on the continuous time horizon [0, 1], and for every stopping time and W, () is bounded below. A necessary and sufficient condition will be given for the existence of a probability measure Q which is equivalent to the original measure and such that each process in W is a martingale under Q. If the processes in W represent the discounted prices of available securities, then the condition given here for the existence of a martingale measure can be interpreted as absence of “free lunch” in the securities market. This is a familiar kind of theorem from the finance literature; the novelty of this paper is that the security prices are not required to be in LP for some 1 p, nor are they assumed to be continuous. Also, the concept of free lunch is invariant under the substitution of the original probability measure by an equivalent probability measure. the assumption that () is bounded below for every W and stopping time is quite natural since prices are nonnegative. We shall define a class of admissible subjective probability measures and assume that each agent in the economy has selected a subjective probability measure from (hat class. Subjective free lunch for an agent will be defined using his or her subjective probability measure. It will be shown that under an additional condition the existence of free lunch is equivalent to the existence of a common subjective free lunch simultaneously for all possible agents in the economy.

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