Game-Theoretic Foundations for Probability and Finance

13Emergence of Randomness in Idealized Financial Markets

This chapter shows how Brownian motion emerges game‐theoretically in continuous time. By trading in a security in the small – trading with higher and higher frequency – a trader can multiply the capital he risks infinitely as soon as the security's price path, modulo a time change, fails to behave locally like Brownian motion. This follows from the game‐theoretic Dubins–Schwarz theorem, a central result of this chapter.

This chapter's protocol involves a single security and two players, a market and a trader. The trader moves first, announcing a trading strategy. Then the market moves, announcing a continuous price path. In Section 13.1 we explain how a strategy for the trader is constructed and how it determines his capital process. We also define the concept of instant enforcement: we say that the trader can instantly force an event if he has a strategy that multiplies the capital he risks infinitely at any time at which the event fails. In Section 13.2, we show how the trader can instantly force some of the best‐known properties of Brownian motion: the existence of quadratic variation, the absence of isolated crossings, the absence of strict monotonicity, and nondifferentiability.

In Section 13.3, we define game‐theoretic upper expected value for the chapter's protocol and state one version of the protocol's Dubins–Schwarz theorem: a variable that is unchanged when the price path is changed by the addition of a constant ...