利用杠杆时，赌徒不会破产
With Leverage, the Gambler is Not the One Who Gets Ruined
AI总结：
该研究探讨引入金融杠杆后赌徒破产现象的变化，证明适度杠杆下赌徒可长期盈利不破产，给出最优策略成功概率公式，成果用于CMU随机算法课程开发。
摘要：
在本注记中，我们探讨了一个简单问题：若允许赌徒使用金融杠杆，经典的“赌徒破产”现象会如何变化？我们证明，对于理想化的赌徒，即使只进行公平赌博，只要使用适度杠杆，就有可能在极长的时间内维持看似不可能的利润，且永远不会破产。当杠杆比率L=10（低于雷曼兄弟1995-2008年的水平）时，赌徒在不破产的情况下将资金增值1000倍的概率约为50%。更一般地，在L倍杠杆下，若目标是将资金增值r>1倍，最优赌博策略的成功概率约为r^(-1/L)。本注记的结果是卡内基梅隆大学（CMU）随机算法课程开发的一部分。
英文摘要：
In this note we explore the following simple question: How does the classic \emph{gambler's ruin} phenomenon change if we allow the gambler \emph{financial leverage}? We show that, with a moderate amount of leverage, it is possible for an (idealized) gambler to sustain seemingly improbable profits over a remarkably long time horizon without ever going bankrupt, \emph{even} if the gambler makes only fair bets. With a leverage ratio of $L = 10$ (less than what Lehman Brothers Holdings had from 1995--2008), the probability of multiplying one's money by $1000\times$ without going bankrupt is roughly $50\%$. More generally, given $L$-fold leverage, and given the goal of multiplying one's money by a factor of $r > 1$, the optimal gambling strategy has probability roughly $r^{-1/L}$ of success. The results in this note are designed as part of the course development for the CMU randomized algorithms course.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
