Infinite ergodicity for geometric Brownian motion

Abstract

Geometric Brownian motion is an exemplary stochastic processes obeying multiplicative noise, with widespread applications in several fields, e.g. in finance, in physics and biology. The definition of the process depends crucially on the interpretation of the stochastic integrals which involves the discretization parameter $\alpha$ with $0\le \alpha \le 1$ , giving rise to the well-known special cases $\alpha =0$ (Itô), $\alpha =1/2$ (Fisk-Stratonovich) and $\alpha =1$ (Hänggi-Klimontovich or anti-Itô). In this paper we study the asymptotic limits of the probability distribution functions (PDFs) of geometric Brownian motion and some related generalizations. We establish the conditions for the existence of normalizable asymptotic distributions depending on the discretization parameter $\alpha$. Using the infinite ergodicity approach, recently applied to stochastic processes with multiplicative noise by E. Barkai and collaborators, we show how meaningful asymptotic results can be formulated in a transparent way.

arXiv comment12 pages, 8 figures

References in corpus (12)

• Anomalous diffusion and ergodicity breaking in heterogeneous diffusion processes
• Infinite Ergodic Theory for Heterogeneous Diffusion Processes
• Random Walks in Logarithmic and Power-Law Potentials, Nonuniversal Persistence, and Vortex Dynamics in the Two-Dimensional XY Model
• Generalized Fokker-Planck equation: Derivation and exact solutions
• Generalised geometric Brownian motion: Theory and applications to option pricing
• From Non-normalizable Boltzmann-Gibbs statistics to infinite-ergodic theory
• First-passage and first-exit times of a Bessel-like stochastic process
• Fourier's law from a chain of coupled planar harmonic oscillators under energy conserving noise
• Infinite ergodic theory meets Boltzmann statistics
• Instantons and the path to intermittency in turbulent flows
• Active noise-driven particles under space-dependent friction in one dimension
• Multiplicative Noise: Applications in Cosmology and Field Theory

Cited by in corpus (4)

• Short-time expansion of one-dimensional Fokker-Planck equations with heterogeneous diffusion
• The effective diffusion constant of stochastic processes with spatially periodic noise
• Explicit noise and dissipation operators for quantum stochastic thermodynamics
• The stability of the multivariate geometric Brownian motion as a bilinear matrix inequality problem