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Taming singular stochastic differential equations: A numerical method

Khoa Lê, Chengcheng Ling

TL;DR

This work develops a robust numerical framework for multidimensional SDEs with integrable drifts under the Krylov–Röckner regime, by introducing a tamed Euler–Maruyama scheme that uses drift approximations $b^n$ and a lattice time grid $k_n(t)$. The authors prove explicit strong convergence rates that depend on the drift-approximation topology, diffusion regularity, and discretization, with rates approaching the benchmark $1/2$ when the truncation/regularization parameters are optimally tuned (e.g., via a parameter $\chi$). A Zvonkin transformation via a PDE solution $U$ and stochastic sewing techniques are employed to manage the rough drift, yielding convergence bounds involving $\varpi_n(\bar p)$ and the topology of $b^n\to b$. The results extend to applications in stochastic transport equations with singular vector fields, illustrating the scheme’s practicality for physics-informed problems where the drift is not explicitly known. Overall, the paper provides a quantitative framework for reliable numerical approximation of SDEs with rough drifts in multiple dimensions, supported by detailed analytic and probabilistic estimates.

Abstract

We consider a generic and explicit tamed Euler--Maruyama scheme for multidimensional time-inhomogeneous stochastic differential equations with multiplicative Brownian noise. The diffusive coefficient is uniformly elliptic, Hölder continuous and weakly differentiable in the spatial variables while the drift satisfies the strict Ladyzhenskaya--Prodi--Serrin condition, as considered by Krylov and Röckner (2005). In the discrete scheme, the drift is tamed by replacing it by an approximation. A strong rate of convergence of the scheme is provided in terms of the approximation error of the drift in a suitable and possibly very weak topology. A few examples of approximating drifts are discussed in detail. The parameters of the approximating drifts can vary and -- under suitable conditions -- be fine-tuned to achieve a strong convergence rate which is arbitrarily close to the benchmark $0.5$ rate. The result is then applied to provide numerical solutions for stochastic transport equations with singular vector fields satisfying the aforementioned condition.

Taming singular stochastic differential equations: A numerical method

TL;DR

This work develops a robust numerical framework for multidimensional SDEs with integrable drifts under the Krylov–Röckner regime, by introducing a tamed Euler–Maruyama scheme that uses drift approximations and a lattice time grid . The authors prove explicit strong convergence rates that depend on the drift-approximation topology, diffusion regularity, and discretization, with rates approaching the benchmark when the truncation/regularization parameters are optimally tuned (e.g., via a parameter ). A Zvonkin transformation via a PDE solution and stochastic sewing techniques are employed to manage the rough drift, yielding convergence bounds involving and the topology of . The results extend to applications in stochastic transport equations with singular vector fields, illustrating the scheme’s practicality for physics-informed problems where the drift is not explicitly known. Overall, the paper provides a quantitative framework for reliable numerical approximation of SDEs with rough drifts in multiple dimensions, supported by detailed analytic and probabilistic estimates.

Abstract

We consider a generic and explicit tamed Euler--Maruyama scheme for multidimensional time-inhomogeneous stochastic differential equations with multiplicative Brownian noise. The diffusive coefficient is uniformly elliptic, Hölder continuous and weakly differentiable in the spatial variables while the drift satisfies the strict Ladyzhenskaya--Prodi--Serrin condition, as considered by Krylov and Röckner (2005). In the discrete scheme, the drift is tamed by replacing it by an approximation. A strong rate of convergence of the scheme is provided in terms of the approximation error of the drift in a suitable and possibly very weak topology. A few examples of approximating drifts are discussed in detail. The parameters of the approximating drifts can vary and -- under suitable conditions -- be fine-tuned to achieve a strong convergence rate which is arbitrarily close to the benchmark rate. The result is then applied to provide numerical solutions for stochastic transport equations with singular vector fields satisfying the aforementioned condition.

Paper Structure

This paper contains 10 sections, 48 theorems, 408 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Regularizing properties of Brownian paths---a case study
  5. Analysis of the discrete paths
  6. Analytic estimates
  7. Moment estimates
  8. Analysis of the continuum paths
  9. Proof of the main results
  10. Application: stochastic transport equations

Key Result

Theorem 2.2

Assume that con.Acon.B hold. Let $(X_t^n)_{t\in[0,1]}$ be the solution to eqn.EMscheme and $(X_t)_{t\in[0,1]}$ be the solution to SDE sde00. Then for any $\bar{p}\in(1,p)\cap(1,\frac{2}{d}(p\wedge p_0))$ and any $\gamma\in(0,1)$, there exists a finite constant $N(K_1,K_2,K_3,K_4,\alpha,p_0,q_0,p,q,d

Theorems & Definitions (104)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Corollary 2.4
  • proof
  • Remark 2.5
  • Remark 2.6
  • Lemma 3.1
  • proof
  • Lemma 3.2: Stochastic sewing
  • ...and 94 more