Truncated Euler Method for SDEs

• Truncated Euler Method is a numerical scheme that modifies the Euler update by applying a truncation function to bound drift and diffusion terms, controlling super-linear growth.
• It employs bounded, truncated coefficients to secure stability and strong convergence under local Lipschitz and Khasminskii-type conditions.
• Applications span stochastic, delay, and multiscale systems, with the method effectively preserving qualitative properties like positivity and invariance.

The Truncated Euler Method encompasses a family of numerical strategies designed to approximate solutions of stochastic differential equations (SDEs), deterministic differential equations, and related classes of functional or delay equations by combining an explicit one-step time discretization (typically the Euler–Maruyama or Euler method) with a truncation or projection of state variables and/or coefficients. The primary motivation is to ensure boundedness, avoid algorithmic divergence due to non-global Lipschitz or super-linear growth in coefficients, and, where needed, preserve crucial qualitative properties—such as positivity or invariance—of the true solution. The method arises as an essential tool in the numerical treatment of SDEs and other stochastic or deterministic systems with coefficients that violate classical numerical stability constraints.

1. Core Definition and Variants

The truncated Euler method modifies the basic explicit Euler update formula by replacing the original drift and diffusion (or deterministic function) coefficients with bounded, truncated versions. For an SDE of the Itô form

$dx(t) = f(x(t))dt + g(x(t))dB(t)$

with initial value $x(0) = x_0$, the standard Euler–Maruyama update is

$x_{k+1} = x_k + f(x_k)\Delta + g(x_k)\Delta B_k,$

where $\Delta B_k = B(t_{k+1}) - B(t_k)$ and $\Delta = t_{k+1} - t_k$.

In the truncated Euler method, one introduces a truncation function or mapping—typically built from a strictly increasing function $\mu(\cdot)$, a step-size-dependent threshold $h(\Delta)$, and their (local) inverses such as

$$x \mapsto (|x| \wedge \mu^{-1}(h(\Delta))) \cdot \frac{x}{|x|}.$$

The truncated coefficients are defined by

$$f_\Delta(x) = f(\text{trunc}(x)), \quad g_\Delta(x) = g(\text{trunc}(x)),$$

so that for large arguments, the coefficients are evaluated at a bounded point. This produces the truncated update

$$x_{k+1} = x_k + f_\Delta(x_k)\Delta + g_\Delta(x_k)\Delta B_k.$$

There exist modifications such as the modified truncated Euler–Maruyama (MTEM) method, which allows the truncation radius to vary with step size; partially truncated methods, in which truncation applies only to selected “super-linear” terms; and logarithmic (Lamperti-transformed) versions for positivity-preservation.

Truncation in the context of deterministic Euler time stepping, as in the approximation of ODEs or in the approximation of multiple stochastic integrals, also appears, though the primary motivation in the stochastic setting is to enforce stability and strong convergence.

2. Theoretical Foundations and Error Analysis

The convergence analysis for truncated Euler-type schemes relies on the construction of globally bounded, globally Lipschitz continuous effective coefficients—regardless of the behavior of the original drift and diffusion. In the SDE context, the method admits convergence under only local Lipschitz and (generalized) Khasminskii-type conditions on the coefficients, in contrast to the global conditions required for the classical Euler–Maruyama scheme.

The strong convergence rate in $L^q$ is of order $1/2$ under appropriate conditions: $$\mathbb{E}|x(T) - x_{\Delta}(T)|^q \leq C \Delta^{q/2}.$$ Some earlier works included an additional infinitesimal factor $h(\Delta)$ due to the truncation, leading to suboptimal convergence rates; more recent analyses eliminate this factor and recover the optimal order (Hu et al., 2 Apr 2025). For drift and diffusion coefficients depending on time, or with only Hölder continuity in the temporal variable, the rate is limited by the minimum of the Hölder exponents and $1/2$, minus an arbitrarily small $\epsilon > 0$ (Liu et al., 2018).

In the context of pathwise approximation (sample-wise convergence), the truncation error can be bounded by

$$\|\delta(\tau_k, \tau_{k+1}, z)\| = O\left( (\tau_{k+1} - \tau_k)^{\gamma + 1/2 - \epsilon} \right),$$

with a global error scaling as $h^{\gamma - \epsilon}$ for maximum step size $h$, and with $\gamma = 1/2$ (Shardlow et al., 2014).

Further refinements include the use of adaptive time-stepping, where time-steps are chosen in a pathwise fashion to limit the truncation error in each coordinate direction or in terms of multiple stochastic integral increments, yielding improved error constants and robustness (Shardlow et al., 2014).

3. Applications to Delay, Functional, and Time-Changed Equations

The truncated Euler methodology extends to stochastic differential delay equations (SDDEs), stochastic delay equations with Markovian switching, stochastic functional differential equations (SFDEs), and time-changed SDEs (with inverse subordinators). In the SDDE and SFDE settings, truncation is applied component-wise to the present and delayed values, ensuring that super-linear growth in either argument is controlled (Guo et al., 2017, Cong et al., 2018, Deng et al., 2021, Li et al., 2022).

In time-changed SDEs of the form

$dY(t) = f(t, Y(t)) dE(t) + g(t, Y(t)) dB(E(t)),$

where $E(t)$ is a stochastic time change, the standard truncation mapping is combined with discretization of the time change process. The method attains strong convergence rates analogous to those for classical SDEs with super-linear coefficients, with the error analysis leveraging moment bounds and the duality principle between classical and time-changed SDEs (Liu et al., 2018, Li et al., 2021).

4. Structural and Qualitative Properties

Beyond stability, truncated Euler-type methods are designed to preserve essential qualitative properties of the true solution. In the case of SDEs with positive solutions, tailored truncation mappings (such as clipping to a positive interval) or logarithmic transformations ensure that the numerical solution remains positive—even in high-dimensional settings (Hu et al., 30 Dec 2024, Deng et al., 8 Oct 2024). For models with invariant domains or boundary constraints (common in finance, population dynamics, and epidemiology), this property distinguishes truncated methods from unmodified explicit schemes.

Stability properties such as mean-square and $H_{\infty}$ stability are also preserved by the truncated (and partially truncated) methods. Under general Khasminskii-type conditions, the scheme inherits the long-time exponential decay of the true solution, as formalized in upper estimates for

$$\limsup_{t \to \infty} \frac{1}{t} \log \mathbb{E}|x(t)|^2 \leq -\gamma,$$

and similar inequalities for the discrete numerical solution (Deng et al., 2021, Cong et al., 2018).

5. Adaptive, Multi-Level, and Multiscale Extensions

Adaptive time-stepping strategies based on real-time control of the Brownian increments or multiple stochastic integrals have been established in the pathwise approximation framework (Shardlow et al., 2014). The next time step is selected so as to bound the local error term, with strategies such as: $$\max_{j} \|q_{ij}(y_n)\|^{1/2}|W_i(t) - W_i(\tau_n)| \leq \alpha h^{1/2}$$ for suitable constants and coefficients $q_{ij}$, and for each component.

The integration of truncated Euler–Maruyama schemes with the Multi-Level Monte Carlo (MLMC) method provides computational efficiency for approximating expectations of functionals of the solution: $\mathbb{E}[f(X(T))] \approx \sum_{l=0}^L Y_l,$ where level differences are estimated using truncated EM approximations at varying resolutions. With appropriately balanced bias, variance, and computational cost, one can achieve the target accuracy at a computational complexity of $O(\epsilon^{-4})$ (Guo et al., 2016).

Multiscale truncated Euler approaches, incorporating heterogeneous multiscale framework, are suited for slow-fast SDEs and extend the above advantages to problems governed by coupled time scales (Cui et al., 2023).

6. Extension to Other Equations and Structures

The truncated Euler method has broader application to polynomial approximation methods as well as to deterministic equations. For instance, truncated Euler polynomials, defined via modified generating functions, generalize classical Euler polynomials and are connected to identities for hypergeometric Bernoulli polynomials (Komatsu et al., 2018). This structure allows efficient implementation of high-order approximation schemes and provides a link between numerical truncation and the theory of polynomial-based representation and error control.

Analogously, in the spectral approximation of fluid dynamics problems (e.g., Galerkin truncation of Euler equations), the notion of truncation appears both in the suppression of high-frequency modes and in regularization, such as wavelet-based denoising at each time step, to maintain physical realism and control equipartition (Farge et al., 2017, Kan et al., 2021).

7. Practical Implications, Limitations, and Recommendations

Truncated Euler-type schemes provide explicit, computationally attractive alternatives to implicit methods for SDEs and complex dynamical systems with non-global Lipschitz coefficients, super-linear growth, and/or bounded or invariant domains. They allow rigorous convergence and stability proofs under weaker assumptions than classical explicit schemes.

However, practical implementation requires the careful selection of truncation functions $\mu, h(\Delta)$, and related parameters. Too aggressive truncation can introduce bias, whereas insufficient truncation risks loss of stability. For adaptive or pathwise implementations, additional computational overhead arises from the need to simulate bounded increments or solve auxiliary maximization problems at each step.

In high-dimensional, multiscale, or nonlinear settings—such as delay equations, SDDEs with switching, or highly irregular coefficients—truncated schemes (including partially truncated and logarithmically truncated variants) remain at the forefront of rigorous, explicit numerical methods. Numerical experiments on stochastic volatility models, population processes, delayed logistic equations, and interest rate models consistently show the advantage of these methods in both qualitative behavior and quantitative error.

In summary, the truncated Euler method forms an essential part of the modern numerical analyst’s toolkit for SDEs and related systems where classical assumptions are not satisfied, combining strong theoretical guarantees with practical flexibility and extensibility (Shardlow et al., 2014, Guo et al., 2016, Lan et al., 2017, Guo et al., 2017, Cong et al., 2018, Liu et al., 2018, Deng et al., 2021, Deng et al., 8 Oct 2024, Hu et al., 30 Dec 2024, Hu et al., 2 Apr 2025).