# Lattice Random Walk Discretisations of Stochastic Differential Equations

## TL;DR

This paper presents a novel lattice random walk discretisation method for SDEs that simplifies computations to binary or ternary steps, enabling efficient stochastic computing and robust handling of complex drifts.

## Contribution

The paper introduces a new lattice-based discretisation scheme for SDEs that reduces complexity and enhances compatibility with stochastic computing architectures.

## Key findings

- Proves weak convergence of the discretisation scheme
- Demonstrates robustness to quantisation errors
- Shows effectiveness on various SDEs including diffusion models

## Abstract

We introduce a lattice random walk discretisation scheme for stochastic differential equations (SDEs) that samples binary or ternary increments at each step, suppressing complex drift and diffusion computations to simple 1 or 2 bit random values. This approach is a significant departure from traditional floating point discretisations and offers several advantages; including compatibility with stochastic computing architectures that avoid floating-point arithmetic in place of directly manipulating the underlying probability distribution of a bitstream, elimination of Gaussian sampling requirements, robustness to quantisation errors, and handling of non-Lipschitz drifts. We prove weak convergence and demonstrate the advantages through experiments on various SDEs, including state-of-the-art diffusion models.