DescriptionImagine: you have K pounds and are invited to play a sequence of bets, each of which you can win with probability p>0.5, independently for individual attempts. Every time you play, you are free to bet any amount you want, so that if you bet £a and win, your fortune increases by £a; however, if you bet £a and lose, your fortune decreases by the size of your bet. Since the game is to your advantage, you decide, at every step, to bet a fixed fraction f of your current fortune, so that initially you bet fK pounds, and if you win, your next bet size becomes f(1+f)K pounds, and so on. What value f delivers the fastest rate of long-time growth of your fortune?The answer to this question is known as the Kelly formula or the Kelly criterion, named after J.L.Kelly, Jr., who derived the result in his famous 1956 paper. The optimal fraction f maximises the long-run geometric mean of individual outcomes, and betting the Kelly way is claimed to be popular among gamblers, investors etc. Surprisingly, a similar result was derived more than two centuries earlier by Daniel Bernoulli in his 1738 treatment of the St.Petersburg paradox. The aim of the project is to mathematically study a class of related models, in particular, to establish optimality of the Kelly criterion for a general problem of the type described above. As a natural complement, one might wish to also explore these models via numerical simulations. Prerequisites2H Probability is essential, 3H Probability is helpful.ResourcesA good starting point is the Google search or the Wiki page. You might wish to have a look at the original Kelly paper, D.Bernoulli's article, or the popular book on the history of Kelly betting by William Poundstone.Further references might be suggested once the project is underway. |
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