Stochastically Discontinuous CIR Process

• Stochastically discontinuous CIR process is a model that incorporates scheduled or random state-dependent jumps while ensuring nonnegative dynamics.
• It extends the classical CIR diffusion by adding a pure jump component that preserves the affine property essential for tractable pricing and simulation.
• The model is applied in interest rate modeling to capture abrupt changes at events like central bank meetings, meeting rigorous mathematical and practical constraints.

A stochastically discontinuous Cox–Ingersoll–Ross (CIR) process is an extension of the classical CIR model that incorporates scheduled or random jumps in the process' trajectory, resulting in deterministic or stochastic discontinuities in its path while preserving critical qualities such as nonnegativity and, often, the affine property. These extended models address empirical requirements in applications—especially short-rate modeling—where discontinuities align with events like central bank policy meetings or regime shifts. Such processes admit nontrivial mathematical formulations, ensure regularity properties via admissibility and non-accumulation constraints, and deeply connect to broader theories of affine semimartingales and infinite divisibility.

1. Formal Definition and Mathematical Structure

The stochastically discontinuous CIR process generalizes the classical diffusion

$$dX_t = \kappa(\theta - X_t) \,dt + \sigma \sqrt{X_t} \,dW_t,$$

by augmenting it with a pure jump component: $$dX_t = \kappa(\theta - X_t) \,dt + \sigma \sqrt{X_t} \,dW_t + dJ_t,$$ where the jump process $J_t$ accrues at deterministic times $\mathcal{S} = \{s_n : n \geq 1\}$. The cumulative effect of these jumps is given by

$J_t = \sum_{n: s_n \leq t} F_n(X_{s_n^-}, Z_n),$

with each $$F_n: \mathbb{R}_+ \times [0,1] \rightarrow \mathbb{R}$$ a measurable transportation function mapping the pre-jump state and a random variable $Z_n$ (uniformly distributed on $[0,1]$, independent of $W$ and the other $Z_k$) to a jump size. The state-dependence of $F_n$ allows for complex, autocorrelated jump dynamics, including both upward and downward movements. Crucially, to ensure nonnegativity, one must have for all $x \geq 0$ and almost every $z$,

$F_n(x, z) \geq -x.$

This guarantees that $X$ stays in $\mathbb{R}_+$ even after a negative jump.

The process is defined on a filtered probability space supporting both $W$ and the $(Z_n)$, and $\mathcal{S}$ is required to be non-accumulating in finite time, i.e., $s_n \rightarrow \infty$ as $n \to \infty$ (Fontana et al., 19 Sep 2025).

2. Motivation and Financial Context

The drive for stochastically discontinuous CIR models arises from empirical features in overnight rates and other term structure objects, where deterministic or random jumps are observed at known calendar dates—typically central bank meetings or regulatory events (Fontana et al., 19 Sep 2025, Fontana et al., 2018). In these contexts, traditional stochastically continuous short-rate modeling fails to account for abrupt, predictable changes in rate levels, impairing pricing and risk management—especially in post-crisis multi-curve frameworks where interbank spread jumps are structurally important.

In the affine semimartingale/HJM frameworks, stochastic discontinuities can be modeled via affine semimartingales with atoms (i.e., jumps) at scheduled dates: $$dX_t = \kappa(\theta - X_t) \,dt + \sigma \sqrt{X_t} \,dW_t + \sum_n \Delta X_{T_n} \mathbf{1}_{\{t=T_n\}},$$ where $\Delta X_{T_n}$ are random variables possibly conditioned on $X_{T_n^-}$ (Fontana et al., 2018). This structure is critical for consistent arbitrage-free modeling of multiple curves when discontinuities in OIS or other risk-free curves must be accurately represented.

3. Construction: Existence, Admissibility, and Affine Property

The existence and uniqueness of a nonnegative strong solution is established under two main constraints (Fontana et al., 19 Sep 2025):

• No accumulation of jump times: $\lim_{n \to \infty} s_n = \infty$.
• Admissibility requirement: For almost every $z$ and all $x \geq 0$, $F_n(x, z) \geq -x$.

Given admissible $(F_n)$ and a Brownian driver $W$, there exists a unique càdlàg, nonnegative strong solution to the SDE, even in the presence of both upward and downward jumps. If the continuous part satisfies the Feller boundary condition $2\kappa\theta \geq \sigma^2$ (preventing instant absorption at $0$), the solution is non-explosive and strictly positive a.s. unless forced to $0$ by a jump (Fontana et al., 19 Sep 2025).

In order for the process to inherit the affine property, that is, for the conditional characteristic function to preserve the exponential-affine structure,

$$\mathbb{E}[e^{u X_t} \mid \mathcal{F}_s] = \exp\left(\phi(t-s, u) + \psi(t-s, u) X_s\right),$$

the conditional characteristic function of each jump $F_n(X_{s_n^-}, Z_n)$ given $\mathcal{F}_{s_n^-}$ must be exponential affine in $X_{s_n^-}$: $$\mathbb{E}[e^{u F_n(X_{s_n^-}, Z_n)} \mid \mathcal{F}_{s_n^-}] = \exp\left(\gamma_{n,0}(u) + \gamma_{n,1}(u) X_{s_n^-}\right),\quad u \in i\mathbb{R}.$$ Necessary and sufficient conditions relate to analytic extension of $\gamma_{n,0}$, $\gamma_{n,1}$ to a neighborhood of zero and suitable uniform bounds, ensuring affine transference throughout the process’ lifespan (Fontana et al., 19 Sep 2025, Liu et al., 2013).

4. Practical Construction and Illustrative Examples

Two canonical examples illuminate the framework:

• Zeroing and Resetting Example: At deterministic time $s_n$, set $F_n(x, z) = -x + f_n(x, z)$ with $f_n(x, z) \geq 0$. The process drops to zero, then jumps up by $f_n(x, z)$. If $f_n$ is calibrated so that the post-jump value follows a Gamma distribution with parameters depending on $x$, admissibility and affine property are preserved (Fontana et al., 19 Sep 2025).
• Deterministic Time-Change: Take a standard continuous CIR process $Y_t$ and define a strictly increasing, càdlàg clock $\tau(t) = t + H(t)$ with $H$ piecewise constant, jumping by $\Delta_n$ at each $s_n$. Set $X_t = Y_{\tau(t)}$. The increments $Y_{\tau(s_n)} - Y_{\tau(s_n^-)}$ serve as jump sizes at $s_n$ and—since the CIR process is affine—the resulting time-changed process inherits both nonnegativity and affine structure; jump sizes have shifted noncentral chi-square distributions (Fontana et al., 19 Sep 2025, Liu et al., 2013).

This construction method via time-change is particularly tractable for simulation and for analytical characterizations of jump-size laws.

5. Infinite Divisibility and Semimartingale Structure

Maintaining infinite divisibility is crucial for tractability and alignment with the affine framework. If, for each $n$, the law of $F_n(x, Z_n)$ admits a Lévy-Khintchine representation (i.e., is infinitely divisible), the full process is infinitely divisible as an affine semimartingale: $$\mathbb{E}[e^{u F_n(X_{s_n^-}, Z_n)} \mid \mathcal{F}_{s_n^-}] = \exp\left\{ \beta_n(X_{s_n^-}) u + \int_{\mathbb{R}_+} (e^{u \xi} - 1) \nu_n(d\xi, X_{s_n^-}) \right\}$$ with $\beta_n(x) = \beta_{n,0} + \beta_{n,1}x$ and Lévy measures $\nu_n(d\xi, x)$ supported in $[0, \infty)$ and linearly parameterized in $x$ (Fontana et al., 19 Sep 2025). One must enforce that the Gaussian part vanishes, $\beta_{n,0} \geq 0$, $\beta_{n,1} \geq -1$, and that $\nu_n$ is a measure on $\mathbb{R}_+$ for all $x$. This ensures the process is an infinitely divisible affine semimartingale, suitable for analytical tractability and simulation.

6. Connections, Context, and Implications

The stochastic discontinuous CIR framework provides an essential extension to traditional short-rate models (Fontana et al., 2018, Fontana et al., 19 Sep 2025). Beyond overnight rates and central bank event modeling, such frameworks generalize to multidimensional affine semimartingales where each component can incorporate stochastic discontinuities (across time, factors, or both).

Alternative approaches for modeling jumps and discontinuities include:

• Jump-diffusion CIR models driven by Lévy subordinators (see affine jump-diffusion models, e.g., (Barczy et al., 2016, Barski et al., 2019)), where jumps are modeled as a compound Poisson or stable process and affect the process via state-dependent or state-independent jump sizes.
• Stochastically discontinuous volatility models such as Heston–CIR under Variance Gamma or more general Lévy processes (with the CIR specification for variance and/or interest rate) (Ascione et al., 2022).
• Multiscale extensions or models with delay, in which deterministic delays or memory effects are allowed in the drift (e.g., fixed-delay CIR (Flore et al., 2018)), sometimes leading to non-Markovian—but still affine—dynamics.

The stochastically discontinuous CIR process is of central importance for term structure modeling in multi-curve environments, for the design of positivity-preserving numerical schemes (where jumps and non-Lipschitz diffusions must be handled simultaneously (Mishura et al., 2016, Stamatiou, 2018)), and for a deeper understanding of the phase transition between ergodic (CIR) and non-ergodic (squared Bessel) behavior under parameter degeneracies (Mishura et al., 17 Oct 2024).

References Table

Main Feature	Paper(s)	Key Point(s)
Deterministic jumps and structure	(Fontana et al., 19 Sep 2025)	Scheduled, state-dependent jumps preserving affine/nonnegativity; time-change construction
Term structure with deterministic jumps	(Fontana et al., 2018)	Affine semimartingale extension to scheduled jumps in HJM/multi-curve models
Jump-diffusion CIR	(Barczy et al., 2016, Barski et al., 2019)	Lévy-driven CIR, affine property, explicit bond pricing, maximum likelihood estimation
Tractable option pricing under jumps	(Liu et al., 2013, Ascione et al., 2022)	Sum-of-squares representations, pricing with generalized chi-square, calibration to American options

Conclusion

Stochastically discontinuous CIR processes form a mathematically rigorous generalization of classical interest rate and volatility models, capable of reproducing empirically observed jumps at deterministic times while preserving nonnegativity, the affine property, and infinite divisibility. These models align with the demands of modern fixed income and FX markets, support tractable pricing and simulation, and offer deep analytic connections to foundational stochastic analysis and the general theory of affine semimartingales.