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Strong convergence and stability of implicit numerical methods for stochastic differential equations with non-globally Lipschitz continuous coefficients
Xuerong Mao, Lukasz Szpruch
TL;DR
The paper addresses strong convergence and almost sure stability of EM-type approximations for SDEs with nonlinear, non-Lipschitz coefficients. It analyzes implicit θ-EM and FBEM schemes under monotone conditions and introduces a stochastic discrete LaSalle principle. The results establish strong convergence and stability properties for numerical methods covering highly nonlinear SDEs.
Problem
The paper asks how to obtain strong convergence and stability for numerical SDE approximations when coefficients are not globally Lipschitz, including cases where standard EM can fail.
Method
The paper analyzes implicit θ-EM and FBEM schemes under monotone conditions and uses a stochastic counterpart of the discrete LaSalle principle for stability.
Results
The paper proves strong convergence results and derives almost sure asymptotic stability properties for EM-type schemes in highly nonlinear SDE settings.
Takeaways & Limitations
The framework extends numerical convergence and stability analysis beyond globally Lipschitz coefficients to highly nonlinear SDEs under mild time-step assumptions.
Takeaways & Limitations
The implicit analysis requires a one-sided Lipschitz condition for unique solvability, and finding the inverse used by the method may be computationally expensive.
Abstract
from arXiv · showhide
We are interested in the strong convergence and almost sure stability of Euler-Maruyama (EM) type approximations to the solutions of stochastic differential equations (SDEs) with non-linear and non-Lipschitzian coefficients. Motivation comes from finance and biology where many widely applied models do not satisfy the standard assumptions required for the strong convergence. In addition we examine the globally almost surely asymptotic stability in this non-linear setting for EM type schemes. In particular, we present a stochastic counterpart of the discrete LaSalle principle from which we deduce stability properties for numerical methods.
1 Introduction
The paper studies strong convergence and stability of computable EM-type approximations for SDEs with non-globally Lipschitz coefficients. It develops implicit schemes for highly nonlinear settings and uses a stochastic discrete LaSalle principle to analyze almost sure asymptotic stability.
- Motivation and objectives: The primary objective is to establish strong convergence and stability for numerical approximations when drift and diffusion coefficients are not globally Lipschitz continuous.The motivation includes computable approximations for Monte Carlo simulations and the need to control moments under monotone conditions.
- Motivation and objectives: Super-linearly growing coefficients can cause explicit EM approximations to fail to converge strongly or weakly, motivating modified implicit schemes.The cited nonlinear example satisfies the monotone condition even though the standard EM method can fail.
- Convergence analysis: The paper targets multidimensional strong convergence under a general monotone condition, supporting variance reduction and implying weak and pathwise convergence.The authors describe the condition as potentially optimal for boundedness of moments of implicit schemes.
- Stability analysis: A stochastic counterpart of the discrete LaSalle principle is used to investigate almost surely asymptotic stability of nonlinear numerical schemes.The stability analysis is motivated by error propagation and possible divergence of simulated paths or biased Monte Carlo functionals.
- Scope and comparison: The proposed framework covers highly nonlinear SDEs under mild time-step assumptions represented by the A(α)-stability concept.Compared with earlier work, the analysis relaxes linear growth requirements on diffusion coefficients and uses the BEM scheme.
- Numerical methods: The θ-EM scheme, including BEM when θ = 1, is analyzed for moment bounds and strong convergence, while FBEM addresses measurability difficulties without Malliavin calculus.The paper compares FBEM and θ-EM in the Lp-sense and extends compact-domain convergence results to the whole domain.
2 Existence and Uniqueness of Solution
Under local Lipschitz and monotone-type assumptions, the paper establishes existence, uniqueness, and globality of SDE solutions together with moment bounds. These bounds support the later convergence analysis.
- Assumptions: The coefficients are required to satisfy local Lipschitz continuity and a monotone condition.These assumptions form the basis for the existence, uniqueness, and boundedness results.
- Moment bounds: The finite-time compact-domain probability bound for the solution is introduced specifically to support the paper's main convergence theorem.This bound complements the global existence and moment estimates.
- Existence and uniqueness: Under Assumption 2.1, the SDE has a unique global solution for every deterministic initial value x(0) = x0 ∈ Rn.The result is stated for any given initial value and establishes global existence.
- Moment bounds: The proof applies Itô's formula to V(x, t) = ∥x∥2 and uses Gronwall's inequality to derive the required bounds.The resulting estimates are used in the subsequent convergence analysis.
3 The θ-Euler-Maruyama Scheme
The θ-EM scheme uses implicitness to approximate non-globally Lipschitz SDEs under one-sided Lipschitz and polynomial-growth conditions. For θ ≥ 0.5 and sufficiently small time steps, its second moments remain uniformly bounded over finite time intervals.
- Scheme definition: The θ-EM scheme introduces a parameter θ ∈ [0,1] controlling implicitness, with the analysis requiring θ ≥ 0.5.The scheme is defined on a time partition and can be implemented through the inverse of F(x) = x − θf(x)∆t.
- Well-posedness: A one-sided Lipschitz condition on the drift ensures that each implicit update has a unique solution when ∆t < 1/(θL).The update can therefore be represented using an inverse function F^−1; for complicated drifts, root-finding methods may be used.
- Moment analysis: The second-moment analysis uses stopping times, overshoot control, discrete Gronwall estimates, and a polynomial-growth assumption on the coefficients.The stopping-time argument addresses the possibility that a discrete process crosses the stopping level between successive time points.
- Moment bound: The moment bound applies only for time steps satisfying ∆t ≤ ∆t∗, where ∆t∗ is restricted by the one-sided Lipschitz and coefficient-growth constants.The restriction arises from the implicitness of the numerical scheme.
- Moment bound: Under Assumptions 2.1, 3.1, and 3.5 with θ ≥ 0.5, the θ-EM scheme has uniformly bounded second moments over finite horizons.There exists C(T) > 0 such that sup_{∆t≤∆t∗} sup_{0≤t_k≤T} E∥X_tk∥^2 < C(T), subject to the stated time-step restriction.
4 Forward-Backward Euler-Maruyama Scheme
The paper introduces the FBEM to obtain an adapted continuous-time representation while analyzing the θ-EM scheme under non-globally Lipschitz conditions. It proves strong convergence, extends the method through partial implicitness, and reports numerical evidence consistent with strong order one-half.
- Scheme construction: The continuous extension of θ-EM is not adapted, so FBEM is introduced to avoid Malliavin calculus while preserving gridpoint values.The continuous and discrete FBEM schemes coincide at gridpoints.
- Convergence analysis: For θ ≥ 0.5, FBEM and θ-EM remain close on compact domains and have controlled first-passage probabilities under the stated assumptions.These estimates support extending convergence from compact domains to the whole domain.
- Strong convergence: Theorem 4.4 establishes strong convergence of θ-EM to the SDE solution under the paper’s monotone-coefficient assumptions.The proof combines compact-domain convergence, scheme-comparison estimates, and bounds preventing explosion on finite intervals.
- Computational considerations: The implicit method may be computationally expensive because each step can require evaluating the inverse of F(x) = x − θf(x)∆t.For the numerical example, the inverse is obtained by solving a cubic equation explicitly, so the added complexity is avoided.
- Partial implicitness: Partial implicitness treats a split drift by applying implicitness only to f1 while evaluating f2 explicitly, and the convergence results extend under modified assumptions.The paper notes that this formulation can address highly nonlinear examples while requiring only f1 to satisfy the relevant implicitness condition.
- Numerical example: The numerical experiment uses BEM and Monte Carlo estimates, with the error plot comparing timestep refinements against a fine reference solution.The observed behavior is consistent with a strong convergence order of one-half.
5 Stability Analysis
The stability analysis develops a stochastic discrete analogue of LaSalle’s principle for θ-EM and uses Lyapunov arguments to establish almost sure asymptotic behavior. The discrete result parallels the continuous stability theorem under the stated timestep and coefficient assumptions.
- 5.1 Continuous Case: The continuous analysis uses V(x) = ∥x∥2 and a stochastic LaSalle theorem to show decay of z(x(t)) almost surely.When z(x) = 0 only at x = 0, the solution converges almost surely to the origin.
- 5.1 Continuous Case: The proof uses a Doob decomposition and martingale convergence argument to control the nonnegative stochastic process arising from the Lyapunov analysis.The paper identifies this lemma as combining Doob decomposition with the martingale convergence theorem.
- 5.2 Almost Sure Stability: Theorem 5.3 provides a discrete counterpart of the continuous LaSalle theorem for the θ-EM scheme under assumptions including a bounded timestep.The theorem is stated for ∆t < (max{L, 2β}θ)^−1.
- 5.2 Almost Sure Stability: The discrete Lyapunov argument yields lim k→∞ z(X_tk) = 0 almost surely for the numerical approximation.This is obtained by constructing the martingale decomposition required by the stochastic convergence lemma.
A Proof of Lemma 3.3
The proof bounds coefficient terms on compact domains and controls Brownian increments using moment estimates. These bounds yield the lemma’s required assertion through an earlier convergence result.
- Coefficient bounds: Under the stated assumptions, the drift and diffusion terms are bounded on regions with ∥x∥ < m.The proof introduces a constant controlling ∥F(x)∥p and ∥g(x)∥ on the compact region.
- Moment estimates: Brownian increment moments satisfy E∥∆w_tk−1∥2p < C(p), enabling the required stochastic moment estimate.The argument combines this estimate with the coefficient bounds.
- Conclusion: The lemma follows by applying Lemma 3.2 after establishing the preceding bounds.The proof explicitly concludes through that earlier lemma.
B Proof of Theorem 4.2
The proof of Theorem 4.2 controls the probability of exiting a bounded region and compares the continuous FBEM and θ-EM stopping events. Choosing the truncation level and timestep appropriately yields the desired estimate.
- Conclusion: Combining the stopping-time estimates completes the proof of the theorem’s convergence bound.The final bound follows after combining the intermediate inequalities.
- Stopping-time comparison: Stopping-time comparisons use lower bounds on the norm at the first passage event and estimates from Lemmas 3.2 and 4.1.The argument distinguishes which continuous approximation reaches the threshold first.
- Stopping-time control: The stopping-time probability is bounded by [∥x0∥2 + 2αT + C(m,T)∆t1/2] exp(4βT).This estimate controls the relevant exit event over the finite interval [0,T].
- Parameter selection: The proof chooses a sufficiently large truncation level m and then a timestep ∆t0(m) small enough to satisfy the required bounds.The choices are made sequentially for any given ǫ > 0.
C Proof of Lemma 4.3
The proof follows the classical globally Lipschitz case while adapting estimates to constants that depend on the moment parameter. Hölder’s, Burkholder–Davis–Gundy, and Gronwall inequalities complete the argument.
- The proof adapts the classical globally Lipschitz argument to the setting where the coefficient-dependent constant C(T, m) depends on m.The exposition nevertheless provides a proof sketch for completeness.
- For any T1 ∈[0, T ], the argument invokes Hölder’s and Burkholder–Davis–Gundy’s inequalities.
- Assumption 2.1 supplies a constant C(m) used in the proof’s estimates.
- The same reasoning underlying estimate (B.6) yields the required intermediate bound.
- The theorem follows by applying the Gronwall inequality.