Euler-Maruyama Scheme for SDEs

• Euler-Maruyama scheme is a numerical method that approximates solutions to Itô stochastic differential equations using discretized drift and diffusion increments.
• It achieves a strong convergence order of 1/2 under Lipschitz conditions and adapts to complex models with jumps, discontinuities, or degenerate coefficients.
• Variants such as accelerated, logarithmic, and adaptive schemes enhance stability, preserve domain constraints, and improve accuracy in simulating stochastic systems.

The Euler-Maruyama scheme is a fundamental numerical method for approximating solutions to stochastic differential equations (SDEs) of the Itô type. It is widely used across probability theory, quantitative finance, stochastic analysis, and applied mathematics for simulating paths and estimating statistics of SDEs with various drift and diffusion regularity. The scheme’s theoretical properties and practical extensions have been developed for models ranging from classical Lipschitz SDEs to degenerate, discontinuous, and jump-driven equations.

1. Formulation of the Euler-Maruyama Scheme

Consider the general SDE in $\mathbb{R}^N$: $dX_t = a(X_t)\,dt + b(X_t)\,dW_t,\quad X_0=x_0$ where $a:\mathbb{R}^N\to\mathbb{R}^N$ is the drift and $b:\mathbb{R}^N\to\mathbb{R}^{N\times d}$ is the diffusion, and $(W_t)_{t\ge0}$ is $d$-dimensional Brownian motion. For time discretization, take a grid $t_i = i\Delta t,$ with $\Delta t = T/n$. The Euler-Maruyama updates are: $$X^n_{i+1} = X^n_i + a(X^n_i)\,\Delta t + b(X^n_i)\,\Delta W_i,\quad X^n_0 = x_0$$ with increments $\Delta W_i = W_{t_{i+1}} - W_{t_i}$ (Tanaka et al., 2012), and analogous recursions hold for SDEs with jumps, Lévy noise, degenerate coefficients, or measure-dependent (McKean–Vlasov) dynamics.

2. Strong and Weak Convergence Orders

Under global Lipschitz and linear growth assumptions for $a$ and $b$, for both Brownian and many jump SDEs, the Euler-Maruyama scheme achieves strong convergence order $1/2$: $\E\left[|X_T - X^n_T|^p\right]^{1/p} \leq C n^{-1/2}$ for any $p \ge 1$ and constant $C$ depending on the problem parameters (Müller-Gronbach et al., 2018, Wang et al., 2020). This rate remains robust for piecewise Lipschitz drift (with non-degenerate diffusion at discontinuity points), Dini continuous drift/diffusion (Wang et al., 2020), and many cases with discontinuous or Sobolev–Slobodeckij regular drift (Neuenkirch et al., 2019).

For SDEs with $\alpha$-stable jumps and $\beta$-Hölder drift, strong $L^p$ rates take the form $\Delta t^{\gamma}$ with $\gamma = \beta/\alpha$ for $p < \alpha/\beta$, and saturate at $O(\Delta t^{1/p})$ for $p \ge \alpha/\beta$ (Li et al., 2023).

Extensions to weak convergence for functionals of particle systems governed by McKean–Vlasov SDEs achieve optimal rates $O(N^{-1}+h)$ under full regularity (Frikha et al., 28 Mar 2025).

3. Extensions and Variant Schemes

Accelerated and Asymptotic Euler–Maruyama

For SDEs perturbed by a small parameter $\epsilon$, an accelerated scheme

$$\bar Y^{\epsilon,(n)}_t = \bar X^{\epsilon,(n)}_t − \bar X^{0,(n)}_t + X^0_t$$

achieves strong error $O(\epsilon n^{-1/2})$ by bias-cancellation (Tanaka et al., 2012). Analogous acceleration applies to the Milstein scheme.

Logarithmic Euler–Maruyama

For multidimensional stochastic delay equations with jumps, the Log–EM scheme preserves positivity and achieves strong $1/2$ order. Discretization uses exponential increments: $$X_i^{n}(t_{k+1}) = X_i^{n}(t_k) \cdot \exp\left( f_{ii}(S^n(t_k-b))\Delta_k + \sum_j \log(1+g_{ij}(S^n(t_k-b))) N_j^{(k)} \right)$$ guaranteeing non-negativity for all mesh steps (Agrawal et al., 2021).

Adaptive and Nonstandard Schemes

Adaptive EM schemes with step-size refinement near drift discontinuities recover $1/2$ strong order up to logarithmic corrections (Neuenkirch et al., 2018). Nonstandard EM, replacing $\Delta t$ by a nonlinear function $\varphi(\Delta t)$, improves domain invariance and stability while preserving strong consistency (Pierret, 2014). Explicit EM with projection handles non-Lipschitz drift/diffusion by truncating iterates before each update, crucial for strict positivity and stability in models like CIR or Ait-Sahalia (Chassagneux et al., 2014).

4. Jump and Lévy-driven SDEs

For SDEs with Lévy noise, the EM scheme incorporates jump increments, and strong convergence rates depend on the drift/diffusion/compensator regularity:

• Spectrally positive Lévy with Hölder coefficients: $$O\left( h^{p/2} + h^{p\beta(1-\epsilon)} \right)^{1/p}$$, under specific growth and monotonicity assumptions (Li et al., 2017).
• For $\alpha$-stable noise, explicit rates in Wasserstein distance for both direct stable-driven EM and Pareto-based EM schemes are $O(\eta^{1-\epsilon})$ and $O(\eta^{2/\alpha-1})$ respectively, shown to be sharp for Ornstein-Uhlenbeck cases (Chen et al., 2022).

5. Long-time and Invariant Measure Approximation

The Euler–Maruyama scheme approximates invariant measures of dissipative SDEs. For irreducible, one-sided Lipschitz drift and additive noise, the chain is geometrically ergodic in total variation distance, with a decay rate independent of the discretization step: $$\|P_h^n(x,\cdot) - \pi_h\|_{TV} \leq \tau \delta^{-n}$$ (Ye et al., 7 May 2025). Analogous uniform-in-time Wasserstein-$1$ error bounds for SDEs with cylindrical $\alpha$-stable drivers yield $\mathcal{W}_1(\nu, \nu_\eta)\leq C \eta$ (Dang et al., 2024).

6. Infinite-dimensional Applications and SPDEs

The EM scheme extends naturally to stochastic PDEs, including the heat equation on spheres, where forward and backward EM steps are applied mode-by-mode after spectral truncation. The strong mean-square error for spatial and temporal discretization can be characterized via the regularity of initial data and the noise’s spectral decay: $$E\left\| X(T) - X^{(L),N}(T) \right\|^2 \leq C \left[ h^{\min(1, \eta/2, \theta/4)} + L^{-\theta} \right]$$ and second-moment errors converge at twice the strong rate (Lang et al., 2023). Modified semi-implicit EM methods achieve improved rates by incorporating exact Ornstein-Uhlenbeck increments in the noise projection (Lord et al., 2010).

7. Practical Considerations, Stability, and Domain Invariance

Naive EM may violate domain constraints in models with non-globally Lipschitz drift or degenerate diffusion, making projection and truncated or logarithmic updates necessary (e.g., to enforce positivity in CIR or to confine trajectories to the unit ball (Nakagawa et al., 2021, Chassagneux et al., 2014)). Variant schemes, such as nonstandard and adaptive step-size refinements, enhance stability while maintaining strong convergence under relaxed moment and regularity assumptions.

Scheme Variant	Key Innovation	Error Rate/Robustness
Standard EM (Lipschitz)	Classic update on uniform grid	$O(n^{-1/2})$
Logarithmic EM (delay, jumps)	Exponential transforms, positivity	$O(\Delta^{1/2})$
Accelerated EM/Milstein	Bias-cancellation asymptotic expansion	$O(\epsilon n^{-1/2})$, $O(\epsilon n^{-1})$
Nonstandard EM	Weighted step function, domain invariance	$O(\varphi(h) + h^{1/2})$
Projection–Euler	Truncation at growing sequence of sets	$r = \min(\cdots)$, up to order 1
Pareto–EM/Stable–EM (jump SDE)	Tail-matching increment law	$O(\eta^{2/\alpha-1})$, $O(\eta^{1-\epsilon})$

References and Further Directions

The above results are documented in (Tanaka et al., 2012, Müller-Gronbach et al., 2018, Wang et al., 2020, Agrawal et al., 2021, Neuenkirch et al., 2019, Dang et al., 2024, Chen et al., 2022, Li et al., 2023, Neuenkirch et al., 2018, Pierret, 2014, Chassagneux et al., 2014, Li et al., 2017, Frikha et al., 28 Mar 2025, Nakagawa et al., 2021, Lang et al., 2023, Lord et al., 2010, Ye et al., 7 May 2025). For in-depth numerical analysis, implementation details, and new theoretical developments in the context of optimal rates under minimal regularity conditions, one should consult the cited works directly.