Characterization of arbitrage-free markets

Abstract

The present paper deals with the characterization of no-arbitrage properties of a continuous semimartingale. The first main result, Theorem 2.1, extends the no-arbitrage criterion by Levental and Skorohod [Ann. Appl. Probab. 5 (1995) 906–925] from diffusion processes to arbitrary continuous semimartingales. The second main result, Theorem 2.4, is a characterization of a weaker notion of no-arbitrage in terms of the existence of supermartingale densities. The pertaining weaker notion of no-arbitrage is equivalent to the absence of immediate arbitrage opportunities, a concept introduced by Delbaen and Schachermayer [Ann. Appl. Probab. 5 (1995) 926–945].

Both results are stated in terms of conditions for any semimartingales starting at arbitrary stopping times σ. The necessity parts of both results are known for the stopping time σ=0 from Delbaen and Schachermayer [Ann. Appl. Probab. 5 (1995) 926–945]. The contribution of the present paper is the proofs of the corresponding sufficiency parts.