i-CIR: Drift-Implicit Euler for CIR Process

• i-CIR is a numerical scheme using the Lamperti transform and drift-implicit Euler method for discretizing the classical CIR process while preserving positivity.
• It achieves order-1 strong convergence under conditions (a > σ²) and avoids the pitfalls of naive Euler–Maruyama, ensuring computational efficiency.
• The acronym 'i-CIR' has field-specific meanings, emphasizing canonical numerical analysis in finance as well as extended applications in wireless sensing and image retrieval.

Searching arXiv for papers on "i-CIR" and closely related usages to ground the article. In current arXiv usage, i-CIR most explicitly denotes the drift-implicit Euler discretization of the Cox–Ingersoll–Ross (CIR) process, obtained by applying the Lamperti transform $Y_t=\sqrt{X_t}$, discretizing the transformed SDE implicitly in the drift, and mapping back through $X_t=Y_t^2$ (Alfonsi, 2012). The surveyed literature also suggests that the label is not fully standardized: within stochastic-rate modeling it is often extended informally to nearby “improved” or “extended” CIR constructions, while in unrelated fields the acronym CIR instead denotes channel impulse response, composed image retrieval, or container intermediate representation.

1. Terminology and scope

The literature suggests that i-CIR has a canonical numerical-analysis meaning and several context-dependent adjacent meanings. In the strictest sense, it is the Lamperti-transformed, drift-implicit Euler method for the classical CIR diffusion. In broader financial usage, it may be used informally as shorthand for CIR variants that improve positivity, tractability, or market realism. Outside finance, the same acronym collides with unrelated technical objects (Francesco et al., 2021).

Usage	Core object	Representative source
Canonical i-CIR	Drift-implicit Euler scheme for the CIR process	(Alfonsi, 2012)
Broad financial extension	Improved or extended CIR-type models	(Francesco et al., 2021)
Wireless sensing	CIR = channel impulse response	(Kong et al., 13 Oct 2025)
Vision-language retrieval	CIR = composed image retrieval	(Takeda et al., 26 May 2026)
Cloud systems	CIR = Container Intermediate Representation	(Li et al., 12 Apr 2026)

A practical implication is that the term should be interpreted field-specifically. In stochastic numerics, i-CIR is a well-defined scheme. In cross-domain searches, however, the same string may refer to entirely different literatures.

2. Canonical meaning: drift-implicit Euler for the CIR process

The underlying CIR diffusion is written as

$$dX_t = (a-kX_t)\,dt + \sigma \sqrt{X_t}\, dW_t,\qquad X_0=x,$$

with

$$a>0,\qquad k\in\mathbb R,\qquad \sigma>0,\qquad x\ge 0.$$

Under $x>0$ and $\sigma^2\le 2a$, the exact solution is positive (Alfonsi, 2012).

The defining step in i-CIR is the Lamperti transform

$Y_t=\sqrt{X_t},$

which yields the additive-noise SDE

$$dY_t=\left(\frac{a-\sigma^2/4}{2Y_t}-\frac{k}{2}Y_t\right)dt+\frac{\sigma}{2}\,dW_t.$$

The scheme is then defined on the grid $t_k=kT/n$, $h=T/n$, by an Euler step that is implicit only in the drift: $X_t=Y_t^2$0 Because this is a scalar quadratic equation in $X_t=Y_t^2$1, it admits an explicit positive root,

$X_t=Y_t^2$2

The approximation in the original variable is then

$X_t=Y_t^2$3

This construction is the central reason the method is called drift-implicit rather than fully implicit. The diffusion term remains explicit, while positivity is enforced structurally through the square-root variable and the positive quadratic root.

3. Admissibility, positivity, and convergence theory

The scheme is well defined when

$X_t=Y_t^2$4

with no timestep restriction if $X_t=Y_t^2$5 (Alfonsi, 2012). Its main practical advantage over naive Euler–Maruyama is that it remains naturally defined on the nonnegative state space, rather than requiring truncation or post hoc projection.

The convergence theory distinguishes sharply between positivity of the exact CIR process and order-1 strong convergence of i-CIR. Before Alfonsi’s 2012 result, the known strong rate for the same scheme in the CIR setting was order $X_t=Y_t^2$6 under

$X_t=Y_t^2$7

namely

$X_t=Y_t^2$8

The main result improves this to strong order $X_t=Y_t^2$9 under the stricter condition

$$dX_t = (a-kX_t)\,dt + \sigma \sqrt{X_t}\, dW_t,\qquad X_0=x,$$0

proving that for all $$dX_t = (a-kX_t)\,dt + \sigma \sqrt{X_t}\, dW_t,\qquad X_0=x,$$1,

$$dX_t = (a-kX_t)\,dt + \sigma \sqrt{X_t}\, dW_t,\qquad X_0=x,$$2

The stronger restriction comes from the need to control inverse moments of the CIR process near zero, since the transformed drift contains singular terms in $$dX_t = (a-kX_t)\,dt + \sigma \sqrt{X_t}\, dW_t,\qquad X_0=x,$$3 and its derivatives (Alfonsi, 2012).

The same positivity-preserving design principle has also been developed in neighboring schemes. Halidias studies explicit positivity-preserving square-form schemes and exact-simulation-based split schemes for the CIR process; in general these attain logarithmic strong convergence, while in stronger parameter regimes they satisfy $$dX_t = (a-kX_t)\,dt + \sigma \sqrt{X_t}\, dW_t,\qquad X_0=x,$$4, and analogous constructions are extended to the two-factor CIR model (Halidias, 2014). For the fixed-delay CIR process, the Alfonsi idea extends to a drift-implicit delay scheme in the square-root variable, with grid-point strong order $$dX_t = (a-kX_t)\,dt + \sigma \sqrt{X_t}\, dW_t,\qquad X_0=x,$$5 and uniform-in-time error $$dX_t = (a-kX_t)\,dt + \sigma \sqrt{X_t}\, dW_t,\qquad X_0=x,$$6 for the piecewise linear interpolation (Flore et al., 2018).

4. Broader CIR-process extensions often associated with “i-CIR”

Several papers are relevant to i-CIR in a broader, interpretive sense: they do not define the canonical drift-implicit scheme, but they extend the CIR family in ways that often motivate searches for “improved” or “interpretable” CIR models.

Extension	Defining idea	Representative claim
Signed-rate extension	$$dX_t = (a-kX_t)\,dt + \sigma \sqrt{X_t}\, dW_t,\qquad X_0=x,$$7 with two independent CIR factors	Allows negative rates while preserving exponential-affine bond pricing (Francesco et al., 2021)
Stochastic discontinuities	Deterministic jump dates with state-dependent jump sizes	Affinity is characterized by affine jump conditional transforms (Fontana et al., 19 Sep 2025)
Multivariate Lévy noise	Affine short-rate equation driven by Lévy vectors	Independent-coordinate case reduces to stable-coordinate CIR-type form (Barski et al., 2022)
Calibration framework	Translated, piecewise, ARIMA-assisted CIR fitting	Designed for near-zero and negative short rates (Orlando et al., 2018)

The difference-of-two-CIR-factors model defines

$$dX_t = (a-kX_t)\,dt + \sigma \sqrt{X_t}\, dW_t,\qquad X_0=x,$$8

with each factor following a standard CIR diffusion. This preserves nonnegativity at the factor level while allowing negative short rates endogenously, and it retains exponential-affine zero-coupon bond prices of the form

$$dX_t = (a-kX_t)\,dt + \sigma \sqrt{X_t}\, dW_t,\qquad X_0=x,$$9

(Francesco et al., 2021).

The stochastically discontinuous CIR process adds deterministic jump dates $$a>0,\qquad k\in\mathbb R,\qquad \sigma>0,\qquad x\ge 0.$$0 with state-dependent jumps

$$a>0,\qquad k\in\mathbb R,\qquad \sigma>0,\qquad x\ge 0.$$1

while maintaining nonnegativity through the admissibility condition $$a>0,\qquad k\in\mathbb R,\qquad \sigma>0,\qquad x\ge 0.$$2. Affinity is preserved when the conditional transform of jump sizes remains exponential-affine in the pre-jump state (Fontana et al., 19 Sep 2025).

The Lévy-driven generalization studies short-rate equations of the form

$$a>0,\qquad k\in\mathbb R,\qquad \sigma>0,\qquad x\ge 0.$$3

under the requirement that the solution stay nonnegative and generate an affine term structure. In the independent-coordinate case, the admissible models reduce to stable-coordinate CIR-type dynamics; in the spherical case, the structure becomes still more rigid (Barski et al., 2022).

Finally, CIR# is a calibration framework rather than a new closed-form diffusion. It translates observed short-rate data upward when necessary, partitions the sample into local regimes, and calibrates local CIR parameters with ARIMA-assisted innovations. It is explicitly designed to address near-zero and negative short rates while preserving the classical CIR pricing machinery locally (Orlando et al., 2018).

5. Cross-domain overload of the acronym CIR

A common source of confusion is that several recent arXiv literatures use CIR for objects unrelated to the Cox–Ingersoll–Ross process.

In wireless sensing, CIR means channel impulse response. “CIRSense” argues that, although CSI and CIR are Fourier-dual representations, CIR is a more intuitive and principled sensing coordinate system because motion energy is concentrated in the delay domain. On commodity WiFi hardware and $$a>0,\qquad k\in\mathbb R,\qquad \sigma>0,\qquad x\ge 0.$$4 MHz bandwidth, the framework reports approximately 0.25 bpm mean respiration error and 0.09 m mean distance error, and at $$a>0,\qquad k\in\mathbb R,\qquad \sigma>0,\qquad x\ge 0.$$5 m it achieves at least 3× higher average accuracy with more than 4.5× higher computational efficiency than CSI-based baselines (Kong et al., 13 Oct 2025). In a related representation-learning direction, CSI-CLIP treats CIR and CSI as naturally aligned modalities and reports a 22% reduction in mean positioning error relative to supervised baselines (Jiang et al., 17 Feb 2025).

In vision-language research, CIR means composed image retrieval. FAR-Net formulates CIR as retrieving a target image from a reference image and modification text using a late-to-early multi-stage fusion architecture (Park et al., 17 Jul 2025). ConText-CIR adds a Text Concept-Consistency loss that regularizes noun-phrase grounding in the query image and reports state-of-the-art results on CIRR and CIRCO (Xing et al., 27 May 2025). CIRCLED, positioned as infrastructure for interactive or iterative CIR, constructs 22,608 multi-turn sessions across nine subsets to benchmark sequential refinement toward a target image (Takeda et al., 26 May 2026).

In systems research, CIR means Container Intermediate Representation. The corresponding container format stores cross-platform application code together with dependency identifiers and defers platform-specific construction to deployment time. In evaluation, a single CIR reduces image size by 95% relative to conventional images and reduces deployment time by 40–60% relative to pre-built images (Li et al., 12 Apr 2026).

The surveyed literature therefore suggests that any use of i-CIR outside stochastic numerics must be interpreted with care: it may indicate an “impulse-response-centric” direction, an “interactive CIR” retrieval setting, or an intermediate representation, rather than the CIR short-rate process.

6. Conceptual significance and recurrent misconceptions

The most important misconception is that i-CIR is a new stochastic process. In its canonical meaning, it is not: it is a numerical discretization scheme for the existing CIR diffusion. Its innovation lies in the Lamperti transformation, the drift-implicit step, and the positive-root reconstruction, not in altering the underlying SDE.

A second misconception is that the method is “fully implicit.” The defining update is implicit only in the drift term. This distinction matters because it explains why the method is both positivity-preserving and computationally cheap: the step reduces to a scalar quadratic with a closed-form positive solution, rather than a generic nonlinear solve (Alfonsi, 2012).

A third misconception is that the order-$$a>0,\qquad k\in\mathbb R,\qquad \sigma>0,\qquad x\ge 0.$$6 strong convergence result holds throughout the full positivity regime of CIR. It does not. Strict positivity of the exact process follows from the Feller-type condition $$a>0,\qquad k\in\mathbb R,\qquad \sigma>0,\qquad x\ge 0.$$7, but the order-$$a>0,\qquad k\in\mathbb R,\qquad \sigma>0,\qquad x\ge 0.$$8 theorem for i-CIR requires the stronger condition $$a>0,\qquad k\in\mathbb R,\qquad \sigma>0,\qquad x\ge 0.$$9. This gap is one of the central technical lessons of the theory.

The broader literature also suggests a methodological divide. In one branch, “i-CIR” points toward structure-preserving discretization of square-root diffusions. In another, it functions more loosely as a search term for extended CIR model families adapted to negative rates, deterministic jump dates, or Lévy noise. In yet other fields, it is simply an acronym collision. A practical implication is that references to i-CIR are most informative when accompanied by the surrounding vocabulary: Lamperti transform, drift-implicit Euler, channel impulse response, composed image retrieval, or Container Intermediate Representation.