Semi-Discrete in Time Method (SDTM)

• Semi‑Discrete in Time Methods are numerical schemes that discretize time while retaining continuous dynamics within each step via solvable auxiliary SDEs.
• Truncated SDTM variants control superlinear coefficients to ensure strong convergence and asymptotic stability, achieving nearly 1/2 order mean‑square error.
• Lamperti-based methods enhance SDTM by applying coordinate transforms that yield explicit, domain‑preserving updates for models like CIR, Heston, and Wright–Fisher.

Semi-Discrete in Time Method (SDTM) denotes a family of numerical constructions in which time is discretized while part of the continuous problem structure is retained inside each step. In the stochastic differential equation literature, the term usually refers to the semi-discrete (SD) method, which replaces the original SDE on each interval by a partially frozen auxiliary SDE chosen to be explicitly solvable; in later work, truncated and Lamperti-based variants were developed to handle superlinear coefficients, preserve positivity or boundedness, and reproduce long-time behavior such as asymptotic stability (Halidias et al., 2013, Halidias et al., 2020, Halidias et al., 2021). In other literatures, the same phrase can denote time-discrete, space-continuous transparent boundary operators, DG-in-time integrators, or neural solvers that discretize time before solving a sequence of spatial approximation problems (Zlotnik et al., 2014, Li, 2023, Wang et al., 17 Sep 2025).

1. Terminology and scope

The phrase “semi-discrete” is not uniform across numerical analysis. In the SDE papers by Halidias and collaborators, the standard term is “Semi-Discrete method”, but this is effectively the same concept as Semi-Discrete in Time Method: only part of the equation is discretized in time, the scheme remains continuous within each time step, and a solvable local sub-SDE is used (Halidias et al., 2013). By contrast, in the Schrödinger transparent-boundary-condition literature, “semi-discrete” explicitly means discrete in time and continuous in space (Zlotnik et al., 2014). Other papers use “semi-discrete” for the opposite direction, namely space discretized and time continuous, as in the semidiscrete Unified Transform Method and the semi-analytical finite element treatment of time-fractional diffusion (Cisneros et al., 2021, Sun et al., 2011). In semi-discrete optimal transport for compressible semi-geostrophic equations, the term again has a different meaning: the target measure is discrete, the source remains continuous, and time remains continuous (Bourne et al., 29 Apr 2025).

Context	Meaning of “semi-discrete”	Source
Scalar SDEs	Time partially discretized; within-step SDE retained	(Halidias et al., 2013)
Schrödinger TBCs	Discrete in time, continuous in space	(Zlotnik et al., 2014)
UTM / fractional FEM	Discrete in space, continuous in time	(Cisneros et al., 2021, Sun et al., 2011)
Semi-discrete OT	Continuous source, discrete target, continuous time	(Bourne et al., 29 Apr 2025)

This terminological variability is a recurrent source of confusion. In the SDE-centered SDTM literature, the method is not a generic method-of-lines device but a specifically tailored time-discretization strategy for nonlinear equations whose exact solutions possess structural properties such as positivity, nonnegativity, boundedness, or asymptotic stability.

2. Core construction in stochastic differential equations

The canonical SDTM construction begins from the scalar SDE

$$dx_t=a(t,x_t)\,dt+b(t,x_t)\,dW_t, \qquad x_t=x_0+\int_0^t a(s,x_s)\,ds+\int_0^t b(s,x_s)\,dW_s.$$

On a uniform grid $0=t_0<t_1<\cdots<t_N=T$ with $\Delta=T/N$, one introduces auxiliary functions $f(s,r,x,y)$ and $g(s,r,x,y)$ satisfying

$f(s,s,x,x)=a(s,x),\qquad g(s,s,x,x)=b(s,x).$

The SD approximation is then defined, for $t\in(t_n,t_{n+1}]$, by

$$y_t = y_{t_n} +\int_{t_n}^t f(t_n,s,y_{t_n},y_s)\,ds +\int_{t_n}^t g(t_n,s,y_{t_n},y_s)\,dW_s,$$

with $y_0=x_0$ a.s. In compact form,

$$y_t = y_0+\int_0^t f(\hat s,s,y_{\hat s},y_s)\,ds +\int_0^t g(\hat s,s,y_{\hat s},y_s)\,dW_s,$$

where $0=t_0<t_1<\cdots<t_N=T$0 on $0=t_0<t_1<\cdots<t_N=T$1 (Halidias et al., 2020, Stamatiou et al., 2020).

The defining feature is that one freezes suitable parts of the coefficients at the left endpoint $0=t_0<t_1<\cdots<t_N=T$2, but still solves an SDE inside each step. This is why the method is explicit in a different sense from Euler–Maruyama: there is no nonlinear algebraic solve, yet the one-step object is not merely an increment formula. The classical Euler idea is recovered by the particular choice

$0=t_0<t_1<\cdots<t_N=T$3

but the advantage of SDTM comes from selecting $0=t_0<t_1<\cdots<t_N=T$4 and $0=t_0<t_1<\cdots<t_N=T$5 so that the within-step SDE is linear or otherwise explicitly solvable (Halidias et al., 2013, Stamatiou et al., 2020).

For positive diffusions, the local subproblem is often engineered into a multiplicative linear form such as

$0=t_0<t_1<\cdots<t_N=T$6

If $0=t_0<t_1<\cdots<t_N=T$7, the exact within-step solution is

$0=t_0<t_1<\cdots<t_N=T$8

Positivity is therefore inherited from the exact local formula rather than imposed by projection or clipping (Halidias et al., 2013).

Under local Lipschitz-type assumptions on $0=t_0<t_1<\cdots<t_N=T$9, existence of stepwise strong solutions, and bounded $\Delta=T/N$0-moments for some $\Delta=T/N$1, the SD approximation converges strongly on finite time intervals: $\Delta=T/N$2 The 2013 and 2020 analyses establish this convergence in $\Delta=T/N$3 for scalar SDEs, including non-autonomous coefficients (Halidias et al., 2013, Stamatiou et al., 2020).

3. Truncation, convergence rates, and asymptotic stability

For superlinear coefficients, the SDE literature introduces a truncated SD method. One first chooses an increasing function $\Delta=T/N$4 such that

$\Delta=T/N$5

and a decreasing function $\Delta=T/N$6 satisfying

$\Delta=T/N$7

Truncation is applied in the frozen argument through

$\Delta=T/N$8

and

$\Delta=T/N$9

The truncated coefficients then obey

$f(s,r,x,y)$0

which supplies a globally controlled Lipschitz scale and enables both convergence and stability arguments (Stamatiou et al., 2020, Halidias et al., 2020).

The truncated SD scheme is

$f(s,r,x,y)$1

With

$f(s,r,x,y)$2

the paper on convergence rates proves

$f(s,r,x,y)$3

which it summarizes as an $f(s,r,x,y)$4-convergence order arbitrarily close to $f(s,r,x,y)$5 (Stamatiou et al., 2020). The earlier motivation for SDTM as a qualitative method is thus complemented by a quantitative strong-order result.

The 2020 stability paper adds a long-time result. For the exact SDE, asymptotic stability is expressed by

$f(s,r,x,y)$6

under the coercive condition

$f(s,r,x,y)$7

where $f(s,r,x,y)$8 is continuous, nondecreasing, $f(s,r,x,y)$9, and $g(s,r,x,y)$0 for $g(s,r,x,y)$1. For the truncated SD method, if the grid-point update admits a square decomposition

$g(s,r,x,y)$2

with

$g(s,r,x,y)$3

and if the deterministic component satisfies

$g(s,r,x,y)$4

then

$g(s,r,x,y)$5

The proof is a Lyapunov-type square-norm argument with a martingale remainder, using summability of the negative drift terms and the nonnegative semimartingale convergence theorem (Halidias et al., 2020).

A central limitation is explicit in the theorem itself: this is not a universal stability result for every SD discretization. The one-step square-decomposition condition must be verified for the chosen semi-discretization.

4. Lamperti semi-discrete methods

A major development of SDTM is the Lamperti Semi-Discrete (LSD) method. The underlying idea is to apply a Lamperti transform so that the diffusion becomes constant, then design the SD step in the transformed variable. In the 2020 stability note, this appears first as a problem-specific construction motivated by the superlinear SDE

$g(s,r,x,y)$6

for which the substitution $g(s,r,x,y)$7 yields

$g(s,r,x,y)$8

A within-step SD equation in the transformed variable then leads to an explicit update, and transforming back preserves positivity in the original variable. The paper concludes that LSD is attractive because “there is no need for an exponential calculation,” while also stating that a fuller theoretical study is deferred (Halidias et al., 2020).

The dedicated LSD paper generalizes this strategy to the CIR process, the CEV process, the Heston $g(s,r,x,y)$9-model, the Aït-Sahalia model, and the Wright–Fisher model (Halidias et al., 2021). Its abstract formulation is: $f(s,s,x,x)=a(s,x),\qquad g(s,s,x,x)=b(s,x).$0 For CIR, for example, the transform

$f(s,s,x,x)=a(s,x),\qquad g(s,s,x,x)=b(s,x).$1

reduces the diffusion to a constant and produces a transformed drift of the form

$f(s,s,x,x)=a(s,x),\qquad g(s,s,x,x)=b(s,x).$2

For Heston $f(s,s,x,x)=a(s,x),\qquad g(s,s,x,x)=b(s,x).$3, the transform

$f(s,s,x,x)=a(s,x),\qquad g(s,s,x,x)=b(s,x).$4

again produces a constant-diffusion equation with drift of the same structural type. For Wright–Fisher, the Lamperti coordinate

$f(s,s,x,x)=a(s,x),\qquad g(s,s,x,x)=b(s,x).$5

converts the state constraint $f(s,s,x,x)=a(s,x),\qquad g(s,s,x,x)=b(s,x).$6 into a trigonometric transformed drift and leads to updates that remain in $f(s,s,x,x)=a(s,x),\qquad g(s,s,x,x)=b(s,x).$7 after the inverse map (Halidias et al., 2021).

The explicit claim of the LSD paper is that the method is domain preserving and seems to converge strongly to the solution process with order $f(s,s,x,x)=a(s,x),\qquad g(s,s,x,x)=b(s,x).$8 and no extra restrictions on the parameters or the step-size (Halidias et al., 2021). The wording “seems” is essential: the paper presents extensive numerical evidence but does not furnish a full general strong-order theorem of order $f(s,s,x,x)=a(s,x),\qquad g(s,s,x,x)=b(s,x).$9. A plausible implication is that the Lamperti step moves the approximation problem into a coordinate system where the SD construction is algebraically simpler and the state constraint is easier to preserve, but the paper itself remains primarily constructive and experimental on this point.

5. Model classes, preserved structure, and applications

The SDE-oriented SDTM literature is motivated by scalar equations whose exact solutions live in constrained domains and whose coefficients are nonlinear or superlinear. Typical examples include the CIR process, the Heston $t\in(t_n,t_{n+1}]$0-volatility model, CEV-type equations, the Wright–Fisher model, the Aït-Sahalia model, and cubic superlinear test equations (Halidias et al., 2013, Halidias et al., 2021). The common theme is that positivity, nonnegativity, boundedness, or asymptotic stability is part of the model specification rather than a secondary numerical convenience.

Model	Structural property emphasized	SDTM/LSD role
CIR	Positivity / nonnegativity	Explicit domain-preserving updates
Heston $t\in(t_n,t_{n+1}]$1	Positivity	SD and LSD target positive volatility dynamics
CEV	Positivity	Lamperti transform yields tractable constant diffusion
Wright–Fisher	$t\in(t_n,t_{n+1}]$2	LSD preserves bounded state interval
Aït-Sahalia	Positivity	Lamperti-based algebraic SD variants

The inadequacy of standard explicit methods is part of the original rationale. The 2013 paper stresses two difficulties: explicit Euler does not preserve positivity because its increments are conditionally Gaussian, and Euler can diverge for super-linear coefficients. It also states that the Tamed Euler method is explicit and avoids explosion, but still does not preserve positivity for the positive super-linear SDEs under consideration (Halidias et al., 2013). In the Heston $t\in(t_n,t_{n+1}]$3 example, SD produces the explicit strictly positive iteration

$t\in(t_n,t_{n+1}]$4

and the paper proves strong mean-square convergence for this scheme (Halidias et al., 2013).

The 2020 stability paper supplies a complementary long-time example with

$t\in(t_n,t_{n+1}]$5

Here the exact solution is positive, Assumption D holds with

$t\in(t_n,t_{n+1}]$6

and the authors compare SD, exponential SD, truncated SD, truncated exponential SD, Lamperti SD, and truncated Euler–Maruyama. The paper reports that TEM is asymptotically stable only for sufficiently small steps, specifically $t\in(t_n,t_{n+1}]$7, whereas the truncated TSD, exponential truncated TSD, and Lamperti LSD “work for all $t\in(t_n,t_{n+1}]$8,” and TEM does not preserve positivity (Halidias et al., 2020). These comparisons capture a central practical distinction: finite-time strong convergence alone does not guarantee faithful long-time dynamics.

The 2021 Lamperti paper extends the model list and emphasizes applications in financial mathematics and population dynamics. Its numerical sections repeatedly compare LSD with direct SD and implicit Lamperti-based schemes and report mean-square convergence behavior close to order $t\in(t_n,t_{n+1}]$9 together with preservation of positivity or boundedness across the tested models (Halidias et al., 2021).

6. Broader time-semidiscrete frameworks and open issues

Outside the SDE literature, SDTM denotes a broader class of time-discretization strategies. In the Schrödinger half-axis problem, the “semi-discrete TBC” is a transparent boundary operator for the time-discrete, space-continuous Crank–Nicolson problem, and one of the paper’s main results is the exact identity

$$y_t = y_{t_n} +\int_{t_n}^t f(t_n,s,y_{t_n},y_s)\,ds +\int_{t_n}^t g(t_n,s,y_{t_n},y_s)\,dW_s,$$0

showing that for $$y_t = y_{t_n} +\int_{t_n}^t f(t_n,s,y_{t_n},y_s)\,ds +\int_{t_n}^t g(t_n,s,y_{t_n},y_s)\,dW_s,$$1 the fully discrete transparent-boundary convolution kernel coincides with the semi-discrete one and does not depend on the spatial mesh width $$y_t = y_{t_n} +\int_{t_n}^t f(t_n,s,y_{t_n},y_s)\,ds +\int_{t_n}^t g(t_n,s,y_{t_n},y_s)\,dW_s,$$2 (Zlotnik et al., 2014). In DG computation for hyperbolic conservation laws, the SDG method is a DG-in-time discretization of the semi-discrete spatial ODE system, and the paper proves that explicit SDG and explicit SDC with $$y_t = y_{t_n} +\int_{t_n}^t f(t_n,s,y_{t_n},y_s)\,ds +\int_{t_n}^t g(t_n,s,y_{t_n},y_s)\,dW_s,$$3 correction iterations are

$$y_t = y_{t_n} +\int_{t_n}^t f(t_n,s,y_{t_n},y_s)\,ds +\int_{t_n}^t g(t_n,s,y_{t_n},y_s)\,dW_s,$$4

(Li, 2023). In high-order conservation-law time integration, semi-implicit SDC and SI-MLSDC are likewise built as time integrators for ODE systems obtained after spatial discretization, with multilevel FAS acceleration and linear temporal stability analysis (Pfister et al., 25 Apr 2025).

A recent neural formulation uses the acronym explicitly: the paper on random neural basis proposes an SDTM in which classical time integrators generate target functions $$y_t = y_{t_n} +\int_{t_n}^t f(t_n,s,y_{t_n},y_s)\,ds +\int_{t_n}^t g(t_n,s,y_{t_n},y_s)\,dW_s,$$5, and each time level is approximated by a random-basis network solving a spatial least-squares problem (Wang et al., 17 Sep 2025). The framework is

$$y_t = y_{t_n} +\int_{t_n}^t f(t_n,s,y_{t_n},y_s)\,ds +\int_{t_n}^t g(t_n,s,y_{t_n},y_s)\,dW_s,$$6

followed by a boundary-penalized least-squares fit in space. This usage is fully consistent with the phrase “semi-discrete in time,” but it belongs to a different methodological lineage from the SD method for SDEs.

Two persistent limitations emerge across the literature. First, the term itself is ambiguous: some “semi-discrete” papers are actually space-semidiscrete and are therefore not SDTM in the usual time-discrete sense (Cisneros et al., 2021, Sun et al., 2011, Bourne et al., 29 Apr 2025). Second, within the SDE-specific SDTM tradition, the strongest general theoretical results remain concentrated in the scalar case, the truncated method, and problem-dependent verifications of one-step solvability or stability inequalities (Halidias et al., 2020, Stamatiou et al., 2020). The 2020 stability note states explicitly that the work is for scalar SDEs, and the 2021 LSD paper presents extensive formulas and experiments but not a complete general convergence theory of the Lamperti variants (Halidias et al., 2020, Halidias et al., 2021).

In that sense, SDTM is best understood not as a single algorithm but as a design principle: discretize time in a way that preserves enough of the original continuous structure to make each step analytically tractable, qualitatively faithful, and computationally explicit. In the SDE literature this principle yields within-step solvable auxiliary SDEs, truncation mechanisms for superlinear growth, and Lamperti-based domain-preserving schemes; in other fields it yields time-discrete transparent boundary operators, DG-in-time collocation corrections, multilevel deferred corrections, and sequential spatial approximation frameworks.