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Analyzing Sequential Betting with a Kelly-Inspired Convective-Diffusion Equation.

Darrell Velegol1,2, Kyle J M Bishop3

  • 1Department of Chemical Engineering, Penn State University, University Park, PA 16802, USA.

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|July 26, 2024
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Summary
This summary is machine-generated.

This study models sequential betting using a convective-diffusion equation (CDE), offering a new perspective beyond the Kelly Criterion. The CDE approach provides insights into bankroll dynamics, optimal betting fractions for any quantile, and ruin probability.

Keywords:
Kelly criterionbetinnovation portfolioinvestmentruin

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Area of Science:

  • Quantitative Finance
  • Mathematical Modeling
  • Probability Theory

Background:

  • The Kelly Criterion is a strategy for determining optimal bet sizes.
  • Sequential betting involves a series of independent wagers over time.
  • Understanding bankroll dynamics and ruin probability is crucial in betting strategies.

Purpose of the Study:

  • To model a sequence of independent bets using a convective-diffusion equation (CDE).
  • To reframe the Kelly Criterion derivation as a CDE in the limit of many bets.
  • To analyze the impact of steady growth and random fluctuations on bankroll prediction.

Main Methods:

  • Modeling sequential bets with a convective-diffusion equation (CDE).
  • Utilizing a binomial distribution for wins and losses in the limit of many bets.
  • Introducing an absorbing boundary condition to model ruin.

Main Results:

  • The CDE clarifies the roles of steady growth (velocity U) and random fluctuations (diffusion coefficient D).
  • The CDE formulation allows optimization of betting fractions for any bankroll quantile, not just the median.
  • Ruin probability is linked to the dimensionless Péclet number, representing the ratio of convection to diffusion.

Conclusions:

  • Reframing the Kelly Criterion with CDE provides new analytical possibilities.
  • The CDE approach offers a powerful tool for analyzing sequential betting problems.
  • Insights from chemico-physical literature can be applied to sequential betting.