Paper: arXiv 2212.08509

Authors: Michael E. Mura

Abstract

A novel approach called Moate Simulation is presented to provide an accurate numerical evolution of probability distribution functions represented on grids arising from stochastic differential processes where initial conditions are specified. Where the variables of stochastic differential equations may be transformed via Itô-Doeblin calculus into stochastic differentials with a constant diffusion term, the probability distribution function for these variables can be simulated in discrete time steps. The drift is applied directly to a volume element of the distribution while the stochastic diffusion term is applied through the use of convolution techniques such as Fast or Discrete Fourier Transforms. This allows for highly accurate distributions to be efficiently simulated to a given time horizon and may be employed in one, two or higher dimensional expectation integrals, e.g. for pricing of financial derivatives. The Moate Simulation approach forms a more accurate and considerably faster alternative to Monte Carlo Simulation for many applications while retaining the opportunity to alter the distribution in mid-simulation.

Complexity vs Empirical Score

  • Math Complexity: 6.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper introduces an innovative numerical technique (Moate Simulation) combining Itô-Doeblin calculus, drift transforms, and FFT convolutions for density propagation. While mathematically sound and conceptually novel compared to standard Monte Carlo methods, it lacks extensive empirical backtesting and real-world market data validation.

Research Flowchart

  flowchart TD
    A[Research Goal: Fast, accurate numerical simulation of SDE probability distributions] --> B[Inputs: Initial SDEs with specified initial conditions]
    B --> C[Step 1: Transform SDEs via Itô-Doeblin Calculus to constant diffusion]
    C --> D[Step 2: Apply Drift directly to distribution volume elements]
    D --> E[Step 3: Apply Stochastic Diffusion via FFT/DFT Convolution]
    E --> F[Outcomes: Faster, more accurate alternative to Monte Carlo for derivative pricing]