Gambler’s ruin estimates on finite inner uniform domains

Abstract In this paper, we will consider an optimal shape problem of heat insulation introduced by [D. Bucur, G. Buttazzo and C. Nitsch, Two optimization problems in thermal insulation, Notices Amer. Math. Soc. 64 (2017), 8, 830–835]. We will establish the existence of optimal shapes in the class of 𝑀-uniform domains. We will also show that balls are stable solutions of the optimal heat insulation problem.

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n-point, win-by-k games

Journal of Applied Probability ◽

10.2307/3214385 ◽

1989 ◽

Vol 26 (4) ◽

pp. 807-814 ◽

Cited By ~ 7

Author(s):

Kyle Siegrist

Keyword(s):

Exit Time ◽

Bernoulli Trials ◽

Expected Number ◽

First Exit Time ◽

World Series ◽

Time Space ◽

Gambler’S Ruin ◽

Gambler's Ruin Problem ◽

Positive Integers ◽

Made In

Consider a sequence of Bernoulli trials between players A and B in which player A wins each trial with probability p∈ [0, 1]. For positive integers n and k with k ≦ n, an (n, k) contest is one in which the first player to win at least n trials and to be ahead of his opponent by at least k trials wins the contest. The (n, 1) contest is the Banach match problem and the (n, n) contest is the gambler's ruin problem. Many real contests (such as the World Series in baseball and the tennis game) have an (n, 1) or an (n, 2) format. The (n, k) contest is formulated in terms of the first-exit time of the graph of a random walk from a certain region of the state-time space. Explicit results are obtained for the probability that player A wins an (n, k) contest and the expected number of trials in an (n, k) contest. Comparisons of (n, k) contests are made in terms of the probability that the stronger player wins and the expected number of trials.

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Tests of randomness by the gambler’s ruin algorithm

Applied Mathematics and Computation ◽

10.1016/j.amc.2007.09.060 ◽

2008 ◽

Vol 199 (1) ◽

pp. 195-210 ◽

Cited By ~ 3

Author(s):

Chihurn Kim ◽

Geon Ho Choe ◽

Dong Han Kim

Keyword(s):

Gambler’S Ruin

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On Removability Properties of $$\psi $$ ψ -Uniform Domains in Banach Spaces

Complex Analysis and Operator Theory ◽

10.1007/s11785-016-0566-z ◽

2016 ◽

Vol 11 (1) ◽

pp. 35-55

Author(s):

M. Huang ◽

M. Vuorinen ◽

X. Wang

Keyword(s):

Banach Spaces ◽

Uniform Domains

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On a non-homogeneous difference equation from probability theory

Tatra Mountains Mathematical Publications ◽

10.2478/v10127-009-0027-4 ◽

2009 ◽

Vol 43 (1) ◽

pp. 81-90 ◽

Cited By ~ 1

Author(s):

Jean-Luc Guilbault ◽

Mario Lefebvre

Keyword(s):

Probability Theory ◽

Markov Chain ◽

Difference Equation ◽

Mathematical Expectation ◽

Transition Probabilities ◽

Current State ◽

Constant Coefficients ◽

Gambler’S Ruin ◽

Gambler's Ruin Problem ◽

Ruin Problem

Abstract The so-called gambler’s ruin problem in probability theory is considered for a Markov chain having transition probabilities depending on the current state. This problem leads to a non-homogeneous difference equation with non-constant coefficients for the expected duration of the game. This mathematical expectation is computed explicitly.

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Bi-directional grid absorption barrier constrained stochastic processes with applications in finance & investment

Risk Governance and Control Financial Markets & Institutions ◽

10.22495/rgcv10i3p2 ◽

2020 ◽

Vol 10 (3) ◽

pp. 20-33

Author(s):

Aldo Taranto ◽

Shahjahan Khan

Keyword(s):

Financial Markets ◽

Distributional Analysis ◽

Portfolio Risk ◽

Expected Number ◽

Short Term ◽

Probability Of Ruin ◽

Loss Recovery ◽

Risk Optimization ◽

Gambler’S Ruin ◽

Gambler's Ruin Problem

Whilst the gambler’s ruin problem (GRP) is based on martingales and the established probability theory proves that the GRP is a doomed strategy, this research details how the semimartingale framework is required for the grid trading problem (GTP) of financial markets, especially foreign exchange (FX) markets. As banks and financial institutions have the requirement to hedge their FX exposure, the GTP can help provide a framework for greater automation of the hedging process and help forecast which hedge scenarios to avoid. Two theorems are adapted from GRP to GTP and prove that grid trading, whilst still subject to the risk of ruin, has the ability to generate significantly more profitable returns in the short term. This is also supported by extensive simulation and distributional analysis. We introduce two absorption barriers, one at zero balance (ruin) and one at a specified profit target. This extends the traditional GRP and the GTP further by deriving both the probability of ruin and the expected number of steps (of reaching a barrier) to better demonstrate that GTP takes longer to reach ruin than GRP. These statistical results have applications into finance such as multivariate dynamic hedging (Noorian, Flower, & Leong, 2016), portfolio risk optimization, and algorithmic loss recovery.

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