--- title: Semi-Discrete in Time Method (SDTM) url: https://www.emergentmind.com/topics/semi-discrete-in-time-method-sdtm type: topic --- # Semi-Discrete in Time Method (SDTM) 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 [1309.3189], [2008.03148], [2104.06149]. 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 [1406.5102], [2312.09592], [2509.13751]. ## 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 [1309.3189]. By contrast, in the Schrödinger transparent-boundary-condition literature, “semi-discrete” explicitly means **discrete in time and continuous in space** [1406.5102]. 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 [2112.01631], [1109.0641]. 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 [2504.20807]. | Context | Meaning of “semi-discrete” | Source | |---|---|---| | Scalar SDEs | Time partially discretized; within-step SDE retained | [1309.3189] | | Schrödinger TBCs | Discrete in time, continuous in space | [1406.5102] | | UTM / fractional FEM | Discrete in space, continuous in time | [2112.01631], [1109.0641] | | Semi-discrete OT | Continuous source, discrete target, continuous time | [2504.20807] | 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 \(0=t_00\), the exact within-step solution is \[ y_t = y_{t_n}\exp\left\{\int_{t_n}^t \left(\alpha_s-\frac12\beta_s^2\right)\,ds +\int_{t_n}^t \beta_s\,dW_s\right\}>0. \] Positivity is therefore inherited from the exact local formula rather than imposed by projection or clipping [1309.3189]. Under local Lipschitz-type assumptions on \(f,g\), existence of stepwise strong solutions, and bounded \(p\)-moments for some \(p>2\), the SD approximation converges strongly on finite time intervals: \[ \lim_{\Delta\to0}\sup_{0\le t\le T}|y_t-x_t|^2=0. \] The 2013 and 2020 analyses establish this convergence in \(L^2(\Omega;C([0,T]))\) for scalar SDEs, including non-autonomous coefficients [1309.3189], [2001.07483]. ## 3. Truncation, convergence rates, and asymptotic stability For superlinear coefficients, the SDE literature introduces a **truncated SD method**. One first chooses an increasing function \(\mu:\mathbb R_+\to\mathbb R_+\) such that \[ \sup_{|x|\le u}\left(|f(s,r,x,y)|\vee |g(s,r,x,y)|\right)\le \mu(u)(1+|y|), \qquad u\ge1, \] and a decreasing function \(h:(0,1]\to[\mu(1),\infty)\) satisfying \[ \lim_{\Delta\to0}h(\Delta)=\infty,\qquad \Delta^{1/6}h(\Delta)\le \hat h. \] Truncation is applied in the frozen argument through \[ \pi_\Delta(x):=\left(|x|\wedge \mu^{-1}(h(\Delta))\right)\frac{x}{|x|}, \] and \[ \phi_\Delta(s,r,x,y):=\phi(s,r,\pi_\Delta(x),y), \qquad \phi\equiv f,g. \] The truncated coefficients then obey \[ |f_\Delta(s,r,x,y)|\vee |g_\Delta(s,r,x,y)| \le h(\Delta)(1+|y|), \] which supplies a globally controlled Lipschitz scale and enables both convergence and stability arguments [2001.07483], [2008.03148]. The truncated SD scheme is \[ y_t^\Delta = y_{t_n}^\Delta +\int_{t_n}^t f_\Delta(t_n,s,y_{t_n}^\Delta,y_s^\Delta)\,ds +\int_{t_n}^t g_\Delta(t_n,s,y_{t_n}^\Delta,y_s^\Delta)\,dW_s. \] With \[ \mu(u)=\bar C\,u^{1+\gamma},\qquad h(\Delta)=\bar C+\sqrt{\ln \Delta^{-\varepsilon}},\qquad \varepsilon\in(0,1/3), \] the paper on convergence rates proves \[ \sup_{0\le t\le T}|y_t^\Delta-x_t|^2\le C\,\Delta^{\,1-\varepsilon}, \] which it summarizes as an \(\mathcal L^2\)-convergence order arbitrarily close to \(1/2\) [2001.07483]. 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 \[ \lim_{t\to\infty}x_t=0\qquad \text{a.s.} \] under the coercive condition \[ 2xa(s,x)+|b(x)|^2\le -\kappa(|x|), \] where \(\kappa:\mathbb R_+\to\mathbb R_+\) is continuous, nondecreasing, \(\kappa(0)=0\), and \(\kappa(u)>0\) for \(u>0\). For the truncated SD method, if the grid-point update admits a square decomposition \[ \big(\phi^\Delta(y_n^\Delta,t_n,\Delta,W_n)\big)^2 = (y_n^\Delta)^2+\phi_1^\Delta(y_n^\Delta,t_n,\Delta)+\phi_2^\Delta(y_n^\Delta,t_n,\Delta,W_n), \] with \[ \mathbb E\!\left[\phi_2^\Delta(\cdot)\mid \mathcal F_{t_n}\right]=0, \] and if the deterministic component satisfies \[ \phi_1^\Delta(y_n^\Delta,t_n,\Delta) \le -\kappa_1\!\left(\left|\pi_\Delta(y_n^\Delta)\right|\right), \] then \[ \lim_{n\to\infty}y_n^\Delta=0\qquad \text{a.s.} \] 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 [2008.03148]. 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 \[ x_t=x_0-10\int_0^t x_s^3\,ds+\int_0^t x_s^2\,dW_s, \] for which the substitution \(z=-1/x\) yields \[ z_t=z_0+11\int_0^t (z_s)^{-1}\,ds+\int_0^t dW_s. \] 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 [2008.03148]. The dedicated LSD paper generalizes this strategy to the CIR process, the CEV process, the Heston \(3/2\)-model, the Aït-Sahalia model, and the Wright–Fisher model [2104.06149]. Its abstract formulation is: \[ \text{original SDE} \;\xrightarrow{\text{Lamperti}}\; \text{constant-diffusion SDE} \;\xrightarrow{\text{SDM}}\; \text{explicit/algebraic step} \;\xrightarrow{\text{inverse transform}}\; \text{domain-preserving approximation.} \] For CIR, for example, the transform \[ z=\frac{2}{k_3}\sqrt{x} \] reduces the diffusion to a constant and produces a transformed drift of the form \[ a z^{-1} - b z. \] For Heston \(3/2\), the transform \[ z=\frac{2}{k_3}x^{-1/2} \] again produces a constant-diffusion equation with drift of the same structural type. For Wright–Fisher, the Lamperti coordinate \[ z=2\arcsin(\sqrt{x}) \] converts the state constraint \(x\in(0,1)\) into a trigonometric transformed drift and leads to updates that remain in \((0,1)\) after the inverse map [2104.06149]. 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 \(1\) and no extra restrictions on the parameters or the step-size** [2104.06149]. The wording “seems” is essential: the paper presents extensive numerical evidence but does not furnish a full general strong-order theorem of order \(1\). 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 \(3/2\)-volatility model, CEV-type equations, the Wright–Fisher model, the Aït-Sahalia model, and cubic superlinear test equations [1309.3189], [2104.06149]. 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 \(3/2\) | Positivity | SD and LSD target positive volatility dynamics | | CEV | Positivity | Lamperti transform yields tractable constant diffusion | | Wright–Fisher | \(0