Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients

Martin Hutzenthaler; Arnulf Jentzen

Source-linked AI summary

Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients

Martin Hutzenthaler, Arnulf Jentzen

arXiv:1203.5809v2Mar 26, 2012math.PRmath.NA

TL;DR

Superlinearly growing SDE coefficients can make Euler-Maruyama moments diverge and prevent strong convergence. The paper develops a rare-event-based Lyapunov theory, establishes moment bounds for explicit and implicit schemes, and uses them to prove strong convergence for several nonlinear SDE examples. Its general convergence results remain subject to locally Lipschitz assumptions in key settings, and Euler-Maruyama is not V-stable for a broad class of superlinear SDEs.

Problem
Euler-Maruyama moments can diverge in finite time for SDEs with superlinearly growing drift or diffusion coefficients, creating a need for alternative approximation analyses.
Method
The paper develops a rare-event-based Lyapunov theory for integrability and stability of discrete-time stochastic processes and applies it to explicit and implicit SDE schemes.
Results
The paper proves uniform moment bounds and strong convergence for increment-tamed, fully drift-implicit, partially drift-implicit, and related approximation schemes across several nonlinear SDE examples.
Takeaways & Limitations
The framework supports strong convergence analysis for numerical approximations of nonlinear SDEs beyond the standard globally Lipschitz setting.
Takeaways & Limitations
A key convergence-in-probability result assumes finite-dimensional SDEs with locally Lipschitz coefficients, while Euler-Maruyama is not V-stable for many superlinear SDEs.

Abstract

from arXiv · show

Many stochastic differential equations (SDEs) in the literature have a superlinearly growing nonlinearity in their drift or diffusion coefficient. Unfortunately, moments of the computationally efficient Euler-Maruyama approximation method diverge for these SDEs in finite time. This article develops a general theory based on rare events for studying integrability properties such as moment bounds for discrete-time stochastic processes. Using this approach, we establish moment bounds for fully and partially drift-implicit Euler methods and for a class of new explicit approximation methods which require only a few more arithmetical operations than the Euler-Maruyama method. These moment bounds are then used to prove strong convergence of the proposed schemes. Finally, we illustrate our results for several SDEs from finance, physics, biology and chemistry.

1.1. Notation

The paper develops notation and a rare-event Lyapunov framework for deriving integrability properties of discrete-time stochastic processes. The framework yields uniform estimates on large subevents and, under additional growth assumptions, on the full probability space.

General framework: The general theory studies discrete-time stochastic processes using Lyapunov-type estimates imposed on typically large subevents of the probability space.The subevents are defined through a Lyapunov-type function and a truncation function.
Lyapunov estimates: The framework includes processes whose Lyapunov estimates hold only on nested subevents, with supermartingale arguments providing corresponding integrability bounds.The subevents are typically large and decrease with the time index.
Moment bounds: Uniform moment bounds can first be obtained on complements of rare events and then extended to the full probability space under an additional growth bound.The extension uses a bootstrap argument based on the Lyapunov estimates.
Stability notation: Semi-stability is monotone in its exponent: β-semi V-stability implies α-semi V-stability for every α ∈(0, β], but not conversely in general.This distinction is relevant for Euler-Maruyama schemes with superlinearly growing coefficients.

2.2. Explicit approximation schemes

This section develops stability properties for explicit approximation schemes, especially Euler-Maruyama, using Lyapunov-type conditions and semi-stability. It also identifies settings where full V-stability fails.

The section analyzes stability properties and moment bounds for explicit approximation schemes, beginning with Euler-Maruyama.
Theorem 2.13 gives sufficient conditions for Euler-Maruyama to be α-semi V-stable with respect to Brownian motion.
For the Euler-Maruyama update, the resulting semi-stability parameter is p/(γ1+2(γ0∨γ1)).
Powers of a Lyapunov-type function preserve the analysis, yielding a corresponding stability result for ˆV(x)=(V(x))^q.
With V(x)=1+∥x∥^p, Euler-Maruyama is p/(γ1+2(γ0∨γ1))-semi-stable with respect to Brownian motion.
For SDEs with coefficients growing more than linearly, Euler-Maruyama may be α-semi V-stable but not V-stable.

2.3. Implicit approximation schemes

This section establishes stability and moment bounds for fully and partially drift-implicit approximation schemes. The results provide uniform q-th moment bounds under stated drift, diffusion, and step-size conditions.

The section analyzes stability and moment bounds for fully drift-implicit and partially drift-implicit schemes.
Fully drift-implicit approximation schemes: Corollary 2.27 proves uniform q-th moment bounds for fully drift-implicit Euler approximations with globally one-sided Lipschitz drift.
Fully drift-implicit approximation schemes: Lemma 2.26 establishes stability of the fully drift-implicit Euler scheme under its stated coefficient conditions.
Fully drift-implicit approximation schemes: The fully drift-implicit scheme has a unique family of stochastic processes when N exceeds cT, together with a finite moment-growth constant ρ.
Partially drift-implicit approximation schemes: Lemma 2.28 treats partially drift-implicit schemes for SDEs with at most linearly growing diffusion coefficients.
Partially drift-implicit approximation schemes: Lemma 2.29 establishes a class of linear implicit schemes for one-dimensional SDEs and shows their stability with respect to Brownian motion.

3.1. Setting and assumptions

This section specifies the SDE setting and the approximation processes considered for convergence. The coefficients are locally Lipschitz, the solution remains in an open state domain, and the goal is convergence to that solution.

The setting uses a filtered probability space with Brownian motion and an SDE solution X having continuous paths in an open set D.
The approximation processes are defined through a measurable increment function φ and are required to converge suitably to the solution process X.

3.2. Consistency

This section introduces consistency of numerical increment functions and states that consistency ensures convergence in probability to the SDE solution. It also provides a characterization and points to examples of consistent schemes.

Consistency is introduced as a property of the numerical increment function φ for SDEs driven by standard Brownian motion.
Theorem 3.3 uses this consistency property to ensure convergence in probability of the interpolated approximation processes to the solution.
Lemma 3.2 characterizes consistency through conditions imposed over the compact sets Dv within the domain D.
The section notes that a list of numerical schemes consistent with respect to Brownian motion is given in Section 3.6.

3.3. Convergence in probability

The paper proves convergence in probability for one-step approximation processes whose increment functions are consistent with the SDE coefficients, using localization and comparison with Euler-Maruyama approximations.

Theorem 3.3: Theorem 3.3 establishes convergence in probability for one-step schemes with (µ, σ)-consistent increment functions.The result applies to finite-dimensional SDEs with locally Lipschitz continuous coefficients.
Scope: The locally Lipschitz assumptions ensure uniqueness of the SDE solution up to indistinguishability.The authors expect extensions to infinite-dimensional SDEs and weaker conditions such as local monotonicity.
Proof strategy: The proof localizes the coefficients on compact subsets and controls approximation errors through auxiliary processes, stopping indices, and probability estimates.The argument combines continuity of sample paths with inequalities including Gronwall's lemma and the Burkholder-Davis-Gundy inequality.
Proof strategy: The consistency condition requires the scheme's increment function to remain close to the corresponding Euler-Maruyama increment.The proof estimates both the exact solution versus Euler-Maruyama and Euler-Maruyama versus the proposed approximation.
Proof structure: Theorem 3.3 is assembled from lemmas controlling localization, increments, auxiliary approximations, and the distance between the numerical and exact processes.Corollary 3.9 provides an intermediate estimate used to complete the theorem.

3.4. Strong convergence

The paper combines convergence in probability with moment bounds to obtain strong convergence, then derives conditions yielding strong convergence for increment-tamed Euler-Maruyama schemes and related methods.

Strong convergence from moment bounds: Moment bounds combined with convergence in probability yield strong convergence of the numerical approximation processes.The framework provides final-value and uniform strong convergence results under suitable moment assumptions.
Proof mechanism: The strong-convergence argument uses a modified Fatou lemma to transfer convergence in probability and uniform integrability into Lq-convergence.The result applies for every q ∈ (0, p) under the stated p-moment bounds.
Strong convergence from moment bounds: Consistency and a priori growth bounds, together with semi-stability, imply strong convergence for one-step numerical approximation processes.The result is formulated through Corollary 3.14 under Lyapunov-type conditions.
Increment-tamed Euler-Maruyama: Theorem 3.15 proves strong convergence of an increment-tamed Euler-Maruyama scheme under coefficient, Lyapunov-function, and growth assumptions.The theorem is obtained by combining a moment-bound result, Corollary 3.12, and a consistency lemma.
Increment-tamed Euler-Maruyama: Specialized corollaries establish strong convergence using powers of a Lyapunov-type function and a polynomial-like Lyapunov function.These results include conditions with p ≥ 3 and provide counterparts to earlier moment-bound results.

3.5. Weak convergence

The paper extends convergence analysis from probability to weak convergence, including bounded test functions, unbounded test functions under Lyapunov control, and almost sure convergence for Monte Carlo methods.

Weak convergence: Convergence in probability yields weak convergence for bounded continuous test functions on path space.Corollary 3.19 applies to continuous functions from C([0, T ], Rd) into a separable Banach space.
Unbounded test functions: For unbounded test functions, the analysis restricts convergence initially to events whose probabilities approach one sufficiently fast.The restriction is removed later for Monte Carlo methods under stronger boundedness and independence conditions.
Unbounded test functions: Semi-weak convergence controls continuous test functions with growth relative to a Lyapunov function when the approximation processes are semi-bounded.The framework uses events measurable with respect to the numerical processes and whose probabilities converge to one.
Almost sure convergence: An Lp-convergence rate with order β > 1/p implies almost sure convergence with a reduced order β − 1/p − ε.The result generalizes an earlier lemma to arbitrary measure spaces.
Monte Carlo methods: Proposition 3.22 establishes almost sure convergence for Monte Carlo methods under semi-boundedness, consistency, and sufficiently strong probability controls.Corollary 3.23 specializes this result to the Monte Carlo Euler method under stated coefficient and Lyapunov assumptions.

3.6. Numerical schemes for SDEs

This section develops Brownian-consistent one-step schemes and shows how consistency, taming, and comparison principles yield convergence results for several Euler-type methods.

General scheme construction: The scheme increment is built from a measurable function φ(x,t,y)=η0(x,t,y)+η1(x,t,y)¯µ(x)t+η2(x,t,y)¯σ(x)y.Under suitable conditions, this construction is (µ, σ)-consistent with respect to Brownian motion.
Euler-type schemes: The Euler-Maruyama scheme is recovered by choosing η0=0 and η1=η2=I, and it converges in probability in the stated setting.The result follows by combining the consistency lemma with Theorem 3.3.
Euler-type schemes: The drift-tamed Euler scheme has strong convergence with rate 1/2 under global Lipschitz diffusion and additional regularity and one-sided Lipschitz assumptions on the drift.Moment bounds and strong convergence extend to a more general setting through the same framework used for drift truncation.
Milstein-type schemes: Under the commutativity condition, the scheme becomes the Milstein scheme; strong convergence can fail for superlinear coefficients, whereas appropriately tamed variants have been established for some cases.The untamed scheme still converges in probability under the stated consistency result.
Taming and comparison principles: Increment taming prevents strong divergence and supports moment bounds, while whole-scheme taming provides another route to Brownian consistency.The section also records that comparison and convex-combination principles preserve Brownian consistency.

4.1. Setting and assumptions

The chapter assumes a solution of an SDE with locally Lipschitz coefficients on an open state domain and studies increment-tamed Euler-Maruyama approximations in the strong sense.

Setting: The coefficients are Borel measurable globally but locally Lipschitz on the open domain D, and the solution process remains D-valued.The setting includes a filtered probability space and a standard Brownian motion.
Approximation objective: The chapter’s goal is strong approximation of the solution process using increment-tamed Euler-Maruyama processes.The analysis concentrates on the approximation family introduced earlier.
Illustrative examples: Strong convergence is established under suitable assumptions for examples including stochastic oscillators, Lorenz, Brusselator, SIR, psychology, and predator-prey models.The authors describe these as first strong-convergence results in the literature for the listed examples.

4.2. Stochastic van der Pol oscillator

The stochastic van der Pol oscillator models stable oscillation with additive noise and has a drift that is not globally Lipschitz or globally one-sided Lipschitz, but satisfies a global one-sided linear growth condition.

Model: The stochastic van der Pol oscillator is presented as a stochastic model of stable oscillation with additive noise.The model is introduced as a stochastic version of the van der Pol oscillator.
Coefficient assumptions: Its diffusion coefficient is globally Lipschitz, whereas its drift is neither globally Lipschitz continuous nor globally one-sided Lipschitz continuous.These properties place the model outside standard globally Lipschitz analyses.
Moment analysis: The drift satisfies a global one-sided linear growth condition, allowing a Lyapunov-type function to be used in the analysis.The cited corollary is applied with a specified Lyapunov function.
Result: The resulting analysis obtains the stated moment bound for the stochastic van der Pol oscillator.The supplied passage reports the obtained bound after applying the corollary.

4.3. Stochastic Duffing-van der Pol oscillator

The stochastic Duffing-van der Pol oscillator combines two oscillator models and is considered with affine-linear noise, while its drift violates several standard global conditions.

Model: The Duffing-van der Pol equation unifies the Duffing and van der Pol equations and has applications including aeroelasticity problems.The stochastic version considered uses an affine-linear noise term.
Coefficient assumptions: The diffusion coefficient is globally Lipschitz, but the drift is not globally one-sided Lipschitz and fails the global one-sided linear growth condition.The passage explicitly identifies both drift limitations.
Moment analysis: Despite these drift properties, the analysis applies a Lyapunov-type function to the stochastic Duffing-van der Pol equation.The cited passage reports the resulting bound expression after this application.

4.4. Stochastic Lorenz equation

The stochastic Lorenz equation models atmospheric convection rolls with multiplicative noise. Its diffusion is globally Lipschitz, while its drift is not globally one-sided Lipschitz but satisfies a global one-sided linear growth condition.

The stochastic Lorenz equation is a three-dimensional model of atmospheric convection rolls with chaotic behaviour and multiplicative noise.
Its diffusion coefficient is globally Lipschitz continuous.
Its drift coefficient is not globally one-sided Lipschitz continuous but fulfills the global one-sided linear growth condition (4.8).The cited passage states that Corollary 3.17 applies with a specified Lyapunov-type function.

4.5. Stochastic Brusselator in the well-stirred case

The well-stirred stochastic Brusselator models a trimolecular chemical reaction. Its coefficients fall outside standard global conditions, so a smooth Lyapunov-type function is constructed to apply the convergence theory.

The well-stirred stochastic Brusselator is a stochastic model of a trimolecular chemical reaction.
Its diffusion coefficient is globally Lipschitz, but its drift is neither globally one-sided Lipschitz nor subject to the one-sided linear growth condition (4.8).
The squared sum of the coordinates is a Lyapunov-type function on (0,∞)^2, but applying Corollary 3.17 requires constructing a suitable function on R^2.
Corollary 3.17 applies with the constructed Lyapunov-type function V in (4.25).
The choice in (4.25) is made to satisfy the smoothness and growth assumptions, although other Lyapunov-type functions are possible.

4.6. Stochastic SIR model

The stochastic SIR model describes susceptible, infected, and recovered populations. Because both coefficients can grow superlinearly and the drift fails a standard growth condition, the analysis constructs a suitable Lyapunov-type function.

The stochastic SIR model tracks susceptible, infected, and recovered individuals on the state space (0,∞)^3.
Both the drift and diffusion coefficients grow superlinearly, and the drift fails the one-sided linear growth condition (4.8).
An appropriate Lyapunov-type function on R^3 is constructed so that Corollary 3.17 can be applied.
Corollary 3.17 applies with the Lyapunov-type function in (4.33).
The function in (4.33) is selected to meet the smoothness and growth assumptions, while other choices are possible.

4.7. Experimental psychology model

This section applies the paper’s Lyapunov-based convergence analysis to several models, including experimental psychology, Lotka-Volterra, and volatility processes. The examples show how model-specific coefficient properties and Lyapunov functions determine applicability and moment conclusions.

4.7. Experimental psychology model: The experimental psychology model is a transformed stochastic version of a model introduced for deterministic and stochastic settings.
4.7. Experimental psychology model: Its diffusion is globally Lipschitz, its drift satisfies the global one-sided linear growth bound (4.8), and ⟨x,µ(x)⟩=0.
4.7. Experimental psychology model: Corollary 3.17 applies with the Lyapunov-type function in (4.41).
4.9. Stochastic Lotka-Volterra equations: The Lotka-Volterra drift contains a quadratic term and is therefore not globally Lipschitz continuous.
4.9. Stochastic Lotka-Volterra equations: A smooth Lyapunov-type function is constructed from auxiliary functions φ and ψ to apply Corollary 3.17.
4.9. Stochastic Lotka-Volterra equations: Corollary 3.17 applies to the stochastic Lotka-Volterra system with the function V in (4.50).
4.10. Volatility processes: For the squared volatility process, the moment parameter p0 determines the range of p for which the stated bounds hold, with q restricted to q<q0.
4.10. Volatility processes: In the Cox-Ingersoll-Ross and simplified Ait-Sahalia examples, p0=q0=∞, yielding the stated conclusion for all q∈(0,∞).

4.11. Langevin equation

For Langevin equations, the paper considers convergence under assumptions on the potential and an associated Lyapunov-type function. Its general convergence results apply when the force has suitable one-sided Lipschitz, growth, and regularity properties.

Assumptions: The Langevin setting assumes a continuously differentiable potential U with a locally Lipschitz continuous derivative.The domain is D = R^d with equal state and Brownian dimensions, d = m.
Scope: Corollary 3.17 does not cover completely arbitrary Langevin equations, but requires the stated assumptions on the potential.
Assumptions: The analysis also requires a twice differentiable Lyapunov-type function V with a locally Lipschitz continuous second derivative.
Model: Under these assumptions, the Langevin equation is represented within the paper’s general SDE setting.
Scope: Existing strong-convergence results apply when the force −▽U is globally one-sided Lipschitz and satisfies suitable growth and regularity conditions.