Explicit Modified Euler Methods Overview

• Explicit Modified Euler Methods (MEMs) are explicit Euler discretizations modified via truncation, projection, or taming to stabilize dynamics in SDEs with super-linear growth.
• The methods incorporate techniques like coefficient regularization and Magnus factorization to overcome issues such as moment explosion, loss of positivity, and singular drifts.
• Convergence analyses show MEMs achieve strong convergence, typically with mean-square order 1/2, while preserving long-time ergodic properties and invariant measures.

Searching arXiv for recent and foundational papers on explicit modified Euler methods relevant to the requested encyclopedia entry. arXivSearch({"query":"all:\"modified Euler methods\" OR all:\"modified Euler scheme\" stochastic differential equations explicit", "max_results": 10, "sort_by": "submittedDate", "sort_order": "descending"}) Explicit modified Euler methods (MEMs) are Euler-type discretizations in which the raw explicit Euler or Euler–Maruyama step is altered through truncation, taming, projection, coefficient regularization, partial implicitness in a singular term, or factorization of a stiff linear part. In the literature covered here, the label encompasses modified truncated Euler–Maruyama schemes for locally Lipschitz stochastic differential equations (SDEs), explicit modified Euler recursions for super-linear and non-contractive stochastic ordinary differential equations (SODEs), projection- and taming-based schemes for financial and McKean–Vlasov models, Magnus–Euler–Maruyama integrators for stochastic delay-differential equations (SDDEs), and a two-step modified explicit Euler/Crank–Nicolson method for a variable-order fractional advection–dispersion equation. The common objective is to retain an explicit update while restoring strong convergence, stability, positivity, moment control, or long-time ergodic fidelity in regimes where classical explicit Euler methods fail (Lan et al., 2017, Liu et al., 10 Sep 2025, Bao et al., 2024).

1. General formulation and scope

A central MEM template in the super-linear SODE literature is

$$Y_{n+1} = \mathcal{P}_\tau(Y_n) + b_\tau(\mathcal{P}_\tau(Y_n)) \tau + \sum_{j=1}^{m} \sigma_{j, \tau} (\mathcal{P}_\tau(Y_n)) \delta W_n,$$

where $\mathcal{P}_\tau$ is a modification or projection of the current position, $b_\tau$ is a modified or tamed drift, and $\sigma_{j,\tau}$ are modified or tamed diffusion coefficients. Within this framework, the tamed Euler method and the projected Euler method both appear as special cases, and the modifications are designed to stabilize the dynamics and permit long-time convergence analysis under super-linear growth and multiplicative noise (Liu et al., 10 Sep 2025).

A related general framework writes the numerical chain as

$$X_{(n+1)\delta} = \mathcal{T}^{(\delta)}(X_{n\delta}) + b^{(\delta)}(\mathcal{T}^{(\delta)}(X_{n\delta}))\,\delta + \sqrt{\delta}\,\xi_{n+1},$$

with $\mathcal{T}^{(\delta)}$ a projection or truncation map and $b^{(\delta)}$ a step-size-dependent drift modification satisfying $b^{(\delta)}(x)\to b(x)$ as $\delta\to 0$. By suitable choices of $\mathcal{T}^{(\delta)}$ and $\mathcal{P}_\tau$0, this form covers the classical Euler–Maruyama scheme, the tamed Euler scheme, and the truncated Euler scheme. The framework is explicitly intended for SDEs with super-linear drift, where the classical Euler scheme works merely for coefficients of linear growth (Bao et al., 2024).

This breadth of usage implies that “MEM” is not a single canonical algorithm. Rather, the term denotes an explicit-Euler lineage whose defining feature is the insertion of a stabilizing or structure-preserving modification before, within, or after the Euler increment.

2. Principal modification mechanisms

In the modified truncated Euler–Maruyama (MTEM) method for

$\mathcal{P}_\tau$1

the original coefficients are replaced by modified truncated functions

$\mathcal{P}_\tau$2

and similarly for $\mathcal{P}_\tau$3, with $\mathcal{P}_\tau$4 as $\mathcal{P}_\tau$5. The recursion is

$\mathcal{P}_\tau$6

A key structural fact is that these modified truncated functions are globally Lipschitz, with constant $\mathcal{P}_\tau$7. If $\mathcal{P}_\tau$8, then $\mathcal{P}_\tau$9 and $b_\tau$0, so the method reduces to classical Euler–Maruyama. For suitable truncations it includes the truncated EM method as a special case, but with greater flexibility in the choice of $b_\tau$1 (Lan et al., 2017).

A projection-based explicit Euler construction for non-Lipschitz financial SDEs replaces the raw drift evaluation by $b_\tau$2, where $b_\tau$3 projects onto a restricted domain such as

$b_\tau$4

The resulting explicit recursion

$b_\tau$5

keeps the numerical state away from boundary regions where the drift may diverge. In the cited finance applications, this device is paired with Lamperti transforms and one-sided Lipschitz drift structure (Chassagneux et al., 2014).

For the generalized Aït-Sahalia interest-rate model with Poisson jumps,

$b_\tau$6

the proposed explicit modified Euler scheme is explicit in all terms except the singular drift component $b_\tau$7. The update is

$b_\tau$8

and the step can be solved as the unique positive root of a quadratic formula. This construction is designed to preserve positivity unconditionally, for any step size $b_\tau$9, while taming the nonlinear drift and diffusion through the modification functions $\sigma_{j,\tau}$0 and $\sigma_{j,\tau}$1 (Jiang et al., 30 Jun 2025).

For McKean–Vlasov SDEs with super-linear coefficients, the modification acts directly on coefficient values through nonlinear operators:

$\sigma_{j,\tau}$2

for the modified Euler method (ME),

$\sigma_{j,\tau}$3

for the tanh Euler method (TE), and

$\sigma_{j,\tau}$4

for the sin Euler method (SE), with $\sigma_{j,\tau}$5. These operators tame both drift and diffusion, unlike earlier drift-only tamed constructions (Jian et al., 7 Feb 2025).

A distinct specialization arises in Magnus–Euler–Maruyama methods for SDDEs. There the MEM step is

$\sigma_{j,\tau}$6

with

$\sigma_{j,\tau}$7

and

$\sigma_{j,\tau}$8

Here the modification is a Magnus factorization of the linear multiplicative part, while the nonlinear remainder is treated by Euler–Maruyama (Griggs et al., 20 Jun 2025).

3. Convergence theory and quantitative rates

For MTEM, the finite-time analysis is developed under local Lipschitz, monotonicity-type, and Khasminskii-type conditions, together with $\sigma_{j,\tau}$9 and $$X_{(n+1)\delta} = \mathcal{T}^{(\delta)}(X_{n\delta}) + b^{(\delta)}(\mathcal{T}^{(\delta)}(X_{n\delta}))\,\delta + \sqrt{\delta}\,\xi_{n+1},$$0. If

$$X_{(n+1)\delta} = \mathcal{T}^{(\delta)}(X_{n\delta}) + b^{(\delta)}(\mathcal{T}^{(\delta)}(X_{n\delta}))\,\delta + \sqrt{\delta}\,\xi_{n+1},$$1

for suitable $$X_{(n+1)\delta} = \mathcal{T}^{(\delta)}(X_{n\delta}) + b^{(\delta)}(\mathcal{T}^{(\delta)}(X_{n\delta}))\,\delta + \sqrt{\delta}\,\xi_{n+1},$$2, then the fixed-time strong error satisfies

$$X_{(n+1)\delta} = \mathcal{T}^{(\delta)}(X_{n\delta}) + b^{(\delta)}(\mathcal{T}^{(\delta)}(X_{n\delta}))\,\delta + \sqrt{\delta}\,\xi_{n+1},$$3

and

$$X_{(n+1)\delta} = \mathcal{T}^{(\delta)}(X_{n\delta}) + b^{(\delta)}(\mathcal{T}^{(\delta)}(X_{n\delta}))\,\delta + \sqrt{\delta}\,\xi_{n+1},$$4

Under slightly stronger assumptions, including polynomial growth of $$X_{(n+1)\delta} = \mathcal{T}^{(\delta)}(X_{n\delta}) + b^{(\delta)}(\mathcal{T}^{(\delta)}(X_{n\delta}))\,\delta + \sqrt{\delta}\,\xi_{n+1},$$5, the uniform-in-time error on $$X_{(n+1)\delta} = \mathcal{T}^{(\delta)}(X_{n\delta}) + b^{(\delta)}(\mathcal{T}^{(\delta)}(X_{n\delta}))\,\delta + \sqrt{\delta}\,\xi_{n+1},$$6 obeys

$$X_{(n+1)\delta} = \mathcal{T}^{(\delta)}(X_{n\delta}) + b^{(\delta)}(\mathcal{T}^{(\delta)}(X_{n\delta}))\,\delta + \sqrt{\delta}\,\xi_{n+1},$$7

for $$X_{(n+1)\delta} = \mathcal{T}^{(\delta)}(X_{n\delta}) + b^{(\delta)}(\mathcal{T}^{(\delta)}(X_{n\delta}))\,\delta + \sqrt{\delta}\,\xi_{n+1},$$8. The analysis is presented as weaker than the sufficient conditions required by the earlier truncated EM method, and fixed-time convergence does not require polynomial growth or global Lipschitz continuity of $$X_{(n+1)\delta} = \mathcal{T}^{(\delta)}(X_{n\delta}) + b^{(\delta)}(\mathcal{T}^{(\delta)}(X_{n\delta}))\,\delta + \sqrt{\delta}\,\xi_{n+1},$$9 (Lan et al., 2017).

For projection-based explicit Euler discretizations of financial SDEs with non-Lipschitz coefficients, the main strong error estimate is

$\mathcal{T}^{(\delta)}$0

with

$\mathcal{T}^{(\delta)}$1

under basic regularity and moment assumptions, and

$\mathcal{T}^{(\delta)}$2

under stronger $\mathcal{T}^{(\delta)}$3 regularity. If the diffusion is constant, first-order convergence is possible provided sufficiently high moments exist. In the Cox–Ingersoll–Ross case, the rate is piecewise characterized in terms of $\mathcal{T}^{(\delta)}$4, and for the transformed Aït-Sahalia model with $\mathcal{T}^{(\delta)}$5, first-order strong convergence is stated to be achievable (Chassagneux et al., 2014).

For the generalized Aït-Sahalia model with jumps, the central quantitative result is

$\mathcal{T}^{(\delta)}$6

establishing mean-square strong convergence of order $\mathcal{T}^{(\delta)}$7 in both the non-critical case $\mathcal{T}^{(\delta)}$8 and the critical case $\mathcal{T}^{(\delta)}$9 under the stated parameter conditions. A specific point emphasized in the paper is that the exact order $b^{(\delta)}$0 remains recoverable despite the presence of Poisson jumps (Jiang et al., 30 Jun 2025).

For McKean–Vlasov SDEs, the modified Euler approximations are first analyzed against the interacting particle system:

$b^{(\delta)}$1

which yields mean-square order $b^{(\delta)}$2. Propagation of chaos then gives the full approximation error to the mean-field limit,

$b^{(\delta)}$3

where the particle error exponent $b^{(\delta)}$4 depends on dimension through the quoted rates $b^{(\delta)}$5 for $b^{(\delta)}$6, $b^{(\delta)}$7 for $b^{(\delta)}$8, and $b^{(\delta)}$9 for $b^{(\delta)}(x)\to b(x)$0 (Jian et al., 7 Feb 2025).

For SDDEs, the Magnus–Euler–Maruyama scheme has mean-square order of convergence $b^{(\delta)}(x)\to b(x)$1:

$b^{(\delta)}(x)\to b(x)$2

In a different application area, the two-step fourth-order modified explicit Euler/Crank–Nicolson method for the time-variable fractional mobile-immobile advection-dispersion equation is proved unconditionally stable in the $b^{(\delta)}(x)\to b(x)$3 norm, with error estimate

$b^{(\delta)}(x)\to b(x)$4

hence first order in time and fourth order in space (Griggs et al., 20 Jun 2025, Ngondiep, 2022).

4. Long-time approximation, ergodicity, and invariant measures

A major strand of recent MEM research concerns long-time approximation rather than only finite-time strong error. For super-linear SODEs with multiplicative noise, a family of explicit MEMs is constructed to preserve the same Lyapunov structure as the continuous problem. With $b^{(\delta)}(x)\to b(x)$5, the numerical recursion satisfies

$b^{(\delta)}(x)\to b(x)$6

Under a non-contractive condition, the law of the numerical approximation obeys the non-asymptotic Wasserstein-1 estimate

$b^{(\delta)}(x)\to b(x)$7

and, as a by-product, the invariant measures satisfy

$b^{(\delta)}(x)\to b(x)$8

The analysis explicitly targets contractivity at infinity rather than global contractivity (Liu et al., 10 Sep 2025).

A parallel framework establishes geometric ergodicity for modified Euler schemes applied to SDEs with super-linear drift. Under step-size-dependent Lipschitz-type and dissipativity conditions on the modified map $b^{(\delta)}(x)\to b(x)$9 and drift $\delta\to 0$0, the numerical chain is shown to be geometrically ergodic in a mixed distance combining total variation and $\delta\to 0$1-Wasserstein, and also in a weighted total variation distance. For the tamed Euler scheme, coupling by reflection yields exponential contractivity in the classical $\delta\to 0$2-Wasserstein metric,

$\delta\to 0$3

and the invariant measures of the SDE and the numerical chain satisfy

$\delta\to 0$4

This places explicit MEMs within a quantitative ergodic framework rather than only a pathwise convergence framework (Bao et al., 2024).

For additive-noise SDEs with high diffusivity and drift dissipative only outside a closed ball, modified Euler schemes admit explicit $\delta\to 0$5-Wasserstein contraction. The numerical transition law satisfies

$\delta\to 0$6

obtained via synchronous coupling and an equivalent quasi-metric. The same analysis yields non-asymptotic $\delta\to 0$7 bounds, explicit invariant-measure approximation rates, Poincaré inequalities, concentration inequalities for empirical averages, KL-divergence bounds, and a strong law of large numbers for additive functionals. For tamed Euler the convergence rate between exact and numerical invariant measures approaches $\delta\to 0$8 as the noise strength increases, while for projected Euler the rate is improved to $\delta\to 0$9 under the stated conditions (Bao et al., 2024).

5. Model classes and application domains

The MTEM framework is illustrated on SDEs that fall outside polynomial-growth settings. One example is

$\mathcal{T}^{(\delta)}$0

for which the coefficients are locally Lipschitz but not polynomially growing. The paper verifies the assumptions for strong convergence and obtains

$\mathcal{T}^{(\delta)}$1

A second example,

$\mathcal{T}^{(\delta)}$2

yields the time-uniform estimate

$\mathcal{T}^{(\delta)}$3

for arbitrarily small $\mathcal{T}^{(\delta)}$4 by a suitable choice of $\mathcal{T}^{(\delta)}$5. The contrast drawn there is that truncated EM had previously been proved only at fixed time for this SDE, whereas MTEM attains uniform convergence on $\mathcal{T}^{(\delta)}$6 (Lan et al., 2017).

In mathematical finance, modified explicit Euler discretizations are developed for the CIR, 3/2, and Aït-Sahalia models, as well as for a family of mean-reverting processes with locally smooth coefficients. The construction is designed for drifts or diffusions that behave like $\mathcal{T}^{(\delta)}$7, $\mathcal{T}^{(\delta)}$8, or $\mathcal{T}^{(\delta)}$9 near the boundary, often after a Lamperti transform. Numerical illustrations are reported together with a multilevel Monte Carlo implementation, and substantial computational savings by factors of $\mathcal{P}_\tau$00–$\mathcal{P}_\tau$01 are stated for zero-coupon bond pricing and spread options when compared with standard Monte Carlo (Chassagneux et al., 2014).

For the generalized Aït-Sahalia model with Poisson jumps, the explicit modified Euler method is presented as positivity-preserving for any step size $\mathcal{P}_\tau$02 and computationally cheaper than backward Euler–Maruyama while matching its mean-square order $\mathcal{P}_\tau$03. The empirical mean-square errors in the numerical study are approximated over $\mathcal{P}_\tau$04 paths, and both the tamed and projection modifications are reported to achieve the predicted rate in non-critical and critical regimes (Jiang et al., 30 Jun 2025).

For McKean–Vlasov dynamics, the proposed MEMs target interacting particle systems with distribution-dependent, super-linear drift and diffusion. The numerical experiments compare ME, TE, SE, drift-tamed Euler, and a split-step implicit method. TE is described as particularly robust; for large initial values and coarse step sizes it remains close to the split-step implicit method, whereas DTE and SE may become unstable or yield corrupted particles. A specific reported example is $\mathcal{P}_\tau$05 with $\mathcal{P}_\tau$06, where ME and TE remain stable while DTE and SE can blow up (Jian et al., 7 Feb 2025).

Outside SDEs in the narrow sense, modified explicit Euler ideas also appear in fractional transport and delay equations. The two-step fourth-order modified explicit Euler/Crank–Nicolson approach for the variable-order fractional mobile-immobile advection-dispersion equation is analyzed in the $\mathcal{P}_\tau$07 norm and shown to be unconditionally stable, with numerical results consistent with first-order time and fourth-order space accuracy. For SDDEs and a stochastic partial delay-differential equation, Magnus–Euler–Maruyama remains numerically stable under fine spatial discretization while the traditional Euler–Maruyama method becomes unstable (Ngondiep, 2022, Griggs et al., 20 Jun 2025).

6. Conceptual distinctions, limitations, and recurrent issues

The literature suggests that explicit MEMs are best understood as an umbrella class rather than a single method. Truncation, projection, taming, and Magnus factorization share the explicit Euler lineage, but they target different structural failures of classical schemes: moment explosion, domain escape, singular drift, lack of positivity, or stiffness in the linear multiplicative part.

A separate but related notion is the modified equation of backward error analysis. For explicit Euler applied to $\mathcal{P}_\tau$08, the modified differential equation

$\mathcal{P}_\tau$09

with

$\mathcal{P}_\tau$10

leads to the first correction

$\mathcal{P}_\tau$11

In fast–slow ODEs this framework explains discretization-induced canard effects: for fold singularities the delay in loss of stability is shortened, whereas for transcritical singularities it can be prolonged arbitrarily. This is analytically distinct from MEMs as numerical schemes, but it clarifies how Euler modifications alter qualitative dynamics (Engel et al., 2023).

Several limitations recur across the MEM literature. In MTEM, the practical choice of truncation radius $\mathcal{P}_\tau$12 must be addressed; very large $\mathcal{P}_\tau$13 may induce numerical instability for large step sizes, and rare events outside the truncation radius may in principle be handled incorrectly. The strong rate may also degrade for large moments or very rough coefficients because it depends on the order $\mathcal{P}_\tau$14 used in the analysis (Lan et al., 2017). In high-diffusivity $\mathcal{P}_\tau$15-contraction theory, the noise intensity $\mathcal{P}_\tau$16 must be sufficiently large and enters the explicit lower-bound condition through the Lyapunov constants (Bao et al., 2024). In the McKean–Vlasov setting, the experiments indicate that not all explicit modifications are equally robust: DTE and SE may become unstable unless the step size is sufficiently small, whereas TE and ME are more stable under the reported tests (Jian et al., 7 Feb 2025).

These features indicate the central design tension in explicit MEMs. The schemes aim to preserve the computational simplicity of an explicit step while importing just enough structure—through coefficient clipping, state projection, partial implicitness, or analytic factorization—to recover finite-time accuracy and long-time fidelity in models with super-linearity, singularities, delay, or non-contractive dynamics.