APPLICATIONES MATHEMATICAE 44,1 (2017), pp. 33–55 Agnieszka Rygiel (Kraków) Lukasz Stettner (Warszawa) REMARKS ON SIMPLE ARBITRAGE ON MARKETS WITH BID AND ASK PRICES Abstract. We consider various kinds of simple investment strategies on markets with bid and ask prices. We formulate necessary and sufficient con- ditions for the absence of arbitrage using those strategies. In the last part of the paper we study the absence of arbitrage for simple strategies without shortselling. 1. Introduction. 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 there exist two R d -valued adapted positive stochastic processes S =(S t ) t∈[0,T ] and S = ( S t ) t∈[0,T ] , with S t (ω) ≤ S t (ω) for all t ∈ [0,T ] and ω ∈ Ω, which model the bid (selling) and ask (buying) prices of risky assets. We buy or sell risky assets during the time interval [0,T ] liquidating our position just after maturity time T at the instant which we shall denote by T + , using the prices S T and S T respectively. We consider the following different classes of trading strategies. Definition 1.1. We call an R d -valued process Θ =(Θ t ) t∈[0,T + ] a simple investment strategy if there exists a positive integer n ≥ 2 and a finite se- quence (τ i ,θ i ) n i=1 of F-stopping times τ i such that 0= τ 1 ≤···≤ τ n = T and of F τ i -measurable random vectors θ i ∈ R d for i =1,...,n − 1 with θ n =0 such that Θ T + = θ n =0 and (1.1) Θ t = n-1  i=1 θ i χ (τ i ,τ i+1 ] (t). 2010 Mathematics Subject Classification : Primary 91G10; Secondary 60G99, 91B26. Key words and phrases : simple strategies, absence of arbitrage, bid and ask prices. Received 27 June 2016; revised 6 November 2016. Published online 19 January 2017. DOI: 10.4064/am2310-11-2016 [33] c  Instytut Matematyczny PAN, 2017