N ov 2 00 1 The gambler ’ s ruin problem in path representation form
نویسنده
چکیده
We analyze the one-dimensional random walk of a particle on the right-half real line. The particles starts at x = k, for k > 0, and ends up at the origin. We solve for the probabilities of absorption at the origin by means of a geometric representation of this random walk in terms of paths on a two-dimensional lattice.
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N ov 2 00 8 Proving the Regularity of the Minimal Probability of Ruin via a Game of Stopping and Control
Abstract: We consider three closely related problems in optimal control: (1) minimizing the probability of lifetime ruin when the rate of consumption is stochastic and when the individual can invest in a Black-Scholes financial market; (2) minimizing the probability of lifetime ruin when the rate of consumption is constant but the individual can invest in two risky correlated assets; and (3) a ...
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Example 1.1 (Gambler Ruin Problem). A gambler has $100. He bets $1 each game, and wins with probability 1/2. He stops playing he gets broke or wins $1000. Natural questions include: what’s the probability that he gets broke? On average how many games are played? This problem is a special case of the so-called Gambler Ruin problem, which can be modelled using a Markov chain as follows. We will b...
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Consider two gamblers A, B with initial integer fortunes a, b. Let m = a + b denote the initial sum of fortunes. In each round of a fair game, one player wins and is paid 1 by the other player: (a, b) 7 → ((a + 1, b − 1) with probability 1/2, (a − 1, b + 1) 00 Assume that rounds are independent for the remainder of this essay. The ruin probability p E for a gambler E is the probability that E's...
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