Semi-Implicit Tamed Method

• Semi-implicit tamed method is a numerical scheme for SDEs and SDAEs that splits the drift into a globally Lipschitz part and a non-globally Lipschitz component, taming only the latter to control growth.
• It achieves strong order convergence of 1/2 by explicitly advancing the stable drift part and applying taming on the superlinear component, ensuring stability where Euler fails.
• For SDAEs, the method implicitly handles the linear constraint while taming the nonlinear drift, balancing computational efficiency with stability in the presence of algebraic constraints.

In the literature represented here, the semi-implicit tamed method denotes a class of numerical discretizations for stochastic equations with non-global Lipschitz drift in which stabilization is obtained by combining a structural split of the drift with a bounded drift increment, and, in some formulations, an implicit treatment of a linear or dissipative component. Two usages are especially important. In the jump–diffusion work of Tambue and Mukam, the closely related semi-tamed Euler scheme splits the drift as $f=u+v$, advances the globally Lipschitz part $u$ by an Euler term, and tames only the superlinearly growing part $v$; despite the terminology occasionally attached to it, this construction is explicit rather than implicit. In the stochastic differential-algebraic equation setting, by contrast, the designation is literal: the linear part $B(t)X(t)$ is handled implicitly, while the nonlinear drift $f(t,X(t))$ is tamed explicitly. Across these settings, the method is designed for regimes in which explicit Euler fails under non-global Lipschitz growth, while fully implicit methods require substantially greater computational effort (Tambue et al., 2015, Tambue et al., 2015, Tsafack et al., 10 Sep 2025).

1. Terminology and scope

A central feature of the subject is terminological ambiguity. In the jump-SDE papers, the relevant model is the $d$-dimensional Itô SDE with Poisson jumps

$$dX(t)=f\big(X(t^{-})\big)\,dt+g\big(X(t^{-})\big)\,dW(t)+h\big(X(t^{-})\big)\,dN(t),\qquad X(0)=X_0,$$

or, in compensated form,

$$dX(t)=f_\lambda\big(X(t^{-})\big)\,dt+g\big(X(t^{-})\big)\,dW(t)+h\big(X(t^{-})\big)\,d\overline{N}(t),\qquad f_\lambda(x)=f(x)+\lambda h(x).$$

Here $W$ is an $m$-dimensional Brownian motion, $u$0 is a scalar Poisson process of intensity $u$1, and $u$2. The drift is decomposed as $u$3, with $u$4 globally Lipschitz and $u$5 non-globally Lipschitz with superlinear growth. In this setting, “semi-tamed” means that only $u$6 is tamed.

In the SDAE setting, the model becomes

$u$7

with singular $u$8, matrix $u$9, nonlinear drift $v$0, and diffusion $v$1. There the method is semi-implicit in the usual sense: the linear component $v$2 is discretized implicitly, while the nonlinear component is tamed explicitly.

Formulation	Core update idea	Implicitness
Semi-tamed Euler for jump SDEs	Euler step on $v$3, taming on $v$4	Explicit
Compensated STS form	Add $v$5, use $v$6	Explicit
Semi-implicit tamed method for SDAEs	Solve $v$7, tame nonlinear $v$8	Linear implicit solve

This distinction removes a common misconception. In the 2015 jump-diffusion literature, “semi-tamed” does not mean that $v$9 appears inside a nonlinear drift evaluation; the papers state explicitly that no implicit evaluation at $B(t)X(t)$0 is performed. A separate misconception is that any semi-implicit stabilization is automatically a tamed method. The rough-path literature provides a counterexample: semi-implicit Taylor schemes can be stabilized purely by implicit drift evaluation under a one-sided Lipschitz condition, without any taming or truncation (Riedel et al., 2020).

2. Split–tamed construction for jump–diffusion SDEs

For a uniform grid $B(t)X(t)$1, $B(t)X(t)$2, the jump-SDE framework uses

$B(t)X(t)$3

with $B(t)X(t)$4 and $B(t)X(t)$5. The semi-tamed scheme of Tambue and Mukam is

$B(t)X(t)$6

with $B(t)X(t)$7 in the convergence theory (Tambue et al., 2015).

The corresponding compensated formulation used in stability analysis is

$B(t)X(t)$8

This form uses centered jump noise and adds the compensation term $B(t)X(t)$9 to the drift (Tambue et al., 2015).

The standing assumptions are monotonicity and polynomial growth rather than global Lipschitz continuity of the full drift. In the convergence paper these are encoded by Assumption A.1–A.4: $f(t,X(t))$0 has finite moments of every order, $f(t,X(t))$1, $f(t,X(t))$2, $f(t,X(t))$3, and $f(t,X(t))$4 are globally Lipschitz, $f(t,X(t))$5 is one-sided Lipschitz,

$f(t,X(t))$6

and $f(t,X(t))$7 satisfies the superlinear growth difference bound

$f(t,X(t))$8

Under these hypotheses, $f(t,X(t))$9 inherits one-sided Lipschitz behavior together with superlinear growth.

The method’s practical meaning is direct. The globally Lipschitz part $d$0 is left untamed, preserving the standard Euler contribution for the stable or linear portion of the drift, while the potentially explosive part $d$1 is rescaled so that the drift increment remains controlled even when $d$2 is large. The scheme therefore retains explicit-Euler cost: one samples $d$3, samples $d$4, and evaluates $d$5, $d$6, $d$7, and $d$8 once per step.

3. Strong convergence theory

The main strong convergence statement for the jump-SDE setting covers the non-compensated tamed Euler scheme, the semi-tamed scheme, and the compensated tamed Euler scheme in a single theorem. If $d$9 denotes the exact solution and $$dX(t)=f\big(X(t^{-})\big)\,dt+g\big(X(t^{-})\big)\,dW(t)+h\big(X(t^{-})\big)\,dN(t),\qquad X(0)=X_0,$$0 the appropriate continuous-time interpolant of one of these schemes, then under Assumption A.1–A.4, for every $$dX(t)=f\big(X(t^{-})\big)\,dt+g\big(X(t^{-})\big)\,dW(t)+h\big(X(t^{-})\big)\,dN(t),\qquad X(0)=X_0,$$1 there exists $$dX(t)=f\big(X(t^{-})\big)\,dt+g\big(X(t^{-})\big)\,dW(t)+h\big(X(t^{-})\big)\,dN(t),\qquad X(0)=X_0,$$2, independent of $$dX(t)=f\big(X(t^{-})\big)\,dt+g\big(X(t^{-})\big)\,dW(t)+h\big(X(t^{-})\big)\,dN(t),\qquad X(0)=X_0,$$3, such that

$$dX(t)=f\big(X(t^{-})\big)\,dt+g\big(X(t^{-})\big)\,dW(t)+h\big(X(t^{-})\big)\,dN(t),\qquad X(0)=X_0,$$4

for $$dX(t)=f\big(X(t^{-})\big)\,dt+g\big(X(t^{-})\big)\,dW(t)+h\big(X(t^{-})\big)\,dN(t),\qquad X(0)=X_0,$$5 (Tambue et al., 2015).

For the semi-tamed scheme, the continuous interpolant on $$dX(t)=f\big(X(t^{-})\big)\,dt+g\big(X(t^{-})\big)\,dW(t)+h\big(X(t^{-})\big)\,dN(t),\qquad X(0)=X_0,$$6 is

$$dX(t)=f\big(X(t^{-})\big)\,dt+g\big(X(t^{-})\big)\,dW(t)+h\big(X(t^{-})\big)\,dN(t),\qquad X(0)=X_0,$$7

The order is therefore the standard strong order $$dX(t)=f\big(X(t^{-})\big)\,dt+g\big(X(t^{-})\big)\,dW(t)+h\big(X(t^{-})\big)\,dN(t),\qquad X(0)=X_0,$$8, not a higher-order correction produced by the jump treatment.

The proof strategy is based on pathwise localization and moment control. The construction introduces control events $$dX(t)=f\big(X(t^{-})\big)\,dt+g\big(X(t^{-})\big)\,dW(t)+h\big(X(t^{-})\big)\,dN(t),\qquad X(0)=X_0,$$9 and auxiliary dominating processes $$dX(t)=f_\lambda\big(X(t^{-})\big)\,dt+g\big(X(t^{-})\big)\,dW(t)+h\big(X(t^{-})\big)\,d\overline{N}(t),\qquad f_\lambda(x)=f(x)+\lambda h(x).$$0, uses Burkholder–Davis–Gundy inequalities for Brownian and compensated Poisson integrals, employs exponential moment bounds and Doob’s maximal inequality, establishes uniform boundedness of moments for the numerical schemes and for drift images such as $$dX(t)=f_\lambda\big(X(t^{-})\big)\,dt+g\big(X(t^{-})\big)\,dW(t)+h\big(X(t^{-})\big)\,d\overline{N}(t),\qquad f_\lambda(x)=f(x)+\lambda h(x).$$1, and then performs a stopped error analysis with Itô’s formula for jump processes followed by discrete Gronwall (Tambue et al., 2015). The additional difficulty relative to pure diffusion is the need to control compensated jump martingales and the drift corrections generated by compensation.

The numerical evidence in the same paper is consistent with the theorem. For

$$dX(t)=f_\lambda\big(X(t^{-})\big)\,dt+g\big(X(t^{-})\big)\,dW(t)+h\big(X(t^{-})\big)\,d\overline{N}(t),\qquad f_\lambda(x)=f(x)+\lambda h(x).$$2

with $$dX(t)=f_\lambda\big(X(t^{-})\big)\,dt+g\big(X(t^{-})\big)\,dW(t)+h\big(X(t^{-})\big)\,d\overline{N}(t),\qquad f_\lambda(x)=f(x)+\lambda h(x).$$3, $$dX(t)=f_\lambda\big(X(t^{-})\big)\,dt+g\big(X(t^{-})\big)\,dW(t)+h\big(X(t^{-})\big)\,d\overline{N}(t),\qquad f_\lambda(x)=f(x)+\lambda h(x).$$4, $$dX(t)=f_\lambda\big(X(t^{-})\big)\,dt+g\big(X(t^{-})\big)\,dW(t)+h\big(X(t^{-})\big)\,d\overline{N}(t),\qquad f_\lambda(x)=f(x)+\lambda h(x).$$5, and for $$dX(t)=f_\lambda\big(X(t^{-})\big)\,dt+g\big(X(t^{-})\big)\,dW(t)+h\big(X(t^{-})\big)\,d\overline{N}(t),\qquad f_\lambda(x)=f(x)+\lambda h(x).$$6, all methods exhibited strong order $$dX(t)=f_\lambda\big(X(t^{-})\big)\,dt+g\big(X(t^{-})\big)\,dW(t)+h\big(X(t^{-})\big)\,d\overline{N}(t),\qquad f_\lambda(x)=f(x)+\lambda h(x).$$7, with log–log slopes approximately $$dX(t)=f_\lambda\big(X(t^{-})\big)\,dt+g\big(X(t^{-})\big)\,dW(t)+h\big(X(t^{-})\big)\,d\overline{N}(t),\qquad f_\lambda(x)=f(x)+\lambda h(x).$$8, and the semi-tamed method performed robustly (Tambue et al., 2015).

4. Linear and nonlinear stability

The stability theory is formulated first on the scalar linear test equation

$$dX(t)=f_\lambda\big(X(t^{-})\big)\,dt+g\big(X(t^{-})\big)\,dW(t)+h\big(X(t^{-})\big)\,d\overline{N}(t),\qquad f_\lambda(x)=f(x)+\lambda h(x).$$9

The exact solution is mean-square stable if and only if

$W$0

For the compensated semi-tamed scheme, the discrete recursion reduces to

$W$1

and the method is mean-square stable if and only if

$W$2

provided $W$3 (Tambue et al., 2015).

This stability region is materially simpler than that of the non-compensated tamed scheme, whose linear mean-square stability depends on the sign of $W$4 and therefore splits into separate regimes. The compensated formulation removes this piecewise complication and is one reason the semi-tamed scheme behaves more regularly in the jump setting.

Nonlinear exponential mean-square stability requires stronger structural assumptions. For the semi-tamed scheme, the drift split must satisfy

$W$5

$W$6

together with

$W$7

If

$W$8

and $W$9, then for

$m$0

there exists $m$1 such that

$m$2

(Tambue et al., 2015).

The experiments reinforce the analytical comparison with fully tamed schemes. In linear tests with $m$3, $m$4, $m$5, $m$6, and with $m$7, $m$8, $m$9, $u$00, the exact solution is stable in both cases, the semi-tamed scheme demonstrated superior stability compared to NCTS and CTS across step sizes, and the stability of NCTS and CTS improved as $u$01. In the nonlinear test

$u$02

the exact solution is exponentially mean-square stable, and the semi-tamed scheme reproduced stability for $u$03, again with better behavior than NCTS and CTS (Tambue et al., 2015).

5. Semi-implicit taming for stochastic differential-algebraic equations

A genuinely semi-implicit tamed method appears in the treatment of index-1 stochastic differential-algebraic equations. The model is

$u$04

where $u$05 is singular. The index-1 hypotheses are that $u$06 for all $u$07 and $u$08, so the noise does not enter the algebraic constraints, and that the algebraic constraints are globally uniquely solvable (Tsafack et al., 10 Sep 2025).

The method decomposes the drift into a linear part $u$09 and a nonlinear part $u$10. On the grid $u$11, $u$12, it is defined by

$u$13

or equivalently

$u$14

Thus the linear component is implicit, while the nonlinear component is tamed explicitly through

$u$15

The analysis uses the Moore–Penrose pseudo-inverse $u$16 and projector matrices $u$17 together with $u$18 such that $u$19. Under Assumptions A2.1–A2.3, $u$20, $u$21, $u$22, $u$23, and $u$24 are bounded and Lipschitz, $u$25 and $u$26 are non-singular with bounded inverses, and $u$27 is non-singular with

$u$28

These conditions guarantee the well-posedness of the linear solve at each step (Tsafack et al., 10 Sep 2025).

A distinctive analytical device is the dual tamed scheme. The solution is represented as $u$29, where $u$30 reconstructs the algebraic component after elimination of the constraints, and the discrete method is rewritten in variables $u$31 and $u$32. The resulting dual formulation is equivalent to the direct semi-implicit scheme, is associated with the inherent SDE obtained by eliminating the constraints, and makes it possible to prove global moment bounds and strong convergence.

The principal convergence theorem states that if $u$33 is the exact solution and $u$34 is the continuous-time interpolation of the direct scheme, then under Assumptions A.1–A.3 and A2.1–A2.3, for all $u$35 there exists $u$36, independent of $u$37, such that

$u$38

The method therefore has strong order $u$39 (Tsafack et al., 10 Sep 2025).

The numerical test problem is a three-dimensional index-1 SDAE with $u$40, $u$41 up to $u$42, and $u$43 Monte Carlo samples. Errors are measured either by $u$44 or by $u$45. The reported log–log plots have slope approximately $u$46, in agreement with the theorem (Tsafack et al., 10 Sep 2025).

6. Relation to adjacent semi-implicit methods and principal limitations

The semi-implicit tamed method should be distinguished from adjacent stabilization strategies. In rough differential equations driven, for example, by fractional Brownian motion, semi-implicit Taylor schemes use implicit drift evaluation but introduce no taming or truncation of the drift. The first-order multiplicative scheme is

$u$47

and its well-posedness follows from the one-sided Lipschitz condition on $u$48: if $u$49, then $u$50 is strongly monotone and hence a homeomorphism with Lipschitz inverse (Riedel et al., 2020). This literature shows that semi-implicitness and taming are analytically distinct mechanisms.

The jump-SDE semi-tamed method is correspondingly limited by the assumptions under which its proofs are carried out. The theory assumes a scalar Poisson process with constant intensity, globally Lipschitz diffusion and jump coefficients $u$51 and $u$52, and a split $u$53 in which $u$54 is globally Lipschitz while $u$55 satisfies one-sided dissipation and superlinear growth bounds. The practical guidance given in the papers is to sample $u$56, sample $u$57, form $u$58 for compensated versions, and choose $u$59, with larger $u$60 tending to improve stability (Tambue et al., 2015).

The SDAE version carries a different set of structural restrictions. It is confined to index-1 problems, requires that noise not enter the algebraic constraints, assumes global solvability of those constraints, uses globally Lipschitz diffusion, and relies on boundedness and Lipschitz continuity of the matrix-valued quantities together with invertibility of $u$61, $u$62, and $u$63. Each time step involves a $u$64 linear solve with coefficient $u$65, so the method avoids nonlinear implicit solves but does not reduce to a purely explicit update (Tsafack et al., 10 Sep 2025).

The open directions stated in these sources reflect the present boundaries of the theory. For jump problems, natural directions include multidimensional jump processes, adaptive time-stepping under large jump intensity, higher-order tamed schemes, and semi-implicit variants. For SDAEs, the corresponding extensions include higher-index systems, multiplicative noise in algebraic constraints, adaptive taming strategies, and weak convergence and multilevel Monte Carlo under constraints (Tambue et al., 2015, Tsafack et al., 10 Sep 2025). These directions suggest that the semi-implicit tamed method is best understood not as a single canonical algorithm but as a design principle: isolate the analytically difficult drift component, control it by taming, and, when the problem structure warrants it, use implicit treatment of the linear or constrained part to recover solvability and stability without the full cost of nonlinear implicit discretization.