Selected problems of mathematical theory of arbitrage

As is known, the no-arbitrage principle is fundamental for mathematical finance. In the talk, along with the dual description of no-arbitrage conditions (the existence of equivalent martingale measures, equivalent supermartingale densities, strictly consistent price processes), we present computationally feasible methods for verification of market no-arbitrage properties. The list of financial models under consideration is presented below.

• 1) Finite discrete time market model with transaction costs. We consider a “constructive” no-arbitrage criterion and recurrence formulas for price bounds of contingent claims (without martingale measures and their counterparts). Here the key role is played by the martingale selection theorem, giving a criterion for the existence of a martingale selector of a prescribed set-valued stochastic process.
  
• 2) Frictionless finite discrete time model. We consider lower bounds of martingale measure densities and the relation of this bounds to the maximization of the expected value $\mathsf E X_N$ of wealth under the loss constraints $\mathsf E |X_N^-|^p$, $1\le p\le\infty$, and also to the Dalang-Morton-Willinger theorem.
  
• 3) Discrete time infinite horizon model, where the set $\mathscr W$ of stochastic wealth processes is subject to a number of axioms with financial interpretation. We obtain necessary and sufficient conditions for the existence of equivalent supermartingale densities and measures for the subset $\mathscr W_+$ of non-negative elements of $\mathscr W$.
  
• 4) Large market model (by Kabanov-Kramkov) as a sequence of the traditional “small” market models with finite number of risky assets. We show that the no-arbitrage properties of the large market are fully determined by the asymptotic behavior of the sequence of the numéraire portfolios related to small markets.