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A fundamental mean-square convergence theorem for SDEs with locally Lipschitz coefficients and its applications
M. V. Tretyakov, Z. Zhang
TL;DR
The paper addresses the loss of convergence of usual SDE methods under nonglobal Lipschitz conditions. It proves a generalized mean-square convergence theorem for polynomial-growth, one-sided-Lipschitz coefficients and applies it to several schemes. The theorem supports the same convergence order in the nonglobal setting under bounded scheme moments, including the drift-implicit scheme and a balanced method of order 1/2.
Problem
Usual numerical methods can lose convergence when global Lipschitz conditions fail, although many relevant SDE coefficients grow faster than linearly at infinity.
Method
The paper generalizes Milstein’s fundamental mean-square convergence theorem to coefficients with polynomial growth and a one-sided Lipschitz condition, then applies it to balanced and implicit schemes.
Results
Under bounded moments of the numerical solution, the scheme retains the same convergence order in the nonglobal Lipschitz case; the drift-implicit scheme has mean-square order 1/2 in the stated example.
Takeaways & Limitations
The theorem provides a framework for establishing strong convergence of numerical SDE methods beyond globally Lipschitz coefficients, including fully implicit methods under relaxed drift conditions.
Takeaways & Limitations
The numerical examples include a positive-probability exploding-trajectory event for every step size that grows with integration time.
Abstract
from arXiv · showhide
A version of the fundamental mean-square convergence theorem is proved for stochastic differential equations (SDE) which coefficients are allowed to grow polynomially at infinity and which satisfy a one-sided Lipschitz condition. The theorem is illustrated on a number of particular numerical methods, including a special balanced scheme and fully implicit methods. Some numerical tests are presented.
1 Introduction
The paper addresses mean-square approximation of SDEs with nonglobal Lipschitz coefficients, motivated by the failure of usual methods when global Lipschitz conditions are violated. It develops a generalized convergence theorem and applies it to balanced, drift-implicit, and fully implicit schemes.
- Motivation: Global Lipschitz assumptions are a significant limitation because many practically relevant SDE coefficients grow faster than linearly at infinity.When the global Lipschitz condition is violated, convergence of many usual numerical methods can disappear.
- Motivation: The paper focuses on mean-square (strong) approximation, which is relevant for simulating scenarios, visualizing stochastic dynamics, and filtering.Mean-square approximation also provides theoretical insight into weak-sense schemes.
- Motivation: Weak-sense schemes generally require low per-step cost because they may simulate many trajectories or operate over long time periods, whereas mean-square schemes permit greater interest in efficient implicit methods.The paper contrasts these computational requirements when motivating implicit methods.
- Contribution: A variant of Milstein’s fundamental mean-square convergence theorem is established for coefficients with polynomial growth and a one-sided Lipschitz condition.The paper also gives a corollary on almost sure convergence.
- Applications: The theorem is applied to the drift-implicit Euler scheme, establishing its convergence order, and to a particular balanced method with order 1/2.The paper presents numerical experiments supporting these results.
- Applications: Fully implicit mean-square schemes are extended from globally Lipschitz settings to cases where the drift satisfies a one-sided Lipschitz condition and polynomial growth bound.The paper revisits schemes implicit in both drift and diffusion.
2 Fundamental theorem
The paper extends the fundamental mean-square convergence theorem from globally to nonglobally Lipschitz SDE coefficients, under one-sided Lipschitz and polynomial-growth conditions. The result transfers convergence orders when numerical approximations have bounded moments, and supports drift-implicit and fully implicit methods.
- Theorem 2.1: Theorem 2.1 generalizes Milstein’s fundamental convergence theorem from globally to nonglobally Lipschitz SDEs.The extension targets coefficients that may grow faster than linearly at infinity while satisfying the paper’s stated assumptions.
- Assumptions: The theorem requires finite moments for the approximation, a condition that is often difficult to verify separately for each nonglobal-Lipschitz scheme.Under global Lipschitz coefficients, moment boundedness follows more directly from solution moments and one-step properties.
- Consequences: The method has almost-sure convergence order q − ε under the corollary’s additional condition on p.The result follows from the theorem through Borel-Cantelli-type arguments.
- Applications: The framework also establishes convergence for balanced and fully implicit methods in the nonglobal-Lipschitz setting.The paper proves boundedness of moments for balanced and fully implicit methods, enabling application of the fundamental theorem.
- Theorem 2.1: If the scheme’s moments are bounded and its global-Lipschitz convergence order is q, it retains order q in the nonglobal-Lipschitz setting.This is the theorem’s central practical interpretation.
3 A balanced method
The paper introduces a particular balanced scheme for nonglobal-Lipschitz SDEs and proves the moment and one-step estimates needed for the general convergence theorem. Under the stated assumptions, the scheme achieves mean-square order 1/2.
- Scheme: The paper proposes a particular balanced scheme for SDEs with nonglobal Lipschitz coefficients.The variant is presented as a balanced-type method suitable for the setting considered.
- Numerical tests: Numerical tests compare the balanced scheme with a tamed scheme and report that the balanced scheme is more efficient.The comparison is discussed in connection with the model problem in Section 5.
- Moment bounds: The balanced scheme has bounded moments uniformly in the time-step index under sufficiently strong moment assumptions.The proof uses stopping-time events and estimates independent of h and k.
- One-step analysis: The scheme’s one-step approximation satisfies orders q1 = 3/2 and q2 = 1 under coefficient differentiability and growth conditions.These estimates are the local ingredients used with Theorem 2.1.
- Convergence: The balanced scheme converges with mean-square order 1/2.The result follows from the moment bound, the one-step estimates, and Theorem 2.1.
- Convergence: In the additive-noise case, the balanced scheme’s mean-square order does not improve.The paper states that q1 and q2 remain 3/2 and 1, respectively.
4 Fully implicit schemes
The section analyzes a one-parameter family of fully implicit schemes under one-sided Lipschitz and polynomial-growth conditions, establishing solvability, moment bounds, and mean-square convergence results.
- Method family: The fully implicit family includes explicit Euler at λ = 0, fully implicit Euler at λ = 1, and the mid-point rule at λ = 1/2.The explicit Euler scheme is divergent in the considered setting, while the mid-point rule is derivative free for Stratonovich SDE systems.
- Solvability: For sufficiently small h and λ > 0, the implicit step has a unique solution satisfying bounds derived from uniform monotonicity.The step equation is shown to be uniformly monotone when h ≤ h0, which implies uniqueness.
- Moment bounds: For 1/2 < λ ≤ 1, the fully implicit scheme has bounded moments under Assumption 4.1 for all time steps up to N.The bound holds for p ≥ 1 and sufficiently small h, with a constant independent of h and k.
- Convergence: Under additional regularity assumptions, the method satisfies one-step error inequalities with q1 = 2 and q2 = 1.These estimates support application of the fundamental mean-square convergence theorem.
- Convergence: In the commutative case or for one noise, the mid-point method has q1 = 2 and q2 = 3/2 and therefore mean-square order 1 when its moments are bounded.For λ = 1/2, the result applies under the additional setting described in Remark 4.2.
5 Numerical examples
The numerical tests compare balanced, tamed, drift-implicit, fully implicit, midpoint, and trapezoidal schemes on two nonglobally Lipschitz SDEs, focusing on accuracy and computational cost. Observed convergence rates generally match predicted behavior, while practical efficiency and stability differ substantially across methods.
- Experimental setup: The experiments compare selected balanced, tamed, drift-implicit, fully implicit, midpoint, and trapezoidal schemes on two model SDEs.The first model has nonglobal Lipschitz drift, global Lipschitz diffusion, and two non-commutative noises; the second has nonglobal Lipschitz drift and diffusion.
- Example 5.1: Observed convergence rates for the first example are close to the predicted 1/2, with the midpoint scheme most accurate and the balanced method least accurate at fixed h.For accuracy approximately 0.06–0.07, runtimes were 170 seconds for the drift-tamed scheme, 329 and 723 seconds for midpoint and fully implicit variants, and 1870 seconds for the balanced method.
- Examples 5.1–5.2: The drift-tamed scheme is highly competitive but is not applicable when diffusion grows faster than linearly; the balanced method can then outperform implicit schemes.This comparison motivates the second example, whose diffusion is nonglobally Lipschitz and grows quadratically.
- Example 5.2: In the second example, the midpoint scheme demonstrates first-order convergence, while the other implicit schemes show order 1/2 as expected.Among order-1/2 methods, the balanced method is fastest; the midpoint scheme is most efficient in this commutative case.
A Proof of the fundamental theorem
The proof establishes moment and error estimates for the numerical method by decomposing the one-step error into propagated initial-data error and local error. Conditional estimates, inequality bounds, and Gronwall’s inequality then yield the stated convergence estimate.
- Conclusion of proof: The proof obtains an estimate independent of h and k, then applies Young’s and Gronwall’s inequalities to derive (2.13) for integer p.Jensen’s inequality extends the result to non-integer p.
- Error decomposition: The proof decomposes the numerical error into error propagated from the initial data and a one-step error.The propagated component is rewritten using the process Z, while the local component is denoted r_k+1.
- Error decomposition: The resulting 2p-th moment expansion separates squared propagated error, cross terms, and squared local error.The expansion is applied to E|ρ_k+1|^2p before estimating its terms.
- Term estimates: Conditional variants of the moment and stability estimates control the decomposed terms using measurability and Cauchy–Bunyakovsky inequalities.Young’s inequality and the conditions on the exponents and coefficients are used to combine the bounds.
B Proof of Lemma 2.1
Lemma 2.1 is proved by applying Itô’s formula under the one-sided Lipschitz condition, estimating the resulting terms, and using Gronwall’s inequality. Jensen’s inequality extends the integer-moment result to non-integer orders.
- Proof strategy: The proof introduces the difference process S(s) = X_t,x(s) − X_t,y(s) and an auxiliary process Z(s).These processes measure solution sensitivity to differing initial conditions.
- Proof strategy: Itô’s formula and condition (2.2) yield the estimate (2.15) after applying Gronwall’s inequality.The condition (2.2) also implies condition (2.5), which is used in the estimate.
- Moment estimates: The higher-moment estimate (2.16) is obtained by separately bounding terms with Young’s and Hölder’s inequalities.The proof applies Hölder’s inequality twice to the second term and then combines the bounds.
- Moment estimates: Gronwall’s inequality gives (2.16) for integer p ≥ 1, while Jensen’s inequality extends it to non-integer p > 1.The proof identifies Lemma 2.1 as an analogue of a previously known lemma.