Jean is not happy with the winnings he can make when applying the 'doubling strategy'. Hence, under otherwise the same assumptions and notations as in example 10.6, he triples his bet size after every lost game, starting again with \(€ 1\).

(1) What is his winnings when he loses 5 games in a row and wins the 6th one?

(2) Is \(\left\{X_{1}, X_{2}, \ldots\right\}\) a martingale?

Data from Example 10.6

Transcribed Image Text:

Example 10.6 The structure of X given by (10.14) includes as a special case the winnings (losses) development when applying the doubling strategy: Jean bets 1 on the first game. If he wins, he gets 1. If he loses, his 'winning' are - 1. Hoping to equalize the loss, Jean will bet 2 on the next game. If he wins, he will get 2 and, hence, will have made total winnings of 1. But if he loses he will have total 'winn- ings' of -3, and will bet 4 on the next game and so on. After the first win Jean stops gambling. The following table shows the losses (winnings) development of Jean if he loses 5 times in a row and then wins: game 1 2 3 4 5 6 result loss loss loss loss loss win bet 1 2 4 8 16 32 'total winnings' -1 -3 -7 -15 -31 +1