On the area swept by a biased diffusion till its first-exit time: Martingale approach and gambling opportunities

Abstract

Using martingale theory, we compute, in very few lines, exact analytical expressions for various first-exit-time statistics associated with one-dimensional biased diffusion. Examples include the distribution for the first-exit time from an interval, moments for the first-exit site, and functionals of the position, which involve memory and time integration. As a key example, we compute analytically the mean area swept by a biased diffusion until it escapes an interval that may be asymmetric and have arbitrary length. The mean area allows us to derive the hitherto unexplored cross-correlation function between the first-exit time and the first-exit site, which vanishes only for exit problems from symmetric intervals. As a colophon, we explore connections of our results with gambling, showing that betting on the time-integrated value of a losing game it is possible to design a strategy that leads to a net average win.

14 pages, 5 figures

References in corpus (17)

• Colloquium. Quantum Fluctuation Relations: Foundations and Applications
• New paradoxical games based on Brownian ratchets
• Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces
• Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas
• Decision Making in the Arrow of Time
• Two refreshing views of Fluctuation Theorems through Kinematics Elements and Exponential Martingale
• Thermodynamics of Gambling Demons
• On the area under a continuous time Brownian motion till its first-passage time
• Integral Fluctuation Relations for Entropy Production at Stopping Times
• The first-passage area for drifted Brownian motion and the moments of the Airy distribution
• Universal tradeoff relation between speed, uncertainty, and dissipation in nonequilibrium stationary states
• Ranking influential nodes in networks from partial information
• Protein logic: a statistical mechanical study of signal integration at the single-molecule level
• First-Passage Duality
• Methods and Conversations in (Post)Modern Thermodynamics
• Statistics of the first passage time of Brownian motion conditioned by maximum value or area
• Large deviations of the stochastic area for linear diffusions

Cited by in corpus (2)

• Rare Events and Single Big Jump Effects in Ornstein-Uhlenbeck Processes
• Martingale approach for first-passage problems of time-additive observables in Markov processes