Stochastic C-stability and B-consistency of explicit and implicit Euler-type schemes

1. Introduction

• Stochastic C-stability and B-consistency simplify error analysis without requiring higher moment estimates or continuous-time extensions of the numerical scheme.The framework generalizes C-stability to stochastic problems; the convergence measure uses the L2 norm because the stability proof relies on conditional-expectation orthogonality.Source: §1 · pp. 1–2
• Global monotonicity includes SODEs with superlinearly growing coefficients, for which explicit Euler-Maruyama is known to diverge.Prior work also analyzes backward Euler, tamed Euler, and stopped-tamed Euler-Maruyama variants under related settings.Source: §1 · p. 2
• The paper's strong-convergence notion uses an L2 error measure, rather than stronger alternatives such as a maximum inside the norm or an Lp norm with p > 2.The choice is tied to conditional-expectation orthogonality in the stability-lemma proof.Source: §1 · p. 2
• The split-step backward Euler method is strongly convergent of order γ = 1/2, and the projected Euler-Maruyama method is also strongly convergent.Both methods are included as concrete examples of stochastically C-stable schemes.Source: §1 · pp. 2–3
• The two methods have equal experimental strong errors in non-stiff problems, favoring the simpler explicit PEM; stiff problems may require impractically small explicit-method steps.The experiments illustrate results for the stochastic Ginzburg-Landau equation and the financial 3/2-volatility model.Source: §1 · pp. 2–3
• A related mean-square convergence theorem [25] uses a similar local truncation-error concept, but its stepwise argument does not exploit global Lipschitz properties of the numerical method.The paper also relates its approach methodologically to one-leg theta methods and B-convergence.Source: §1 · p. 3

Problem setting

• The problem setup assumes continuous drift and diffusion coefficients under global monotonicity, with q > 1 excluding the global Lipschitz case.Under the stated assumption, a unique solution to the SODE exists.Source: §2 · p. 4
• The exact solution has a finite p-th moment when the stated global coercivity condition holds.The condition is given as an additional assumption beyond global monotonicity.Source: §2 · p. 4

Numerical methods

• The paper's stated aim is to establish both SSBE and PEM as stochastically C-stable numerical methods.The methods are then analyzed for strong convergence under the paper's assumptions.Source: §2 · pp. 4–5

Split-step backward Euler method

• The SSBE method is strongly convergent of order γ = 1/2, with diffusion coefficients evaluated at the current time point t_i.This evaluation aligns the drift and diffusion terms at the same time in the global monotonicity condition and avoids technical issues in the analysis.Source: §2 · p. 5

Projected Euler-Maruyama method

• PEM applies a projection onto a ball with radius expanding as a negative power of the step size to the standard Euler-Maruyama method.The paper presents PEM as a new explicit stochastic scheme, inspired by a truncation procedure [15, Chap. 2, Theorem 3.4].Source: §2 · p. 5

• The framework defines stochastic one-step methods through adapted, square-integrable grid functions generated by a one-step map.The map is required to satisfy measurability and integrability conditions.Source: §3 · p. 6
• Stochastic B-consistency is introduced as the local truncation-error condition paired with stochastic C-stability in the convergence theorem.The section defines strong convergence by an error bound against the exact solution over the grid.Source: §3 · pp. 6–7
• The stability lemma propagates error through one application of the numerical one-step map, whose stochastic C-stability supplies a global Lipschitz property.The proof then uses a telescopic sum and the discrete Gronwall lemma.Source: §3 · pp. 7–9
• Under an additional one-step bound, stochastic C-stability also yields a bound on the generated grid function's second moment.Source: §3 · p. 9
• Stochastic C-stability and B-consistency of order γ, with the exact initial value, imply strong convergence of order γ.The convergence bound is uniform over the permitted vectors of step sizes.Source: §3 · p. 9

• A continuous map satisfying the stated uniform monotonicity inequality is a homeomorphism with a Lipschitz-continuous inverse.The section cites this result as a standard theorem in nonlinear analysis.Source: §4 · p. 10
• The resulting nonlinear-equation estimates are identified as key to proving C-stability of the split-step backward Euler scheme.Source: §4 · p. 10
• For the split-step backward Euler drift map Fδ(t,x)=x−δf(t,x), the section establishes homeomorphism when the step bound satisfies h<L^-1.The map is defined for δ in (0,h].Source: §4 · pp. 10–11

5. C-stability and B-consistency of the SSBE method

• The SSBE scheme is a stochastic one-step method, with a bound involving a constant depending on L, q, m, and h.Source: §5 · p. 12
• Under Assumption 2.1 and h < L^-1, the SSBE scheme is stochastically C-stable.Source: §5 · p. 13
• SSBE is stochastically B-consistent of order 1/2 when the exact solution has the stated uniform L^(4q−2) moment bound.Source: §5 · p. 16
• Under the stated assumptions and exact-solution moment bound, SSBE is strongly convergent of order 1/2.The result applies for step sizes h < L^-1.Source: §5 · p. 18
• The BEM error argument described here requires higher moment bounds for its grid function uniformly in the step size; the authors do not establish BEM’s stochastic C-stability under Assumption 2.1.The passage distinguishes this argument from a direct mean-square convergence proof cited as [1].Source: §5 · p. 19

6. C-stability and B-consistency of the PEM method

• PEM is presented as stochastically C-stable and B-consistent of order 1/2.Source: §6 · p. 19
• The PEM projection map is globally Lipschitz continuous with Lipschitz constant 1.Source: §6 · p. 20
• The section verifies that PEM is stochastically C-stable.Source: §6 · pp. 21–22
• The section states that PEM’s strong convergence follows from its C-stability and B-consistency results together with the abstract convergence theorem.Source: §6 · p. 24

Ginzburg-Landau equation

• For the Ginzburg-Landau equation, no method has a significant advantage, and fitted experimental orders are 0.55 for SSBE, 0.52 for BEM, and 0.54 for PEM.The estimates use six equidistant step sizes and 10^6 Monte Carlo sample paths.Source: §7 · pp. 26–27
• At step size h = 2^-6, approximately 3.3 percent of PEM trajectories do not coincide with standard Euler-Maruyama trajectories; as h decreases, the number of such samples drops quickly.The passages leave open whether these events arise from standard Euler-Maruyama explosiveness or an intrinsic feature of the exact solution; if the latter, projection may cause more harm than good.Source: §7 · p. 28

3/2-volatility model

• For sufficiently small step sizes, all three methods show strong convergence in line with predicted order γ = 1/2, with no significant behavioral differences; at larger step sizes, BEM outperforms SSBE, while PEM performs much worse.The average experimental orders for SSBE, BEM, and PEM are 0.63, 0.53, and 0.77, respectively; theoretical results are verified only for small h.Source: §7 · p. 29
• In the volatility model, stronger noise and weaker drift damping accompany better BEM performance; implicit damping of noise contrasts with the undamped latest noise increment in SSBE and the explicit PEM scheme.The exact solution more often leaves PEM’s projection sphere under high noise, and more trajectories consequently undergo projection.Source: §7 · p. 29

Overall observations

• PEM is generally cheaper to simulate, so its step size can be reduced when projection-event frequency signals that a smaller step may be needed.The experiments suggest choosing a smaller step when the relative frequency of projection events is too high.Source: §7 · p. 30