We study capital process behavior in the fair-coin and biased-coin games in the framework of the game-theoretic probability of Shafer and Vovk (2001). We show that if Skeptic uses a Bayesian strategy with a beta prior, the capital process is lucidly expressed in terms of the past average of Reality's moves. From this it is proved that the Skeptic's Bayesian strategy weakly forces the strong law of large numbers (SLLN) with the convergence rate of O( log n/n) and if Reality violates SLLN then the exponential growth rate of the capital process is very accurately described in terms of the Kullback divergence between the average of Reality's moves when she violates SLLN and the average when she observes SLLN. We also investigate optimality properties associated with Bayesian strategy.Recently [6], motivated by Takeuchi's works ([14], [15]), proved that a very simple single strategy, based only on the past average of Reality's moves, is weakly forcing SLLN with the convergence rate of O( log n/n), which is a substantial improvement over the original strategy of Shafer and Vovk. Versions of SLLN for unbounded moves by Reality is obtained in Kumon, Takemura and Takeuchi (2007) [7].In this paper for general biased-coin games, we consider a class of Bayesian strategies for Skeptic. As in we prove that Bayesian strategies in the class weakly force SLLN with the convergence rate of O( log n/n). Furthermore we establish the important fact that if Skeptic uses a Bayesian strategy and Reality violates SLLN, then the exponential growth rate of the Skeptic's capital process is very accurately described in terms of the Kullback divergence between the average of Reality's moves when she violates SLLN and the average when she observes SLLN.In the protocol of the coin-tossing game of Shafer and Vovk (2001), there is no probabilistic assumption on the behavior of Reality. In the games, Skeptic tries to become rich and Reality tries to prevent it. However in the Bayesian strategy, Skeptic simply and naively assumes that Reality behaves probabilistically and for choosing his moves Skeptic uses the Bayesian prediction of Reality's moves. It is a remarkable fact that this naive Bayesian prediction by Skeptic actually works and forces SLLN even if Reality's moves are not probabilistic at all and Reality tries to beat Skeptic as an adversary. Furthermore Skeptic achieves an optimal growth rate if Reality violates SLLN in a way accounted for by the prior. As in the standard statistical decision theory (e.g. Berger [2] and Robert [8]), this optimality is inherent in Bayesian procedures. However in the setting of the present paper, only a very simple protocol of the game is assumed and no other modeling assumptions are made on Reality's moves. In this sense, we believe that the optimality considered in this paper has much broader conceptual implications than those offered by the standard statistical decision theory.The organization of this paper is as follows. In Section 2 we formulate coin-tossing games and set up some notations. In Section...

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We introduce a new formulation of asset trading games in continuous time in the framework of the game-theoretic probability established by Shafer and Vovk (Probability and Finance: It's Only a Game! (2001) Wiley). In our formulation, the market moves continuously, but an investor trades in discrete times, which can depend on the past path of the market. We prove that an investor can essentially force that the asset price path behaves with the variation exponent exactly equal to two. Our proof is based on embedding high-frequency discrete-time games into the continuous-time game and the use of the Bayesian strategy of Kumon, Takemura and Takeuchi (Stoch. Anal. Appl. 26 (2008) 1161-1180) for discrete-time coin-tossing games. We also show that the main growth part of the investor's capital processes is clearly described by the information quantities, which are derived from the Kullback-Leibler information with respect to the empirical fluctuation of the asset price.An asset trading game is a complete information game between an investor and the market. Following Chapter 9 of Shafer and Vovk [15], we denote these two players as "Investor" and "Market". In our formulation, Market moves continuously, but Investor moves in discrete times, depending on the past path of Market. The trading times of Investor need not be equally spaced. In this paper, we mainly consider "limit order" strategy (rather than the "market order" strategy) of Investor. In the limit order strategy, Investor trades a financial asset when the asset price or the increment of the asset price hits a certain level. We shall prove that by a high-frequency limit order type Bayesian strategy, Investor can essentially force the variation exponent of two in the price path of Market. The precise definition of essential forcing will be given in Section 2.In an infinitely repeated series of fair betting games, a gambler cannot make gain with certainty. This fact has been formulated and proven in the theory of martingales. But when the games are favorable to a gambler, for example, if the results of the games are stochastically independent with positive expected value, to what extent can he exploit the situation and what would be a good strategy to adopt? Several years after the advent of Shannon's celebrated work [16], this problem was first systematically studied by Kelly [9] in relation to the betting game interpretation of Shannon's mutual information quantity. In this spirit, betting games have been investigated by information theorists, which led to the notion of Cover's universal portfolios [2,3]. One of the present authors also wrote a note on it about forty years ago in Japanese, presenting the results in [18].Recently, Shafer and Vovk originated a new, attractive field of game-theoretic probability and finance [15]. The most important point concerning their approach is that stochastic behavior of Market is not assumed a priori, but follows from the protocol of the game between Investor and Market. Shafer and Vovk established the general fact that i...

We consider strong law of large numbers (SLLN) in the framework of game-theoretic probability of Shafer and Vovk (Shafer, G. and Vovk, V. 2001, Probability and Finance: It's Only a Game! (New York: Wiley)). We prove several versions of SLLN for the case that Reality's moves are unbounded. Our game-theoretic versions of SLLN largely correspond to standard measure-theoretic results. However game-theoretic proofs are different from measure-theoretic ones in the explicit consideration of various hedges. In measure-theoretic proofs existence of moments is assumed, whereas in our game-theoretic proofs we assume availability of various hedges to Skeptic for finite prices.

Azuma-Hoeffding-Bennett inequality, Capital process, Game-theoretic probability, Large deviation,