Martingale Methods in the Problem of Existence of Survival Strategies

Dedicated to Albert Nikolaevich Shiryaev on the occasion of his 90th birthday We consider a model of a financial market with endogenous asset prices. An agent's strategy is called survival if its share of the total market wealth is bounded away from zero over the infinite time horizon regardless of the strategies used by the other agents. We obtain necessary and sufficient conditions for survival of strategies, which generalize known results that deal with construction of such strategies or only with necessary conditions for survival. The proofs rely mainly on martingale convergence theorems.