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

TL;DR

The paper tackles the challenge of approximating continuous-time, continuous-space stochastic processes with discrete-time Markov chains while preserving key moment characteristics. It develops a nonuniform-grid framework that, under explicit inequalities, yields transition probabilities $\lambda_{i,L},\lambda_{i,C},\lambda_{i,R}$ ensuring $\mathbb{E}[r(k)]=Mk$ and $\mathrm{Var}[r(k)]=Vk$ for all $k$, via a time-scaling factor mapping discrete time to physics time. The approach is demonstrated on stochastic heat diffusion and geometric Brownian motion, with corollaries showing moment matching for heat and a GBM formulation using $M=\eta\tau$, $V=\sigma^2\tau$; empirical results include Wasserstein-1 distance analyses and GBM histograms that validate long-horizon accuracy. The method enables simulation-based numerical analysis that preserves essential process characteristics and offers potential extensions to Levy processes and regime-switching models, enhancing flexibility and computational efficiency for stochastic modeling. The nonuniform-grid design, combined with a trinomial-like transition structure, provides improved approximation quality over uniform grids in many regimes, especially at long horizons.

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.

Figures (5)

• Figure 1: Comparison of the probability density function of $\ln(s_{k\tau}/s_0)$ for $k\tau=2$ and histogram of $r(k)$ for $k=10000$.
• Figure 2: Sample and average trajectories of $s_0 e^{r(k)}$ compared with $\mathbb{E}[s_{k\tau}]$.
• Figure 3: Uniform and nonuniform grids with equal point density
• Figure 4: Comparison of Wasserstein $1$-distances obtained with uniform and nonuniform grids. Top: 10000 simulations, Bottom: 100000 simulations.
• Figure 5: Probability density function of log-return $\ln(s_{k\tau} / s_0)$ compared to histograms of $r(k)$. Top-Left: Uniform grid for $k=10$, Top-Right: Nonuniform grid for $k=10$, Bottom-Left: Uniform grid for $k=10000$, Bottom-Right: Nonuniform grid for $k=10000$

Theorems & Definitions (11)

• Theorem 1
• Remark 1
• Remark 2
• Lemma 1
• proof
• proof : Proof of Theorem \ref{['thrm:markov']}
• Corollary 1
• proof : Proof of Corollary \ref{['corollary:Heat']}
• Corollary 2
• proof