### A biased roulette

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We present a very simple proof of the existence of the value for 'Big Match' first shown by Blackwell and Ferguson (1968).

Let ${\left({M}_{n}\right)}_{n\ge 0}$ be a zero-mean martingale with canonical filtration ${\left({\mathcal{F}}_{n}\right)}_{n\ge 0}$ and stochastically ${L}_{2}$-bounded increments ${Y}_{1},{Y}_{2},...,$ which means that $P\left(\right|{Y}_{n}|>t|{\mathcal{F}}_{n-1})\le 1-H\left(t\right)$ a.s. for all n ≥ 1, t > 0 and some square-integrable distribution H on [0,∞). Let ${V}^{2}={\sum}_{n\ge 1}E\left({Y}_{n}^{2}\right|{\mathcal{F}}_{n-1})$. It is the main result of this paper that each such martingale is a.s. convergent on V < ∞ and recurrent on V = ∞, i.e. $P({M}_{n}\in [-c,c]i.o.|V=\infty )=1$ for some c > 0. This generalizes a recent result by Durrett, Kesten and Lawler [4] who consider the case of only finitely many square-integrable increment distributions....

It was shown in [2] that a Langevin process can be reflected at an energy absorbing boundary. Here, we establish that the law of this reflecting process can be characterized as the unique weak solution to a certain second order stochastic differential equation with constraints, which is in sharp contrast with a deterministic analog.

This paper develops and analyzes a time-dependent optimal stopping problem and its application to the decision making process concerning organ transplants. Offers (organs for transplant) appear at jump times of a Poisson process. The values of the offers are i.i.d. random variables with a known distribution function. These values express the degree of histocompatibility between the donor and the recipient. The sequence of offers is independent of the jump times of the Poisson process. The decision...