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On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients
Martin Hutzenthaler, Arnulf Jentzen
TL;DR
The paper addresses perturbation estimates for stochastic differential equations when global monotonicity is too restrictive. It bounds global errors through local characteristic differences, then establishes strong convergence rates for SODE and SPDE approximations with non-globally monotone coefficients.
Problem
Global monotonicity is too restrictive for many applied nonlinear equations, and a general technique for estimating global perturbation errors from local errors is lacking.
Source: §1 · p. 2Method
The theory compares an SDE solution with an Itô process by estimating their strong distance through differences in initial values, drift, and diffusion.
Source: §1 · p. 2Results
Strong convergence rates are established for suitable numerical approximations of a large class of non-globally monotone SODEs and for spectral Galerkin approximations of semilinear SPDEs.
Source: §1 · pp. 4–5Takeaways & Limitations
The perturbation estimate also supports local Lipschitz continuity of SDE solutions with respect to initial values under suitable exponential-moment conditions.
Source: §1 · p. 4The stated SODE rate theorem is restricted to finite-dimensional R^d states and R^m noise, with smooth coefficients and further assumptions.
Source: §1 · p. 4
Abstract
from arXiv · showhide
We develope a perturbation theory for stochastic differential equations (SDEs) by which we mean both stochastic ordinary differential equations (SODEs) and stochastic partial differential equations (SPDEs). In particular, we estimate the $ L^p $-distance between the solution process of an SDE and an arbitrary It\^o process, which we view as a perturbation of the solution process of the SDE, by the $ L^q $-distances of the differences of the local characteristics for suitable $ p, q > 0 $. As application of our perturbation theory, we establish strong convergence rates for numerical approximations of a class of SODEs with non-globally monotone coefficients. As another application of our perturbation theory, we prove strong convergence rates for spectral Galerkin approximations of solutions of semilinear SPDEs with non-globally monotone nonlinearities including Cahn-Hilliard-Cook type equations and stochastic Burgers equations. Further applications of the perturbation theory include the regularity of solutions of SDEs with respect to the initial values and small-noise analysis for ordinary and partial differential equations.
1 Introduction
The paper develops perturbation bounds for stochastic differential equations that compare a solution with an Itô process using differences in their local characteristics. It applies this theory to numerical SODE and SPDE approximations and to further questions including initial-value regularity and small-noise analysis.
1 Introduction
- The perturbation theory estimates the global distance between an SDE solution and an Itô-process perturbation using differences in initial values, drift, and diffusion.The comparison applies to any stochastic process that is an Itô process with respect to the Wiener process.Source: §1 · p. 2
- Global monotonicity is too restrictive for many applied nonlinear equations, and the authors report no general method beyond it for bounding global perturbation errors by local ones.Earlier results for non-globally monotone coefficients covered some initial-value and one-dimensional cases, but the authors knew of no result for multidimensional SODE problem (ii).Source: §1 · pp. 2–3
- Exponential integrating factors and a pathwise perturbation formula enable estimates that reduce global errors to local characteristic errors.The distance function can be chosen beyond squared Hilbert-space distance, which is useful for examples including CIR processes and the Cahn–Hilliard–Cook equation with space-time white noise.Source: §1 · p. 3
- Further applications include local Lipschitz regularity with respect to initial values and small-noise analysis for nonlinear ordinary and partial differential equations.The initial-value regularity application requires suitable conditions ensuring finiteness of an exponential moment.Source: §1 · pp. 4–5
- A strong convergence rate of 1/2 is established for suitable numerical approximations of a large class of finite-dimensional SODEs with non-globally monotone coefficients.The authors identify this as the first such rate result for time-discrete approximations of multidimensional SODEs in this coefficient class.Source: §1 · p. 4
- The perturbation estimates yield strong convergence rates for spectral Galerkin approximations of semilinear SPDEs with non-globally monotone nonlinearities.The applications include Cahn–Hilliard–Cook type equations and stochastic Burgers equations with bounded diffusion coefficients.Source: §1 · p. 5
2 A perturbation theory for stochastic differential equations (SDEs)
The section develops perturbation formulas and estimates for comparing stochastic processes. It applies Itô’s formula, exponential integrating factors, and localization to obtain estimates that also cover Hilbert-space distances and comparisons of semilinear SPDE solutions.
2.1 Itˆo’s formula and an exponential integrating factor
- Itô’s formula and an exponential integrating factor provide the basis for the section’s perturbation estimates.A subsequent formulation adds and subtracts the generator and noise operator.Source: §2.1 · p. 7
2.2 A perturbation formula
- The perturbation formula includes the special case where the process under study is a pairing of two stochastic processes.A further special case takes one process to solve the SDE with the specified drift and diffusion coefficients.Source: §2.2 · p. 8
2.3 Perturbation estimates
- The perturbation estimates address the V-distance between a solution process and an Itô process, with drift and diffusion mismatches appearing in the bound.The section uses localization and Hölder’s inequality to estimate the supremum of the V-distance.Source: §2.3 · p. 9
2.4 Perturbation estimates in the case of Hilbert space distances
- The Hilbert-space-distance results include estimates for differences between solutions of semilinear SPDEs with possibly different coefficient functions.The section also develops its estimates using localization, expectation bounds, and Hölder’s inequality.Source: §2.4 · pp. 11, 13, 15
3.1 Numerical approximations of SODEs
This section applies the perturbation theory to establish strong convergence rates for stopped-tamed Euler–Maruyama approximations of SODEs with non-globally monotone coefficients. It verifies the assumptions for Langevin dynamics and the stochastic Duffing oscillator, yielding a uniform-in-time approximation bound.
3.1 Numerical approximations of SODEs
- Under suitable assumptions, Proposition 3.3 establishes strong convergence rates for stopped-tamed Euler-Maruyama approximations in [30] for SDEs with possibly non-globally Lipschitz drift and diffusion; Corollary 3.4 treats the special case of globally Lipschitz diffusion.Source: §3.1 · p. 22
3.1.2 Stochastic Lorenz equation with bounded noise
- The passage states that condition (93) is satisfied, so Corollary 3.4 applies and yields constants C_r ∈ [0, ∞) and r ∈ (0, ∞).Source: §3.1.2 · p. 23
3.1.3 Stochastic van der Pol oscillator
- Conditions (92) and (93) are satisfied, so Corollary 3.4 applies and yields constants C_r ∈ [0, ∞) and r ∈ (0, ∞) for the stated bound across n and θ.Source: §3.1.3 · p. 23
3.1.4 Stochastic Duffing-van der Pol oscillator
- The example fulfills (92) and (93), so Corollary 3.4 applies, yielding constants C_r ∈ [0,∞) and r ∈ (0,∞) for the stated estimate over n ∈ N and θ ∈ PT.Source: §3.1.4 · p. 24
3.1.5 Experimental psychology model
- After verifying (92) and (93), the authors apply Corollary 3.4, obtaining constants C_r ∈ [0, ∞) and r ∈ (0, ∞) for the stated bound.Source: §3.1.5 · p. 24
3.1.6 Brownian dynamics (Overdamped Langevin dynamics)
- The passage applies Corollary 3.4 and states that parameters C_r and r exist, with an assertion involving sup_{t∈[0,T]} ||X_t−Y^θ|| for n∈N and θ∈PT.Source: §3.1.6 · p. 25
- The authors expect Proposition 2.9 to yield a strong convergence rate of 1 for additive-noise SDEs and higher-order rates for suitable higher-order schemes.These expectations concern SDEs with possibly non-globally monotone coefficients.Source: §3.1.6 · p. 25
3.1.7 Langevin dynamics and stochastic Duffing oscillator
- The next subsection begins setting up Langevin dynamics and a stochastic Duffing oscillator as additional SODE examples.The supplied passage gives the model setup but no convergence result for these examples.Source: §3.1.7 · p. 25
- The assumptions include the stochastic Duffing oscillator with additive noise, with m = 1 and V(x) = 1/4 x^4.Source: §3.1.7 · p. 25
- Corollary 3.4 yields a uniform-in-time bound involving sup_{t∈[0,T]} ∥X_t − Y^θ∥, with C_r ∈ [0,∞) and r ∈ (0,∞), for all r ∈ (0,∞), n ∈ N, and θ ∈ P_T.The supplied passages truncate the full bound expression.Source: §3.1.7 · p. 25
3.2 Galerkin approximations of stochastic partial differential equations (SPDEs)
The section applies the paper’s perturbation estimates to spectral Galerkin approximations of semilinear SPDEs. It develops approximation estimates for Cahn–Hilliard–Cook type equations and the stochastic Burgers equation.
3.2 Galerkin approximations of stochastic partial differential equations (SPDEs)
- Corollary 3.5 specializes the general perturbation result to estimate errors in Galerkin approximations of SPDE solutions.The section then applies this framework to two semilinear examples with non-globally monotone nonlinearities.Source: §3.2 · pp. 25, 26, 29
3.2.1 Setting
- The setting uses finite-dimensional operators P_N and compares an SPDE solution X with processes X_N in the corresponding projected spaces.The SPDE is specified using an analytic semigroup and fractional-power spaces.Source: §3.2.1 · p. 29
3.2.2 Cahn-Hilliard-Cook type equations
- For Cahn–Hilliard–Cook type equations, the authors apply Proposition 3.6 to spectral Galerkin processes and derive estimates for their differences as the projection dimensions increase.The argument passes from estimates comparing X_N and X_M, for M>N, to bounds for the approximation sequence using Fatou’s lemma.Source: §3.2.2 · pp. 32–33
3.2.3 Stochastic Burgers equation
- For the stochastic Burgers equation, the authors apply Corollary 3.7 to spectral Galerkin processes and derive estimates for differences between projection levels.The estimates are developed for M>N and then extended to the limiting approximation sequence using Fatou’s lemma.Source: §3.2.3 · pp. 35–36
3.3 Analysis of SDEs with small noise
The section studies deterministic ordinary and partial differential equations perturbed by small noise. It applies the perturbation theory to bound the distance between the unperturbed and noise-perturbed processes, with applications to nonlinear equations.
3.3 Analysis of SDEs with small noise
- The section applies Corollary 2.12 to study deterministic ordinary and partial differential equations perturbed by a small noise term.The result can be applied to nonlinear examples, including equations discussed in the paper’s SPDE sections.Source: §3.3 · p. 36
- Corollary 3.8 estimates the distance between processes when the comparison process has drift μ and noise scaled by ε.The result assumes suitable integrability conditions and allows the comparison to stop at a stopping time.Source: §3.3 · p. 36
- With suitable exponential integrability, the right-hand side of Corollary 3.8’s estimate can be further bounded, yielding Corollary 3.9.The paper identifies nonlinear ordinary and partial differential equations perturbed by small noise as applications.Source: §3.3 · p. 36