Source-linked AI summary
Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients
Martin Hutzenthaler, Arnulf Jentzen, Peter E. Kloeden
TL;DR
The paper tackles the lack of strong convergence of explicit Euler for SDEs with superlinearly growing drift, while implicit Euler requires extra computation. It proposes a tamed explicit Euler method and proves standard strong order-one-half convergence under specified coefficient assumptions. Simulations report substantially faster implementation than implicit Euler.
Problem
Explicit Euler may fail strong convergence for SDEs with superlinearly growing drift, whereas implicit Euler converges but requires solving a nonlinear equation at every time step.
Method
The paper tames the explicit Euler drift by making it uniformly bounded, producing an explicit and easily implementable numerical scheme.
Results
The tamed Euler scheme converges strongly with the standard order 1/2 under a globally one-sided Lipschitz drift, polynomially growing drift derivative, and globally Lipschitz diffusion.
Takeaways & Limitations
The tamed Euler method provides an explicit scheme that is strongly convergent in this superlinear setting and is considerably faster than implicit Euler in the simulations.
Takeaways & Limitations
The result assumes a globally one-sided Lipschitz drift with at most polynomially growing derivative and a globally Lipschitz diffusion coefficient.
Abstract
from arXiv · showhide
On the one hand, the explicit Euler scheme fails to converge strongly to the exact solution of a stochastic differential equation (SDE) with a superlinearly growing and globally one-sided Lipschitz continuous drift coefficient. On the other hand, the implicit Euler scheme is known to converge strongly to the exact solution of such an SDE. Implementations of the implicit Euler scheme, however, require additional computational effort. In this article we therefore propose an explicit and easily implementable numerical method for such an SDE and show that this method converges strongly with the standard order one-half to the exact solution of the SDE. Simulations reveal that this explicit strongly convergent numerical scheme is considerably faster than the implicit Euler scheme.
1. Introduction and main result.
The paper addresses the failure of explicit Euler strong convergence for SDEs with superlinearly growing drift by proposing a tamed explicit scheme. Under stated coefficient assumptions, the scheme achieves standard strong order one-half while retaining easy implementation.
- Problem: Explicit Euler can fail strong mean square convergence for SDEs with superlinearly growing, globally one-sided Lipschitz drift.Its finite-time absolute moments may also diverge, whereas the implicit Euler scheme converges strongly but requires solving a nonlinear equation each step.
- Method: The tamed Euler method modifies the drift so it is uniformly bounded while remaining almost identical to explicit Euler.This preserves an explicit, easily implementable method for the nonglobally Lipschitz setting.
- Main result: Under a globally one-sided Lipschitz drift with polynomially growing derivative and globally Lipschitz diffusion, the tamed scheme converges strongly with order 1/2.The theorem gives strong Lp convergence with the supremum over the time interval inside the expectation.
- Main result: The proof relies on a priori moment bounds for the tamed approximations, which explicit Euler lacks in the superlinear setting.Adding a second-order term changes the numerical method's moment behavior enough to support the convergence argument.
- Simulation: On the reported computer, the tamed Euler scheme was more than one thousand times faster than implicit Euler while achieving three-decimal precision.The measured runtimes were about 0.0212 seconds for tamed Euler and 35.1 seconds for implicit Euler.
2. Further examples.
Further simulations examine the tamed Euler scheme on scalar and multidimensional SDEs, comparing its runtime and approximation behavior with implicit Euler methods. The reported examples show faster tamed-Euler implementations and dimension-dependent runtime advantages.
- Stochastic Ginzburg–Landau equation: The stochastic Ginzburg–Landau example uses T = d = m = 1, ξ = 1, μ(x) = x−x^3, and σ(x) = x.The resulting SDE has multiplicative noise and a cubic drift.
- Stochastic Ginzburg–Landau equation: Cardano’s method makes the implicit Euler implementation explicit for the one-dimensional cubic drift in the stochastic Ginzburg–Landau example.This avoids using a numerical root finder for that example.
- Stochastic Ginzburg–Landau equation: 0.0836 seconds versus 0.0467 seconds: tamed Euler is almost two times faster than the explicitly computed implicit Euler scheme at three-decimal precision.These runtimes are reported for N = 2^17 on the same computer.
- Multidimensional Langevin equation: The multidimensional Langevin example uses μ(x) = x−∥x∥^2·x, σ(x) = I, T = 1, and d = 10.Its SDE is dX_t = (X_t−∥X_t∥^2·X_t)dt + dW_t with X_0 = 0.
- Multidimensional Langevin equation: Both numerical approximations apparently converge with rate 1 for the d = 10 Langevin equation, presumably because its noise is additive.Figure 3 compares root mean square approximation errors as functions of runtime.
- Multidimensional Langevin equation: Quadratic versus linear dimension dependence: implicit Euler runtime increases quadratically with d, whereas tamed Euler runtime is linear apart from coefficient-evaluation costs.Figures 4 and 5 are reported to suggest these runtime dependencies.
3. Proof of Theorem 1.1.
The proof assembles auxiliary lemmas for the tamed Euler approximations and their continuous interpolations, then completes Theorem 1.1 using moment and approximation estimates. Runtime comparisons are also framed through dimension-dependent figures.
- The proof introduces mappings and dominating stochastic processes used to analyze the tamed Euler approximations.
- Figures 4 and 5 examine the runtime of implicit and tamed Euler approximations as functions of the dimension d.
- The continuous interpolations are adapted processes with continuous sample paths, providing the time-continuous objects needed for the main convergence result.
- Uniform moment bounds and auxiliary inequalities support the analysis of the tamed Euler approximations and the completion of Theorem 1.1.
- The proof of Theorem 1.1 is completed using Lemmas 3.9 and 3.10 after the preceding lemmas are established.
3.1. Proof of Lemma 3.1.
Lemma 3.1 is proved by combining growth and Lipschitz estimates with an induction argument controlling the tamed Euler approximation and related events.
- The induction step uses the preceding estimates and the defining event for the tamed Euler approximation.
- The proof uses global Lipschitz continuity of σ, global one-sided Lipschitz continuity of µ, and a polynomial growth bound on µ′.
- The estimates are combined to control the approximation when its norm lies within an N-dependent range.
- Lemma 3.1 is established by induction over the time-discrete index n.
3.2. Proof of Lemma 3.2.
Lemma 3.2 is established first for one increment and then extended to the general case using independence.
- The proof treats the case n = 1 as the base case for establishing the lemma’s estimate.
- The general case follows from independence after the one-step case has been shown.
- The proof concludes once the estimate in the one-step argument is obtained.
3.3. Proof of Lemma 3.3.
Lemma 3.3 is proved through martingale and submartingale arguments, followed by maximal inequalities and estimates applied uniformly over the discretization parameters.
- The relevant time-discrete stochastic process is a martingale for each z ∈ {−1,1} and every N.
- Its exponential transform is a positive submartingale, enabling the use of Doob’s maximal inequality.
- The argument applies estimates uniformly over N, p, z, x, and the discrete index k.
- The final estimate follows by combining the preceding bounds.
3.5. Proof of Lemma 3.5.
The proof combines Hölder’s inequality with earlier estimates, then uses probability bounds and previously established lemmas to conclude the result.
- Hölder’s inequality is applied first to derive an estimate valid for all N ∈ N and p ∈ [1,∞).
- Combining the resulting estimate with Lemmas 3.3, 3.4, and 5 completes the proof of Lemma 3.5.
- Subadditivity of the probability measure and Markov’s inequality provide a bound for all N ∈ N and q ∈ [1,∞).
3.8. Proof of Lemma 3.8.
The proof of Lemma 3.8 uses the discrete Burkholder–Davis–Gundy type inequality to obtain the stated estimate.
- The Burkholder–Davis–Gundy type inequality in Lemma 3.7 yields the required discrete estimate.
- The estimate holds for all n ∈ {0,1,...,N}, N ∈ N, and p ∈ [2,∞), completing Lemma 3.8.
3.9. Proof of Lemma 3.9.
The proof represents the numerical approximations, applies stochastic and Lipschitz estimates, uses Gronwall’s lemma, and closes the argument through a bootstrap.
- The numerical approximations (8) are first represented in an appropriate form.
- The Burkholder–Davis–Gundy type inequality in Lemma 3.8 is applied to the representation.
- Global Lipschitz continuity of σ supplies a subsequent estimate for the approximation terms.
- Gronwall’s lemma yields an estimate valid for all N ∈ N and p ∈ [2,∞).
- Because estimate (58) still contains N on its right-hand side, a bootstrap using Hölder’s inequality, (58), and Lemma 3.6 establishes the needed bound.
- Combining estimates (59) and (60) completes the proof for all p ∈ [1,∞).
3.10. Proof of Lemma 3.10.
The proof of Lemma 3.10 derives estimates for the drift and diffusion coefficients from Lemma 3.9, then uses these results to establish inequality (11).
- A polynomial-growth bound for μ, together with Lemma 3.9, yields the required estimate.
- A linear-growth bound for σ, again combined with Lemma 3.9, provides an additional estimate.
- Lemmas 3.9 and 3.10 are then used to establish inequality (11).
3.11. Proof of Theorem 1.1.
The proof estimates the error between the exact solution X and the interpolated numerical process, using Itô’s formula, standard inequalities, and martingale bounds. Gronwall’s lemma and supporting lemmas then yield the theorem’s stated estimate.
- The proof targets E[sup_{s∈[0,t]} ∥X_s−Ȳ_s^N∥^p] for t∈[0,T] and p∈[1,∞).
- Itô’s formula is applied to the error process, after relating the exact and numerical processes through equations (2) and (63).
- Cauchy–Schwarz, Hölder, elementary quadratic bounds, and the Burkholder–Davis–Gundy inequality control the resulting terms.
- Gronwall’s lemma and Lemmas 3.9–3.10, together with inequalities (66)–(70), complete the proof of Theorem 1.1.
- The coefficient estimate ∥µ(x)−µ(y)∥≤c(1+∥x∥^c+∥y∥^c)∥x−y∥ supports the bounds for differences in the drift coefficient.