Projected Euler Scheme Analysis

• Projected Euler scheme is a discretization method that incorporates projection or truncation into Euler updates, stabilizing numerical solutions under superlinear growth and non-globally Lipschitz conditions.
• It employs radial or coefficient truncation mechanisms to maintain nonexpansiveness and recover classical convergence properties, often achieving a strong order of 1/2.
• It is widely applied in stochastic delay differential equations, jump-diffusions, and financial models to control blow-up and preserve long-term dynamics in explicit schemes.

Searching arXiv for recent and foundational papers on projected Euler schemes and closely related variants. Projected Euler scheme denotes a family of Euler-type discretizations in which a projection or truncation is inserted into the numerical update so that explicit time stepping remains viable under superlinear growth, non-globally Lipschitz coefficients, state-space constraints, or other structural requirements. In the stochastic-numerical literature, the canonical construction clips the current state to a step-size-dependent radius before evaluating drift and diffusion coefficients; related variants project coefficients onto truncated domains, impose nonexpansive truncation maps on modified Euler chains, or project auxiliary representations such as Wiener chaos expansions. Across these formulations, the central objective is to retain the low cost of explicit Euler while recovering stability, strong convergence, or long-time approximation properties that fail for the naive scheme (Li et al., 2018, Guo et al., 2024, Bao et al., 2024, Chassagneux et al., 2014, Li et al., 2018).

1. Canonical construction

In its most standard stochastic form, the projected Euler idea is a radial truncation of the state prior to coefficient evaluation. For stochastic delay differential equations, the projection is defined by

$x^{\circ}:=\min\bigl(1,h^{-\alpha}|x|^{-1}\bigr)x,$

with projection radius

$R_h=h^{-\alpha}.$

If $|x|\le h^{-\alpha}$, then $x^{\circ}=x$; if $|x|>h^{-\alpha}$, then $x^{\circ}$ is scaled back onto the sphere of radius $h^{-\alpha}$. The projected Euler–Maruyama update for the delay equation

$$dX(t)=f(X(t),X(t-\tau))\,dt+g(X(t),X(t-\tau))\,dW(t)$$

uses projected current and delayed states inside $f$ and $g$: $R_h=h^{-\alpha}.$0 The history values are taken from the prescribed initial segment $R_h=h^{-\alpha}.$1, and the delay is handled by the stored value $R_h=h^{-\alpha}.$2 (Li et al., 2018).

A closely related construction appears for semi-linear SDEs used to approximate random periodic solutions. There the projection map is

$R_h=h^{-\alpha}.$3

and the recursion evaluates every nonlinear occurrence of the state at $R_h=h^{-\alpha}.$4: $R_h=h^{-\alpha}.$5 Here the projection radius grows as $R_h=h^{-\alpha}.$6, so the modification vanishes asymptotically while still regularizing finite-step dynamics (Guo et al., 2024).

Another important variant is coefficient projection. For one-dimensional financial SDEs on $R_h=h^{-\alpha}.$7,

$R_h=h^{-\alpha}.$8

the method introduces a projection operator $R_h=h^{-\alpha}.$9 onto a truncated interval $|x|\le h^{-\alpha}$0, defines $|x|\le h^{-\alpha}$1 and $|x|\le h^{-\alpha}$2, and then applies the explicit update

$|x|\le h^{-\alpha}$3

This is not a projection of the full Euler iterate back into the domain; rather, the coefficients are modified before the step is taken (Chassagneux et al., 2014).

2. Structural assumptions and stability mechanisms

Projected Euler schemes are typically analyzed under monotonicity, coercivity, or dissipativity assumptions that are weaker than global Lipschitz conditions but still compatible with explicit discretization after truncation. In the SDDE setting, the core hypothesis is a global monotonicity condition: $|x|\le h^{-\alpha}$4 with $|x|\le h^{-\alpha}$5, together with polynomial growth and local Lipschitz bounds of polynomial type. The projection map satisfies the key estimate

$|x|\le h^{-\alpha}$6

so projection does not amplify perturbations. This nonexpansiveness is the basic stabilization mechanism in the delayed theory (Li et al., 2018).

For random periodic solutions of semi-linear SDEs, the assumptions combine a dissipative linear part, periodic coefficients, and a coupled coercivity estimate: $|x|\le h^{-\alpha}$7 with $|x|\le h^{-\alpha}$8, where $|x|\le h^{-\alpha}$9 is self-adjoint negative definite and $x^{\circ}=x$0. The projection map again satisfies

$x^{\circ}=x$1

and the clipped coefficients inherit step-size-dependent bounds such as

$x^{\circ}=x$2

These estimates let the superlinear terms be balanced directly against the step size (Guo et al., 2024).

In the ergodic high-diffusivity setting, the projected Euler scheme is embedded in a modified Euler recursion

$x^{\circ}=x$3

where the truncation map $x^{\circ}=x$4 is required to be nonexpansive: $x^{\circ}=x$5 The corresponding hypotheses impose dissipativity outside a ball, a discrete Lyapunov condition, and sufficiently strong noise. In this formulation, projection is part of the Markov kernel itself and is used to establish Wasserstein contraction rather than only finite-horizon mean-square stability (Bao et al., 2024).

For non-Lipschitz financial SDEs, the projected drift $x^{\circ}=x$6 becomes globally Lipschitz with an $x^{\circ}=x$7-dependent constant

$x^{\circ}=x$8

and the projection exponents are constrained by

$x^{\circ}=x$9

This design keeps $|x|>h^{-\alpha}$0 under control when $|x|>h^{-\alpha}$1, which is exactly the regime needed for an explicit strong-error analysis (Chassagneux et al., 2014).

3. Convergence theory

A central analytical framework for projected Euler methods is the stability-consistency paradigm. In the delayed setting, stochastic C-stability measures one-step propagation of conditional means and martingale fluctuations, while stochastic B-consistency of order $|x|>h^{-\alpha}$2 controls the local truncation defect. The general theorem states that if a stochastic one-step method is C-stable and B-consistent of order $|x|>h^{-\alpha}$3, then

$|x|>h^{-\alpha}$4

For the projected Euler–Maruyama method, the authors prove both properties with $|x|>h^{-\alpha}$5, yielding

$|x|>h^{-\alpha}$6

The proof relies on the nonexpansiveness of the projection, global monotonicity, local defect estimates for drift and diffusion increments, and discrete Gronwall (Li et al., 2018).

The same strong-order benchmark appears in several other projected formulations. The compensated projected Euler-Maruyama method for stochastic differential equations with jumps is analyzed under a global monotonicity condition that allows the jump-diffusion coefficient to grow superlinearly, and the method is proved to be convergent with strong order $|x|>h^{-\alpha}$7 on the discrete time level (Li et al., 2018). For random periodic solutions of semi-linear SDEs, the projected Euler method achieves mean-square order $|x|>h^{-\alpha}$8 for multiplicative noise and order $|x|>h^{-\alpha}$9 for additive noise, without relying on a priori high-order moment bounds of the numerical approximations (Guo et al., 2024).

Coefficient-projection schemes can recover the classical Euler rate under suitable integrability and regularity hypotheses. For the truncated explicit method used in financial SDEs, the main estimate has the form

$x^{\circ}$0

with $x^{\circ}$1 determined by moment exponents and local singular behavior. Under stronger assumptions, the rate reaches the classical strong order $x^{\circ}$2; in the constant-diffusion case, the paper proves a first-order strong rate

$x^{\circ}$3

This identifies projection as a device that can restore explicit-scheme convergence even when the original coefficients are non-Lipschitz or singular (Chassagneux et al., 2014).

4. Long-time dynamics, contraction, and invariant structures

Projected Euler constructions are not limited to finite-time strong approximation. In the random-periodic setting, the numerical scheme itself generates a pull-back limit and hence a discrete random periodic solution. Under a step-size restriction, the projected recursion admits a unique random periodic solution $x^{\circ}$4, and the pull-back convergence is quantified by

$x^{\circ}$5

The key estimate is a contraction bound for two projected numerical trajectories driven by the same noise: $x^{\circ}$6 Thus projection is used not merely to suppress blow-up, but to construct the invariant object of interest directly at the discrete level (Guo et al., 2024).

For dissipative SDEs with sufficiently large diffusivity, the projected Euler scheme also admits a precise ergodic analysis in $x^{\circ}$7-Wasserstein distance. The proof is based on synchronous coupling and the quasi-distance

$x^{\circ}$8

which is equivalent to $x^{\circ}$9 because $h^{-\alpha}$0 is bounded. One obtains

$h^{-\alpha}$1

and therefore

$h^{-\alpha}$2

The same framework yields a non-asymptotic invariant-measure approximation bound

$h^{-\alpha}$3

together with

$h^{-\alpha}$4

For the projected Euler specialization, the paper states that the exponent is essentially $h^{-\alpha}$5 under its assumptions, which is presented as a marked improvement over earlier $h^{-\alpha}$6-type rates for related truncation methods (Bao et al., 2024).

5. Representative variants and application domains

Setting	Projection mechanism	Reported guarantee
SDDEs under global monotonicity	Radial truncation $h^{-\alpha}$7 on current and delayed states	Strong order $h^{-\alpha}$8 (Li et al., 2018)
Jump-diffusions	Compensated projected Euler-Maruyama under global monotonicity	Strong order $h^{-\alpha}$9 on the discrete time level (Li et al., 2018)
Random periodic solutions of semi-linear SDEs	Projection $$dX(t)=f(X(t),X(t-\tau))\,dt+g(X(t),X(t-\tau))\,dW(t)$$0 onto the ball of radius $$dX(t)=f(X(t),X(t-\tau))\,dt+g(X(t),X(t-\tau))\,dW(t)$$1	Mean-square order $$dX(t)=f(X(t),X(t-\tau))\,dt+g(X(t),X(t-\tau))\,dW(t)$$2 for multiplicative noise and order $$dX(t)=f(X(t),X(t-\tau))\,dt+g(X(t),X(t-\tau))\,dW(t)$$3 for additive noise (Guo et al., 2024)
Dissipative SDEs with high diffusivity	Nonexpansive truncation map $$dX(t)=f(X(t),X(t-\tau))\,dt+g(X(t),X(t-\tau))\,dW(t)$$4 in a modified Euler chain	Exponential $$dX(t)=f(X(t),X(t-\tau))\,dt+g(X(t),X(t-\tau))\,dW(t)$$5-contraction and invariant-law bias $$dX(t)=f(X(t),X(t-\tau))\,dt+g(X(t),X(t-\tau))\,dW(t)$$6 (Bao et al., 2024)
One-dimensional financial SDEs on $$dX(t)=f(X(t),X(t-\tau))\,dt+g(X(t),X(t-\tau))\,dW(t)$$7	Coefficient projection $$dX(t)=f(X(t),X(t-\tau))\,dt+g(X(t),X(t-\tau))\,dW(t)$$8, $$dX(t)=f(X(t),X(t-\tau))\,dt+g(X(t),X(t-\tau))\,dW(t)$$9	Strong convergence with rates up to $f$0, and order $f$1 for constant diffusion (Chassagneux et al., 2014)

These examples show that projected Euler schemes are used in several distinct regimes. In delay equations, the projection regularizes superlinear drift and diffusion while preserving an explicit update. In the jump setting, compensation and projection are combined under a global monotonicity condition. In long-time stochastic dynamics, projection supports the construction of random periodic solutions and the approximation of invariant laws. In finance, coefficient projection is closely tied to positivity preservation and to the need to keep the simulated process in $f$2 (Li et al., 2018, Guo et al., 2024, Chassagneux et al., 2014).

The numerical evidence reported in these papers is consistent with the theoretical role assigned to projection. For the delay examples

$f$3

and

$f$4

log-log plots show a slope consistent with strong order $f$5. For random periodic solutions, the fitted rates reported are about $f$6 in the multiplicative-noise example and about $f$7 in the additive-noise example. In multilevel Monte Carlo tests for CIR bond pricing, the modified explicit projected scheme yields substantial empirical savings, with a reported savings factor of about $f$8 for the smallest target accuracy tested (Li et al., 2018, Guo et al., 2024, Chassagneux et al., 2014).

6. Terminological scope, neighboring constructions, and common misconceptions

A common misconception is that “projected Euler scheme” always means projecting the full Euler iterate back onto a constraint set. The literature summarized here does not support that identification. In SDDEs and random periodic SDEs, the state is clipped before coefficient evaluation; in financial SDEs, the drift and diffusion are composed with projection operators acting on the input state; in ergodic modified Euler chains, the projection is encoded as a nonexpansive truncation map in the one-step recursion itself. This suggests that the term is best understood structurally—Euler plus a stabilizing projection/truncation—rather than as a single canonical formula (Li et al., 2018, Guo et al., 2024, Bao et al., 2024, Chassagneux et al., 2014).

The same caution applies across adjacent areas. In BSDE numerics, an Euler discretization can be called projected when the random variable $f$9 is projected onto a finite Wiener chaos space: $g$0 Here the projection is not state-space clipping but projection in $g$1 onto a finite chaos basis (Lozano et al., 18 Dec 2025). In IMEX relaxation schemes for the compressible Euler equations, “projection” refers to projection onto the equilibrium manifold,

$g$2

after explicit-implicit evolution of a relaxation system (Thomann et al., 2019).

Conversely, not every Euler method that uses a projection-related geometric object is a projected Euler scheme in the standard sense. The adaptive Euler–Maruyama method for discontinuous drift uses the closest point projection $g$3 onto a hypersurface to define distance-dependent step sizes, but the paper explicitly states that the idea is not a projection in the geometric sense applied directly to the numerical solution (Neuenkirch et al., 2018). Likewise, the nonlinear ODE study under inexact information analyzes the plain explicit Euler method under local one-sided Lipschitz and local Hölder assumptions and states that there is no projection step, no truncation, and no correction operator in the algorithm (Czyżewska et al., 2022).

Within stochastic numerics proper, the projected Euler scheme is therefore best viewed as one branch of a broader stabilization program for explicit time stepping under irregular coefficients. Its most characteristic form is radial or coefficient truncation with a step-size-dependent radius, but the surrounding literature makes clear that the projection may act on states, coefficients, Markov kernels, or auxiliary function spaces, depending on the problem class and the property being targeted.