meaning roughly that the market value of a trading strategy is,at any time, the state-price discounted expected future dividends generated by the strat- egy The gain process G for (8,S)is defined by G=+,the price plus accumulated dividend.Given a deflator y,the deflated gain process G is defined by G?We can think of deflation as a change of numeraire. Theorem.The dividend-price pair (S)admits no arbitrage if and only if there is a state-price density.A deflator n is a state-price density if and only if Sr=0 and the state-price-deflated gain process G"is a martingale. Proof:It can be shown as an easy exercise that a deflator m is a state-price density if and only if Sr=0 and the state-price-deflated gain process G"is a martingale. Suppose there is no arbitrage.Then ST =0,for otherwise the strategy a is an arbitrage when defined by 0=0,t<T,er=-ST.By the previous proposition,there is some deflator such that E0for any strategy 0. We must prove (2),or equivalently,that G"is a martingale.Doob's Optional Sampling Theorem states that an adapted process X is a martingale if and only if E(X,)=Xo for any stopping time TT.Consider,for an arbitrary security n and an arbitrary stopping time r<T,the trading strategy 0 defined by ()=0 for ktn and of)=1,t<T,with om)= 0,t≥T. Since E(∑t-ortd)=0,we have =0 implying that the m-deflated gain process G of security n satisfies Go= E(G).Since T is arbitrary,Gm.is a martingale,and since n is arbitrary, G is a martingale. This shows that absence of arbitrage implies the existence of a state-price density.The converse is easy. The proof is motivated by those of Harrison and Kreps 1979 and Harri- son and Pliska [1981]for a similar result to follow in this section regarding the notion of an "equivalent martingale measure."Ross 1987,Prisman [1985, Kabanov and Stricker [2001],and Schachermayer [2001]show the impact of taxes or transactions costs on the state-pricing model. 7meaning roughly that the market value of a trading strategy is, at any time, the state-price discounted expected future dividends generated by the strat￾egy. The gain process G for (δ, S) is defined by Gt = St + Pt j=1 δj , the price plus accumulated dividend. Given a deflator γ, the deflated gain process Gγ is defined by Gγ t = γtSt + Pt j=1 γjδj . We can think of deflation as a change of numeraire. Theorem. The dividend-price pair (δ, S) admits no arbitrage if and only if there is a state-price density. A deflator π is a state-price density if and only if ST = 0 and the state-price-deflated gain process Gπ is a martingale. Proof: It can be shown as an easy exercise that a deflator π is a state-price density if and only if ST = 0 and the state-price-deflated gain process Gπ is a martingale. Suppose there is no arbitrage. Then ST = 0, for otherwise the strategy θ is an arbitrage when defined by θt = 0, t<T, θT = −ST . By the previous proposition, there is some deflator π such that E( PT t=0 δθ t πt) = 0 for any strategy θ. We must prove (2), or equivalently, that Gπ is a martingale. Doob’s Optional Sampling Theorem states that an adapted process X is a martingale if and only if E(Xτ ) = X0 for any stopping time τ ≤ T. Consider, for an arbitrary security n and an arbitrary stopping time τ ≤ T, the trading strategy θ defined by θ(k) = 0 for k 6= n and θ (n) t = 1,t<τ , with θ (n) t = 0, t ≥ τ . Since E( PT t=0 πtδθ t ) = 0, we have E −S(n) 0 π0 +Xτ t=1 πtδ (n) t + πτS(n) τ ! = 0, implying that the π-deflated gain process Gn,π of security n satisfies Gn,π 0 = E (Gn,π τ ). Since τ is arbitrary, Gn,π is a martingale, and since n is arbitrary, Gπ is a martingale. This shows that absence of arbitrage implies the existence of a state-price density. The converse is easy. The proof is motivated by those of Harrison and Kreps [1979] and Harri￾son and Pliska [1981] for a similar result to follow in this section regarding the notion of an “equivalent martingale measure.” Ross [1987], Prisman [1985], Kabanov and Stricker [2001], and Schachermayer [2001] show the impact of taxes or transactions costs on the state-pricing model. 7