## Optimal control of favorable games with a time limit |

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0/1 utility asymptotic properties bets binomial fortunes binomial stake continuous time problem CONTROL OF FAVORABLE control variable converges to zero Cornell University Corollarv cumulative distribution function derivative of Ue discrete time problem Dubins and Savage expected utility f can according f is binomial f-At FAVORABLE GAMES Follows from theorem fortune f fortune is binary gamble Inequalitv initial fortune integer intuition k-1 losses Kelly criterion left to play lemma Leo Breiman 1961 Let F Let Q(f,t lim Vt limit martingale maximize the expected OPTIMAL CONTROL optimal stake optimal strategy probability measure probability to reach process dXt Proof Proposition random walk Xt Red and Black rt-k section three show that U(f,t standard Brownian motion standard normal distribution strategy is binary supermartingale trivially true true for t=0 U(fj U(gj Ue Xt upper bound Vt r 1-r w)k wt-k want to show Yt-i zero uniformly