Explicit Truncated Euler–Maruyama Schemes

• Explicit truncated Euler–Maruyama schemes are numerical methods for SDEs that apply truncation or projection to handle non-globally Lipschitz and superlinear coefficients.
• They achieve strong convergence by modifying the drift and diffusion evaluations through a step-size–dependent truncation operator, ensuring stable simulation even when standard methods diverge.
• These schemes are practically applied in finance, physics, and biology to simulate SDEs with irregular dynamics while preserving key properties like positivity and computational efficiency.

Explicit truncated Euler–Maruyama schemes are a class of explicit numerical discretizations for stochastic differential equations (SDEs) in which the drift and/or diffusion coefficients are “truncated” or projected before each time step to control the effect of non-globally Lipschitz, superlinear, or otherwise irregular coefficients. This approach extends the applicability of explicit schemes to SDEs where the standard Euler–Maruyama method is known to diverge, enabling stable and strongly convergent simulations for a broad range of models in finance, physics, biology, and engineering.

1. Core Principles and Mathematical Formulation

Central to the explicit truncated Euler–Maruyama schemes is the modification of the forward Euler update by (i) applying a truncation (or projection) to the argument at which the drift $f$ and diffusion $g$ are evaluated, or (ii) truncating the coefficients themselves outside a prescribed, step-size dependent region. Given an SDE

$dX_t = f(X_t)\,dt + g(X_t)\,dW_t,\quad X_0 = x_0,$

the prototypical explicit truncated Euler–Maruyama update reads

$$X_{n+1} = X_n + f_\mathrm{tr}(X_n)\,h + g_\mathrm{tr}(X_n)\,\Delta W_n,$$

where:

• $h$ is the time step, and $\Delta W_n = W_{t_{n+1}} - W_{t_n}$,
• $f_\mathrm{tr}(x) = f(p_h(x))$ and $g_\mathrm{tr}(x) = g(p_h(x))$,
• $p_h$ is a truncation or projection operator designed so that $p_h(x)$ is “tamed” within a ball or interval $D_h$ whose size increases as $h\to 0$.

A common choice is

$$p_h(x) = \min\{ \|x\|, R(h) \}\frac{x}{\|x\|},\quad \text{for }\|x\|>0,$$

or, in one dimension, $p_h(x) = R(h)^{-1} \vee x \wedge R(h)$ for some $R(h) \to \infty$ as $h\to 0$.

This truncation ensures that the numerical method does not experience unbounded growth due to superlinear or non-Lipschitz coefficients, while the “mild” effect of the truncation vanishes for sufficiently small time steps.

2. Strong Convergence and Error Analysis

A major theoretical advance is the rigorous derivation of strong convergence rates for these explicit truncated schemes, even for SDEs with locally Lipschitz and/or one-sided Lipschitz drift and locally Lipschitz or polynomially growing diffusion coefficients. Under appropriate moment and regularity hypotheses—often including a Khasminskii-type integrability condition or a monotonicity property—one obtains

$$\max_{n\leq N} \mathbb{E}\big[|X_{t_n} - \hat X_{t_n}|^2\big]^{1/2} \leq C h^r,$$

where $r$ typically approaches $1/2$ (the standard rate for explicit Euler–Maruyama under globally Lipschitz coefficients), with explicit expressions for $r$ depending on the exponents characterizing the local growth or Hölder continuity of $f$ and $g$ (Chassagneux et al., 2014, Guo et al., 2016, Lan et al., 2017, Liu et al., 2018, Li et al., 2022, Liu et al., 17 Mar 2024, Deng et al., 8 Oct 2024, Hu et al., 30 Dec 2024).

For instance, in the modified explicit Euler–Maruyama scheme of (Chassagneux et al., 2014), the truncation domain and projection exponents are set based on the degree of nonlinearity in $f$ and $g$, and strong rates of $1/2$ (or higher) are achieved for models such as CIR, 3/2, and Aït–Sahalia interest-rate models. The analysis is underpinned by (i) technical lemmas ensuring that the truncated drift retains one-sided Lipschitz properties, and (ii) discrete Gronwall inequalities adapted to the truncated context.

In multilevel Monte Carlo (MLMC) settings, the strong error controls both the bias and variance at each level, yielding an overall computational cost proportional to the inverse fourth power of the target root mean square error, $O(\varepsilon^{-4})$ (Guo et al., 2016).

3. Handling Superlinear, Nonsmooth, and Delay Coefficients

The appeal of explicit truncated schemes lies in their ability to handle SDEs with:

• Superlinear coefficients: By projecting onto balls of radius $h^{-\alpha}$ or similar, such schemes prevent “blow-up” while only modifying the solution far from the typical region (Beyn et al., 2014, Lan et al., 2017, Liu et al., 2018, Li et al., 2022).
• Non-Lipschitz/irregular drift and diffusion: As in (Liu et al., 17 Mar 2024), truncated schemes combined with Yamada–Watanabe approximations permit strong convergence results even under merely Hölder-continuous coefficients, and their convergence rates are proven to be robust to the number of stochastic delay terms.
• Multiplicative and/or degenerate noise: In SDEs with positive solutions (e.g., financial models imposing $X_t>0$), explicit truncation or projection ensures positivity preservation, without the need for Lamperti or logarithmic transformations or recourse to the Milstein scheme (Deng et al., 8 Oct 2024, Hu et al., 30 Dec 2024).

A typical implementation involves adaptive step sizes or truncation radii that depend on a step-size-dependent function $h$ and an increasing function $f$ controlling the coefficient’s growth (Lan et al., 2017, Liu et al., 2018).

4. Algorithmic Structure and Implementation Examples

A canonical algorithm for the projected/truncated Euler–Maruyama scheme is:

• Input: $T>0$, step-size $h$, SDE coefficients $f$, $g$, initial value $X_0$.
• For $n=0,\dots, N-1$
  1. Compute projected state: $Y_n = p_h(X_n)$.
  2. Set $X_{n+1} = X_n + f(Y_n)h + g(Y_n) \Delta W_n$, where $p_h$ is as above.

Python pseudocode:

def truncated_em(f, g, X0, T, N, R): h = T / N X = np.zeros(N+1) X[0] = X0 for n in range(N): Y = np.clip(X[n], 1/R, R) dW = np.random.normal(0, np.sqrt(h)) X[n+1] = X[n] + f(Y)*h + g(Y)*dW return X

For multidimensional or delay SDEs, the projection $p_h$ becomes componentwise or is applied to the full segment in the functional/delay case (Li et al., 2022, Liu et al., 17 Mar 2024). In applications to multiscale SDEs, explicit truncation plays a similar stabilizing role within heterogeneous multiscale frameworks (Cui et al., 2023).

5. Applications in Finance, SPDEs, and High-Dimensional Contexts

Financial SDEs with non-Lipschitz coefficients: The schemes are applied to mean-reverting models (CIR), stochastic volatility models (3/2, Aït–Sahalia), and VIX dynamics, yielding strong convergence with optimal rates and preserving positivity needed for practical simulation (Chassagneux et al., 2014, Guo et al., 2016, Deng et al., 8 Oct 2024, Hu et al., 30 Dec 2024, Liu et al., 17 Mar 2024).

Functional/delay equations and SPDEs: For stochastic functional differential equations (SFDEs), explicit truncated schemes allow the control of high-dimensional delay states and superlinear growth with L$^p$ boundedness and convergence, circumventing the need for implicit methods (Li et al., 2022). In infinite-dimensional SPDEs, spectral truncation can be combined with explicit predictor–corrector steps, resulting in optimal rates for strong, weak, and second-moment convergence (Lang et al., 2023).

Heavy-tailed and jump processes: For SDEs with $\alpha$-stable (or more general cylindrical Lévy) noise, explicit truncation of noise increments and/or coefficients guarantees uniform-in-time error control in Wasserstein distance, facilitating robust simulation even for systems with infinite variance jumps (Chen et al., 2022, Dang et al., 19 Feb 2024).

6. Practical Benefits, Limitations, and Extensions

Advantages:

• Fully explicit, computationally efficient, and simple to implement.

• Versatile: work under weak regularity assumptions, locally Lipschitz or one-sided monotonicity, superlinear growth, irregular delay structure, or high dimensions.
• The effect of the truncation/projection vanishes for small step sizes, so consistency with the original SDE is retained.

Limitations:

• The choice of truncation/domain parameters (radius or bound functions) requires care: too small a domain increases bias; too large may not control the numerical instability. Many schemes choose the domain size to blow up as $h\to 0$, carefully balancing accuracy and stability.
• For certain models (e.g., with extremely strong nonlinearities or singularities), the error rate may be sub-optimal unless additional moment/exponential integrability is enforced.
• Weak convergence rates may lag behind strong rates in the absence of further regularity (Hu et al., 30 Dec 2024).

Extensions:

• Pathwise and adaptive schemes: Advanced algorithms adaptively adjust the local step size or truncation threshold based on pathwise error estimators or the local “difficulty” of the SDE (Shardlow et al., 2014, Neuenkirch et al., 2018).
• Multilevel and variance reduction: In MLMC contexts, these methods enable unbiased variance estimation and significant computational savings through their compatibility with coarse/fine level differences (Guo et al., 2016).

7. Summary Table: Key Properties of Explicit Truncated Euler–Maruyama Schemes

Feature	Typical Value / Property	References
Drift/Diffusion Regularity	Locally Lipschitz, one-sided monotonicity, superlinear allowed	(Chassagneux et al., 2014, Lan et al., 2017)
Projection/Truncation Operation	$$\displaystyle p_h(x) = \min(1, h^{-\alpha}|x|^{-1})x$$	(Beyn et al., 2014, Guo et al., 2016)
Strong Convergence Rate	$r = 1/2$ (or model-dependent analog)	(Chassagneux et al., 2014, Lan et al., 2017)
Positivity preservation	Achieved via componentwise truncation in $[R^{-1}, R]$	(Deng et al., 8 Oct 2024, Hu et al., 30 Dec 2024)
Extension to delays/multiscale/SPDE	Supported (w/ adapted projection or spectral truncation)	(Li et al., 2022, Lang et al., 2023)
MLMC compatibility	Yes, with optimal complexity under strong error control	(Guo et al., 2016)

• (Chassagneux et al., 2014) “An explicit Euler scheme with strong rate of convergence for financial SDEs with non-Lipschitz coefficients”
• (Beyn et al., 2014) “Stochastic C-stability and B-consistency of explicit and implicit Euler-type schemes”
• (Guo et al., 2016) "Multi-level Monte Carlo methods with the Truncated Euler-Maruyama Scheme for Stochastic Differential Equations"
• (Lan et al., 2017) "Strong convergence rates of modified truncated EM method for stochastic differential equations"
• (Guo et al., 2017) "The Truncated Euler-Maruyama Method for Stochastic Differential Delay Equations"
• (Liu et al., 2018) “Truncated Euler-Maruyama method for classical and time-changed non-autonomous stochastic differential equations”
• (Chen et al., 2022) "Approximation of the invariant measure of stable SDEs by an Euler--Maruyama scheme"
• (Li et al., 2022) "An explicit approximation for super-linear stochastic functional differential equations"
• (Lang et al., 2023) "Euler-Maruyama approximations of the stochastic heat equation on the sphere"
• (Cui et al., 2023) "Strong convergence of multiscale truncated Euler-Maruyama method for super-linear slow-fast stochastic differential equations"
• (Liu et al., 17 Mar 2024) "The truncated EM scheme for multiple-delay SDEs with irregular coefficients and application to stochastic volatility model"
• (Deng et al., 8 Oct 2024) "Positivity-preserving truncated Euler and Milstein methods for financial SDEs with super-linear coefficients"
• (Hu et al., 30 Dec 2024) "A positivity-preserving truncated Euler--Maruyama method for stochastic differential equations with positive solutions: multi-dimensional case"