---
title: Explicit Truncated Euler–Maruyama Schemes
url: https://www.emergentmind.com/topics/explicit-truncated-euler-maruyama-schemes
type: topic
---

# Explicit Truncated Euler–Maruyama Schemes

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 $g$ [1405.3561, 1611.07833, 1701.04598, 1812.00683, 2208.10214, 2403.11178, 2410.05614, 2412.20988].

For instance, in the modified explicit Euler–Maruyama scheme of [1405.3561], 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, $O(\varepsilon^{-4})$ [1611.07833].

## 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 [1411.6961, 1701.04598, 1812.00683, 2208.10214].
- **Non-Lipschitz/irregular drift and diffusion**: As in [2403.11178], 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 [2410.05614, 2412.20988].

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 [1701.04598, 1812.00683].

## 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:**
```python
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 [2208.10214, 2403.11178]. In applications to multiscale SDEs, explicit truncation plays a similar stabilizing role within heterogeneous multiscale frameworks [2308.02110].

## 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 [1405.3561, 1611.07833, 2410.05614, 2412.20988, 2403.11178].

**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 [2208.10214]. 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 [2307.07564].

**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 [2205.01342, 2402.12502].

## 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 [2412.20988].

**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 [1409.2362, 1802.04521].
- 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 [1611.07833].

## 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    | [1405.3561, 1701.04598]   |
| Projection/Truncation Operation         | $\displaystyle p_h(x) = \min(1, h^{-\alpha}|x|^{-1})x$            | [1411.6961, 1611.07833]   |
| Strong Convergence Rate                 | $r = 1/2$ (or model-dependent analog)                             | [1405.3561, 1701.04598]   |
| Positivity preservation                 | Achieved via componentwise truncation in $[R^{-1}, R]$            | [2410.05614, 2412.20988]  |
| Extension to delays/multiscale/SPDE     | Supported (w/ adapted projection or spectral truncation)          | [2208.10214, 2307.07564]  |
| MLMC compatibility                      | Yes, with optimal complexity under strong error control           | [1611.07833]              |

## References
- [1405.3561] “An explicit Euler scheme with strong rate of convergence for financial SDEs with non-Lipschitz coefficients”
- [1411.6961] “Stochastic C-stability and B-consistency of explicit and implicit Euler-type schemes”
- [1611.07833] "Multi-level Monte Carlo methods with the Truncated Euler-Maruyama Scheme for Stochastic Differential Equations"
- [1701.04598] "Strong convergence rates of modified truncated EM method for stochastic differential equations"
- [1703.09565] "The Truncated Euler-Maruyama Method for Stochastic Differential Delay Equations"
- [1812.00683] “Truncated Euler-Maruyama method for classical and time-changed non-autonomous stochastic differential equations”
- [2205.01342] "Approximation of the invariant measure of stable SDEs by an Euler--Maruyama scheme"
- [2208.10214] "An explicit approximation for super-linear stochastic functional differential equations"
- [2307.07564] "Euler-Maruyama approximations of the stochastic heat equation on the sphere"
- [2308.02110] "Strong convergence of multiscale truncated Euler-Maruyama method for super-linear slow-fast stochastic differential equations"
- [2403.11178] "The truncated EM scheme for multiple-delay SDEs with irregular coefficients and application to stochastic volatility model"
- [2410.05614] "Positivity-preserving truncated Euler and Milstein methods for financial SDEs with super-linear coefficients"
- [2412.20988] "A positivity-preserving truncated Euler--Maruyama method for stochastic differential equations with positive solutions: multi-dimensional case"

Source: https://www.emergentmind.com/topics/explicit-truncated-euler-maruyama-schemes