Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sampling inverse subordinators and subdiffusions

Ivan Biočić, Daniel E. Cedeño-Girón, Bruno Toaldo

TL;DR

The paper develops exact sampling methods for the finite-dimensional distributions of inverse subordinators and their undershoot/overshoot counterparts, enabling exact simulation of time-changed Feller processes without discretization bias. It establishes Markov/Hunt properties to justify pathwise sampling via dedicated algorithms, and it provides comprehensive complexity analyses. For functionals of time-changed processes, the authors prove central limit and Berry–Esseen bounds for Monte Carlo estimators, and they also treat approximate Euler–Maruyama schemes with explicit strong-error rates when exact sampling is unavailable. The framework integrates CTRW, nonlocal time operators, and subdiffusive dynamics, with practical impact on modeling weak ergodicity breaking and related anomalous diffusion phenomena through tractable, provably accurate simulations and estimators.

Abstract

In this paper, a method to exactly sample the trajectories of inverse subordinators (in the sense of the finite-dimensional distributions), jointly with the undershooting or overshooting process, is provided. The method applies to general strictly increasing subordinators. The (random) running times of these algorithms have finite moments and explicit bounds for the expectations are provided. Additionally, the Monte Carlo approximation of functionals of subdiffusive processes (in the form of time-changed Feller processes) is considered where a central limit theorem and the Berry-Esseen bounds are proved. The approximation of time-changed Itô diffusions is also studied. The strong error, as a function of the time step, is explicitly evaluated demonstrating the strong convergence, and the algorithm's complexity is provided. The Monte Carlo approximation of functionals and its properties for the approximate method is studied as well. An application of our algorithms in the context of weak ergodicity breaking of subdiffusion is also discussed.

Sampling inverse subordinators and subdiffusions

TL;DR

The paper develops exact sampling methods for the finite-dimensional distributions of inverse subordinators and their undershoot/overshoot counterparts, enabling exact simulation of time-changed Feller processes without discretization bias. It establishes Markov/Hunt properties to justify pathwise sampling via dedicated algorithms, and it provides comprehensive complexity analyses. For functionals of time-changed processes, the authors prove central limit and Berry–Esseen bounds for Monte Carlo estimators, and they also treat approximate Euler–Maruyama schemes with explicit strong-error rates when exact sampling is unavailable. The framework integrates CTRW, nonlocal time operators, and subdiffusive dynamics, with practical impact on modeling weak ergodicity breaking and related anomalous diffusion phenomena through tractable, provably accurate simulations and estimators.

Abstract

In this paper, a method to exactly sample the trajectories of inverse subordinators (in the sense of the finite-dimensional distributions), jointly with the undershooting or overshooting process, is provided. The method applies to general strictly increasing subordinators. The (random) running times of these algorithms have finite moments and explicit bounds for the expectations are provided. Additionally, the Monte Carlo approximation of functionals of subdiffusive processes (in the form of time-changed Feller processes) is considered where a central limit theorem and the Berry-Esseen bounds are proved. The approximation of time-changed Itô diffusions is also studied. The strong error, as a function of the time step, is explicitly evaluated demonstrating the strong convergence, and the algorithm's complexity is provided. The Monte Carlo approximation of functionals and its properties for the approximate method is studied as well. An application of our algorithms in the context of weak ergodicity breaking of subdiffusion is also discussed.

Paper Structure

This paper contains 16 sections, 20 theorems, 201 equations, 6 figures, 7 algorithms.

Table of Contents

  1. Introduction
  2. Main results and structure of the paper
  3. Exact simulation of paths of the inverse, undershoot, and overshoot of a subordinator
  4. Exact sampling and computational complexity
  5. Sampling $L_t$, $\gamma_t$ and $\Gamma_t$ at single times, and computational complexity
  6. Parsimonious algorithms for $(A_{L_t}, \Gamma_{t})$ and $(A_{L_t}, \gamma_{t})$
  7. Computational complexity
  8. Exact simulation of time-changed processes and Monte Carlo method
  9. Monte Carlo approximations
  10. Isotropic Lévy processes
  11. Approximate simulation of time-changed processes
  12. Monte Carlo approximations of time-changed diffusion processes
  13. Numerical examples
  14. The limit distribution
  15. Subdiffusion and Weak Ergodicity Breaking
  16. ...and 1 more sections

Key Result

Theorem 2.1

The process $(A_{L_t}, \gamma_t),\, t\geq0$, is a homogeneous Markov process with the transition probabilities satisfying for all $x\in \mathbb{R}$ and $v>0$.

Figures (6)

  • Figure 1: See Remark \ref{['1644']} for details of these boxplot comparisons.
  • Figure 2: Paths for the $\alpha$-stable inverse subordinator and its tempered version in a time interval from $0$ to $2.5$ with $10^3$ equidistant steps with $\alpha =0.75$, $q=1$, and $x=v=r=0$. The green, orange, and blue colors stand for the inverse subordinators, age process (remaining lifetime), and undershooting (overshooting) process, respectively. Time is displayed in black.
  • Figure 3: Sample of the $\alpha$-stable inverse subordinator, the remaining lifetime, the overshoot, the age and the undershoot processes through Algorithm \ref{['alg:triplet']}, left figure, while the right one stands for the tempered version; the parameters are $\alpha=0.75$ and $q=1$.
  • Figure 4: Time-changed Brownian motion. The Brownian motion $B_t$ shown on the top-left is time-changed with the tempered $\alpha$-stable inverse subordinator of the previous figure, i.e. Figure \ref{['Fig1']} (A) right-hand side, and shown in the top-right. The bottom figures show the same Brownian motion time-changed with the undershooting and overshooting processes (left and right side, respectively) of Figure \ref{['Fig1']} (A) and (B) on the right-hand side, respectively.
  • Figure 5: 500 independent error \ref{['31128']} samples for two different diffusions, with a desired tolerance error $\varepsilon=10^{-1}$. Each point represents the maximum error obtained by a sample and its corresponding time point at which the error occurred.
  • ...and 1 more figures

Theorems & Definitions (56)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Remark 2.6
  • ...and 46 more