that Xt is observable at time t.An adapted process X is a martingale if,for any times t and s >t,we have Et(Xs)=Xt. A security is a claim to an adapted dividend process,say 6,with 6t denot- ing the dividend paid by the security at time t.Each security has an adapted security-price process S,so that St is the price of the security,ex dividend,at time t.That is,at each time t,the security pays its dividend ot and is then available for trade at the price St.This convention implies that do plays no role in determining ex-dividend prices.The cum-dividend security price at time t is S:+6t. We suppose that there are N securities defined by an RN-valued adapted dividend process 6=(6(1),...,(N)).These securities have some adapted price process S=(S(),...,S(N)).A trading strategy is an adapted process 0 in RN.Here,represents the portfolio held after trading at time t.The dividend process 6 generated by a trading strategy 0 is defined by 69=0-1…(S:+d)-0·S, (1) with“o_l”taken to be zero by convention. 2.2 Arbitrage,State Prices,and Martingales Given a dividend-price pair (S)for N securities,a trading strategy 0 is an arbitrage if 60>0(that is,if 60>0 and 60).An arbitrage is thus a trading strategy that costs nothing to form,never generates losses,and, with positive probability,will produce strictly positive gains at some time. One of the precepts of modern asset pricing theory is a notion of efficient markets under which there is no arbitrage.This is reasonable axiom,for in the presence of an arbitrage,any rational investor who prefers to increase his dividends would undertake such arbitrages without limit,so markets could not be in equilibrium,in a sense that we shall see more formally later in this section.We will first explore the implications of no arbitrage for the representation of security prices in terms of "state prices,"the first step toward which is made with the following result. Proposition.There is no arbitrage if and only if there is a strictly positive adapted process n such that,for any trading strategy 0, 5that Xt is observable at time t. An adapted process X is a martingale if, for any times t and s>t, we have Et(Xs) = Xt. A security is a claim to an adapted dividend process, say δ, with δt denot￾ing the dividend paid by the security at time t. Each security has an adapted security-price process S, so that St is the price of the security, ex dividend, at time t. That is, at each time t, the security pays its dividend δt and is then available for trade at the price St. This convention implies that δ0 plays no role in determining ex-dividend prices. The cum-dividend security price at time t is St + δt. We suppose that there are N securities defined by an RN -valued adapted dividend process δ = (δ(1),...,δ(N) ). These securities have some adapted price process S = (S(1),...,S(N) ). A trading strategy is an adapted process θ in RN . Here, θt represents the portfolio held after trading at time t. The dividend process δθ generated by a trading strategy θ is defined by δθ t = θt−1 · (St + δt) − θt · St, (1) with “θ−1” taken to be zero by convention. 2.2 Arbitrage, State Prices, and Martingales Given a dividend-price pair (δ, S) for N securities, a trading strategy θ is an arbitrage if δθ > 0 (that is, if δθ ≥ 0 and δθ 6= 0). An arbitrage is thus a trading strategy that costs nothing to form, never generates losses, and, with positive probability, will produce strictly positive gains at some time. One of the precepts of modern asset pricing theory is a notion of efficient markets under which there is no arbitrage. This is reasonable axiom, for in the presence of an arbitrage, any rational investor who prefers to increase his dividends would undertake such arbitrages without limit, so markets could not be in equilibrium, in a sense that we shall see more formally later in this section. We will first explore the implications of no arbitrage for the representation of security prices in terms of “state prices,” the first step toward which is made with the following result. Proposition. There is no arbitrage if and only if there is a strictly positive adapted process π such that, for any trading strategy θ, E X T t=0 πtδθ t ! = 0. 5