Lattice Random Walk Discretisations of Stochastic Differential Equations

2025-08-28

AI Summary

Existing numerical solvers for stochastic differential equations (SDEs) suffer from computationally expensive drift/diffusion evaluations, reliance on Gaussian sampling, sensitivity to quantization errors, and instability with non-Lipschitz drift terms. To address these challenges, this paper proposes a lattice-based random walk discretization method. It replaces continuous drift and diffusion dynamics with 1–2-bit stochastic operations—using binary or ternary increment sampling—thereby eliminating floating-point arithmetic and Gaussian sampling entirely and enabling native compatibility with bitstream probabilistic computing architectures. Theoretically, the method achieves first-order weak convergence, exhibits robustness to quantization errors, and remains stable under non-Lipschitz drift conditions. Empirical evaluation demonstrates its effectiveness on canonical SDEs and state-of-the-art diffusion models, while delivering substantial hardware efficiency gains and computational speedup.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Search and Retrieval-Augmented AI: Large language models for search
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

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.

Problem

• Discretizing SDEs with binary/ternary increments
• Avoiding complex drift and diffusion computations
• Enabling compatibility with stochastic computing architectures

Innovation

• Lattice random walk discretisation for SDEs
• Binary/ternary increments replace complex computations
• Eliminates Gaussian sampling and floating-point arithmetic