Continuity of the shafer-Vovk-Ville operator

Kolmogorov’s axiomatic framework is the best-known approach to describing probabilities and, due to its use of the Lebesgue integral, leads to remarkably strong continuity properties. However, it relies on the specification of a probability measure on all measurable events. The game-theoretic framework proposed by Shafer and Vovk does without this restriction. They define global upper expectation operators using local betting options. We study the continuity properties of these more general operators. We prove that they are continuous with respect to upward convergence and show that this is not the case for downward convergence. We also prove a version of Fatou’s Lemma in this more general context. Finally, we prove their continuity with respect to point-wise limits of two-sided cuts

Arbitrage Pricing of Multi-person Game Contingent Claims

We introduce a class of financial contracts involving several parties by extending the notion of a two-person game option (see Kifer (2000)) to a contract in which an arbitrary number of parties is involved and each of them is allowed to make a wide array of decisions at any time, not restricted to simply `exercising the option'. The collection of decisions by all parties then determines the contract's settlement date as well as the terminal payoff for each party. We provide sufficient conditions under which a multi-person game option has a unique arbitrage price, which is additive with respect to any partition of the contract