APPLICATIONES MATHEMATICAE 39,4 (2012), pp. 379–412 Agnieszka Rygiel (Tarnów) Lukasz Stettner (Warszawa) ARBITRAGE FOR SIMPLE STRATEGIES Abstract. Various aspects of arbitrage on finite horizon continuous time markets using simple strategies consisting of a finite number of transactions are studied. Special attention is devoted to transactions without shortselling, in which we are not allowed to borrow assets. The markets without or with proportional transaction costs are considered. Necessary and sufficient con- ditions for absence of arbitrage are shown. 1. Introduction. Consider a market consisting of d risky assets with the prices given by an R d -valued adapted process X =(X t ) t∈[0,T ] with strictly positive components, and of a bank account. We will assume for simplic- ity that the bank interest rate is equal to zero and we shall use the nota- tion ¯ X t = (1,X t ) ∈ R d+1 for t ∈ [0,T ], where 1 stands for a unit amount in a bank account. Let (Ω, F , P, F =(F t ) t∈[0,T ] ) be a filtered probability space satisfying the usual conditions, i.e. the filtration (F t ) t∈[0,T ] is right continuous, and F 0 contains all the P -null sets of F . Assume that the set of trading dates is a set of F-stopping times {τ i : i =1,...,n} such that 0 ≤ τ 1 ≤···≤ τ n ≡ T and n ≥ 2. We admit only a finite number of trans- actions over a finite time horizon T , which is bounded by a deterministic number n ≥ 2. In other words every admissible strategy should consist of a finite number of transactions, and for every strategy there is a determinis- tic integer n ≥ 2 which bounds the number of transactions. Let the vector θ i =(x i ,y i ) := (x i ,y 1 i ,...,y d i ) ∈ R d+1 be the position of the investor after transactions at time τ i , where x i is the position in a bank account and y j i is the number of j th risky assets held in the portfolio. We allow y j i to have an arbitrary sign (a negative quantity means shortselling), and assume it is F τ i -measurable, i.e. determined using information available at time τ i . The 2010 Mathematics Subject Classification : Primary 91G10; Secondary 60G99, 91B26. Key words and phrases : simple trading strategies, arbitrage, shortsale restrictions. DOI: 10.4064/am39-4-1 [379] c  Instytut Matematyczny PAN, 2012