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Inequalities for Stochastic Processes: How to Gamble If You Must
Posted on 2010-04-10
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{\it L. E. Dubins} and {\it L. J. Savage} [Inequalities for stochastic processes. How to gamble if you must. (1976; Zbl 0359.60002)] gambling problem ${\cal E}=(X,\Gamma,u)$, where X is a set, u: $X\to {\bbfR}$, while $\Gamma$ maps X into sets of finitely additive measures on X; ${\cal E}$ is called analytic if X is a Borel set, the graph of $\Gamma$ is an analytic set (in $X\times Prob(X))$ while u is bounded and upper analytic (i.e. $\{x\vert u(x)>\alpha \}$ is analytic for every real $\alpha)$; u($\sigma$,t) denotes the expected utility under the choice of strategy $\sigma =(\sigma\sb 0,\sigma\sb 1,...)$ (which means $\sigma\sb n(x\sb 1,...,x\sb n)\in \Gamma (x\sb n)$ and stopping time t. The authors deal with the functions $$ V(x):=\sup \{\limsup\sb tu(\sigma,t)\vert \quad \sigma \text{ is a strategy available at }x\} $$ and $V\sb M(x)$ defined as above but with the choice of strategies restricted to universally measurable functions. \ Related Articles
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