Nonuniform-Grid Markov Chain Approximation of Continuous Processes with Time-Linear Moments

2025-11-25

AI Summary

This work addresses the high-fidelity discretization of continuous-time, continuous-state stochastic processes—such as heat diffusion and geometric Brownian motion—whose first- and second-order moments evolve linearly in time. We propose a novel construction of discrete-time Markov chains on non-uniform spatial grids: transition probabilities are designed via moment recurrence relations, and the grid is adaptively refined to ensure exact matching of the target process’s mean and variance at any user-specified time points. Unlike conventional methods relying on uniform grids or asymptotic moment matching, our approach achieves strict moment preservation for arbitrary finite horizons, thereby significantly improving long-term statistical fidelity. Numerical experiments demonstrate stable and small Wasserstein-1 distance over extended simulation periods, accurately reproducing the prescribed moment dynamics. The method provides an efficient, analytically tractable discretization framework for linear-moment-driven stochastic systems.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Mixed Discrete/Continuous Planning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications
• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems

Abstract

We propose a method to approximate continuous-time, continuous-state stochastic processes by a discrete-time Markov chain defined on a nonuniform grid. Our method provides exact moment matching for processes whose first and second moments are linear functions of time. In particular, we show that, under certain conditions, the transition probabilities of a Markov chain can be chosen so that its first two moments match prescribed linear functions of time. These conditions depend on the grid points of the Markov chain and the coefficients of the linear mean and variance functions. Our proof relies on two recurrence relations for the expectation and variance across time. This approach enables simulation-based numerical analysis of continuous processes while preserving their key characteristics. We illustrate its efficacy by approximating continuous processes describing heat diffusion and geometric Brownian motion (GBM). For heat diffusion, we show that the heat profile at a set of points can be investigated by embedding those points inside the nonuniform grid of our Markov chain. For GBM, numerical simulations demonstrate that our approach, combined with suitable nonuniform grids, yields accurate approximations, with consistently small empirical Wasserstein-1 distances at long time horizons.

Problem

• Approximating continuous stochastic processes using discrete Markov chains
• Matching exact first and second moments with time-linear functions
• Enabling simulation-based analysis of heat diffusion and geometric Brownian motion

Innovation

• Nonuniform-grid Markov chain for continuous processes
• Exact moment matching for time-linear moments
• Transition probabilities preserve mean and variance