Information through time
Price dynamics, accumulating evidence, and learning from returns
The static framework connects a payoff to a probability distribution. Time introduces two further questions: how does new information change the price, and how does an investor adapt as outcomes arrive?
These questions share a likelihood language, but concern different objects. A price process compares two consistent probability laws through time. An adaptive allocation also specifies a learning rule, a trading rule, and the capital available for future decisions.
From one payoff to a price process
Start with two equivalent probability laws on the same evolving observations. Their likelihood-ratio process is a positive martingale under the reference law. Expressing prices in units of the reference asset incorporates the numeraire convention from the outset.
The two ways to build a market page explains the connection with the classical route: begin with price dynamics, then recover compatible pricing probabilities. The binomial model makes both directions explicit.
A companion chapter: linear information flows
Independent increments in suitable exponential families give particularly transparent models. The log price is affine in an accumulating statistic, and expected information grows linearly in the stated clock. This is a useful subclass of price models; the general payoff–probability correspondence does not require linear information flow.
The separate Linear information flows chapter contains six model sections: binomial, negative binomial, normal, Poisson, gamma, and inverse Gaussian. Its overview specifies what accumulates and which parameters stay common. The normal example recovers geometric Brownian motion; the jump examples illustrate different patterns of information arrival.
Suggested route: binomial → negative binomial → linear information flow. For continuous-time examples, start with normal / GBM, then explore the remaining families in the catalogue.
The trading clock changes what can be implemented
Observing every binary move and trading after every move is a different opportunity set from trading only before and after a completed shock. The negative-binomial example holds the underlying sequence of trials fixed and makes this distinction visible.
In the Columbia example, the resulting growth gap measures the value of the unavailable intermediate rebalancing. The comparison of three hedging criteria shows the portfolios and the numerical decomposition.
Learning and capital commitment
Under full commitment in the specified prediction-market setting, relative wealth evolves like Bayesian model weights. Partial commitment changes that recursion. Keeping capital in reserve and preserving accumulated evidence are distinct decisions.
The Bayesian learning and commitment page develops this connection, including the role of prefunded restarts. The Oxford presentation provides the broader computational context.