Simulation of a Lévy process, its extremum, and hitting time of the extremum via characteristic functions

ArXiv ID: 2312.03929 “View on arXiv”

Authors: Unknown

Abstract

We suggest a general framework for simulation of the triplet $(X_T,\bar X_ T,τ_T)$ (Lévy process, its extremum, and hitting time of the extremum), and, separately, $X_T,\bar X_ T$ and pairs $(X_T,\bar X_ T)$, $(\bar X_ T,τ_T)$, $(\bar X_ T-X_T,τ_T)$, via characteristic functions and conditional characteristic functions. The conformal deformations technique allows one to evaluate probability distributions, joint probability distributions and conditional probability distributions accurately and fast. For simulations in the far tails of the distribution, we precalculate and store the values of the (conditional) characteristic functions on multi-grids on appropriate surfaces in $C^n$, and use these values to calculate the quantiles in the tails. For simulation in the central part of a distribution, we precalculate the values of the cumulative distribution at points of a non-uniform (multi-)grid, and use interpolation to calculate quantiles.

Keywords: Lévy processes, Conditional characteristic functions, Conformal deformations, Multi-grid interpolation, Extreme value theory, Equities (General)

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats
• Why: The paper is heavily theoretical, involving advanced stochastic calculus (Lévy processes), complex analysis (conformal deformations, analytic continuation), and functional transforms (Wiener-Hopf factorization, Laplace-Fourier transforms), giving it a very high math complexity. However, it proposes a theoretical simulation framework without presenting actual backtests, code, or performance metrics on financial data, making it low in empirical rigor.

  flowchart TD
    A["Research Goal:<br>Simulate (X_T, \bar{"X"}_T, τ_T)<br>via Characteristic Functions"] --> B["Methodology<br>Conformal Deformations &<br>Multi-grid Interpolation"]
    B --> C{"Distribution Region?"}
    C -->|Far Tails| D["Precalc CCF on Multi-grids<br>in Complex Plane C^n"]
    C -->|Central Part| E["Precalc CDF on<br>Non-uniform Grid"]
    D --> F["Compute Quantiles via<br>Inverse Fourier Transform"]
    E --> F
    F --> G["Outcomes:<br>Accurate & Fast Simulation of<br>Extrema, Hitting Times, &<br>Joint Distributions"]