Time-Inhomogeneous Affine Process

• Time-inhomogeneous affine processes are Markov or semimartingale models whose conditional transforms are exponentially affine with explicit time dependence.
• They generalize classical affine models by allowing time-dependent drift, diffusion, and jump parameters through generalized Riccati equations and deterministic time-changes.
• These models are applied in finance, stochastic filtering, and environmental sciences, with efficient numerical methods supporting simulation and pricing.

A time-inhomogeneous affine process is a Markov or semimartingale process characterized by the property that its conditional (Fourier–Laplace or moment generating) transform is exponentially affine in the present state, with the law depending explicitly and nontrivially on both initial and future times. The class generalizes classical time-homogeneous affine processes by allowing model coefficients—drift, diffusion, and jump parameters—to depend on time, deterministic time-changes, stochastic environments, or external processes. Time-inhomogeneous affine processes appear ubiquitously in mathematical finance, stochastic filtering, branching, queuing theory, and modeling of phenomena exhibiting nonstationary or regime-switching behavior.

1. Affine Transform Structure and Generalized Riccati Equations

A process $X$ on a suitable state space $D$, possibly a Borel subset of $\mathbb{R}^d$, is affine if there exist (possibly complex-valued) functions $\phi_{s, t}(u)$ and $\psi_{s, t}(u)$ defined for $0 \leq s \leq t$, such that

$$\mathbb{E}\left[\exp\big(\langle u, X_t \rangle\big) \,\big|\, X_s = x\right] = \exp\big(\phi_{s, t}(u) + \langle\psi_{s, t}(u), x \rangle\big).$$

In the time-inhomogeneous case, the transform depends on both $s$ and $t$. These functions satisfy generalized Riccati equations. Unlike the time-homogeneous setting, where ordinary differential equations suffice,

$$\begin{aligned} \frac{d}{dt}\phi(t, u) &= F(\psi(t,u)),\quad \phi(0,u)=0, \ \frac{d}{dt}\psi(t, u) &= R(\psi(t,u)),\quad \psi(0,u)=u, \end{aligned}$$

the time-inhomogeneous structure requires (possibly integral) equations

$$\begin{aligned} \phi_{s, t}(u) &= \int_s^t F(r, \psi_{r, t}(u))\, dG(r), \ \psi_{s, t}(u) &= u + \int_s^t R(r, \psi_{r, t}(u))\, dG(r), \end{aligned}$$

where $F$ and $R$ are of Lévy–Khintchine type, and $G$ is an increasing deterministic time-change function, which encodes local or global time-inhomogeneity (Waldenberger, 2015). In the strictly regular case, $G(r) = r$ and derivatives exist classically. In general, $F$ and $R$ as functions of $u$ and possibly $r$ take the form

$$F(r, u) = \langle b(r), u \rangle + \tfrac12 \langle u, a(r) u \rangle - c(r) + \int_{\mathbb{R}^{d}\backslash\{0\}} \left(e^{\langle \xi, u \rangle} - 1 - \langle h(\xi), u \rangle\right) m(r, d\xi),$$

with $b(r), a(r), c(r), m(r, \cdot)$ varying in time; $h(\cdot)$ is a truncation function.

Even when the process is not "regular" (i.e., differentiability fails due to irregular time dependence or jump structure), these equations remain valid in an integral or measure-theoretical sense (Keller-Ressel et al., 2018).

2. Pathwise and Time-Change Constructions for Inhomogeneity

Time-inhomogeneous affine processes can also arise as pathwise deterministic time-changes or superpositions. In the time-change framework, $X$ is constructed from independent Lévy processes $Z^{(i)}$ via state-dependent (stochastic) clocks $\theta^{(i)}_t = \int_0^t X^{(i)}_s ds$, as in

$X_t = x + \sum_{i=1}^d Z^{(i)}(\theta^{(i)}_t),$

so that the time-inhomogeneity emerges from feedback between the current state and the operational time (Gabrielli et al., 2014). This extends to multiparameter time-change schemes, e.g., for vector processes on $\mathbb{R}_+^m \times \mathbb{R}^n$ where clocks are constructed for each coordinate, leading to intricate interactions between different components (Caballero et al., 2015).

Such constructions not only ensure the affine transform property but also facilitate limit theorems and simulation via Euler-type schemes, since stability and convergence of the deterministic ODE systems used in the time-change underpin both analytic and computational approaches.

3. Relaxing Stochastic Continuity and Semimartingale Formulations

A salient feature of time-inhomogeneous affine processes is the potential lack of stochastic continuity. The general theory (Keller-Ressel et al., 2018) characterizes affine semimartingales whose characteristics (drift $B$, covariance $C$, and jump compensator $\nu$) admit affine representations modulated by a deterministic function $A$. The canonical semimartingale decomposition reads

$$\begin{aligned} B_t &= \int_0^t \left[ \beta_0(s) + \sum_{i=1}^d X_{s-}^i \beta_i(s) \right] dA_s, \ C_t &= \int_0^t \left[ \alpha_0(s) + \sum_{i=1}^d X_{s-}^i \alpha_i(s) \right] dA_s, \ \nu(ds, dx) &= \left[ \mu_0(s, dx) + \sum_{i=1}^d X_{s-}^i \mu_i(s, dx)\right] dA_s. \end{aligned}$$

Predictable jump times (e.g., scheduled dividends, earnings announcements) can be modeled directly. The conditional Fourier-Laplace transform is governed by generalized Riccati equations in measure or Stieltjes form: $$\begin{aligned} d\phi^\mathrm{c}_t(T,u)/dA_t^\mathrm{c} &= -F(t, \psi_t(T,u)),\ d\psi^\mathrm{c}_t(T,u)/dA_t^\mathrm{c} &= -R(t, \psi_t(T,u)), \end{aligned}$$ augmented by jump conditions at discontinuities of $A$.

The existence of affine Markov processes (for continuous or discrete time) is established under mild admissibility and integrability conditions on $(\beta,\alpha,\mu)$ (Keller-Ressel et al., 2018).

4. Polynomial Processes, Magnus Series, and Numerical Solutions

Every affine process is a special case of a polynomial process, where for any polynomial $f$ of degree at most $m$, $E[f(X_t) | X_s = x]$ is again a polynomial of degree at most $m$ (Hurtado et al., 2018). In the time-homogeneous case, polynomial moments may be propagated using matrix exponentials. For time-inhomogeneous settings, the moment evolution generally requires an operator-valued Magnus series: $$P_{s,t} = \exp\left(\Omega(s, t)\right), \quad \Omega(s, t) = \sum_{k=1}^{\infty} \Omega_k(s, t),$$ with specific expressions for $\Omega_k$ in terms of time-ordered commutators of the generator matrices. This methodology enables efficient computation of moments and hence pricing or estimation in practical applications where explicit solutions are unavailable.

Applications include term-structure and energy models, where the flexibility in time-parameter dependence can capture observed economic or seasonal effects.

5. Extensions: Stochastic Environment, Non-Markovianity, Nonlinearities

Markov-Modulated and Superposed Affine Structures

Extensions involve allowing coefficients to be modulated by an external Markov process ("Markov-Modulated Affine Process", MMAP). Conditional on the environment process $X$, the process $Y$ is affine and time-inhomogeneous: the joint generator splits as $$\mathcal{G} f(x, y) = \mathcal{G}^x f + \mathcal{G}^y f$$, with $\mathcal{G}^y$'s "constant" parameters depending on $x$ (Kurt et al., 2021). The conditional characteristic function preserves the affine structure: $$\mathbb{E}[e^{\langle u, Y_t\rangle}| X_0 = x, Y_0 = y] = \phi(t,x;u)\, e^{\langle \psi(t,u), y\rangle},$$ with $\psi$ solving autonomous Riccati ODEs and $\phi$ linked to a Cauchy problem driven by the generator $\mathcal{G}^x$.

Superpositions and Markovian Lifts

Non-Markovian, long-memory phenomena can be rendered Markovian by "lifting" to infinite-dimensional Markovian systems: $X_t = \int_0^\infty x_t(r) p(dr)$, each $x_t(r)$ solving its own affine SDE indexed by a mean-reversion speed $r$ (Yoshioka, 18 Apr 2025). The resulting process admits generalized Riccati equations for its exponential moments, and numerically exact discretization schemes—preserving crucial nonnegativity and marginal properties—facilitate simulation and calibration.

Stochastic Volterra and Memory-Effect Models

Affine Volterra processes generalize classical affine diffusions to include memory effects by convolution with kernels $K(t, s)$, yielding non-Markovian and time-inhomogeneous behavior: $$X_t = X_0 + \int_0^t K(t,s) b(s, X_s) ds + \int_0^t K(t,s) \sigma(s, X_s) dW_s.$$ Their conditional Fourier-Laplace transforms remain exponential-affine in an appropriately defined path-dependent sense, with the key objects given via inhomogeneous Riccati-Volterra equations (Ackermann et al., 2020).

6. Ergodicity, Stationarity, and Long-Time Limits

If the affine process possesses sufficiently negative drift (subcriticality), and the jump intensities satisfy moment or log-moment constraints, the transition semigroup converges exponentially fast in Wasserstein distance to a unique stationary law (Friesen et al., 2019Friesen et al., 2022). For infinite-dimensional models (e.g., forward curve models in the HJMM framework with stochastic volatility in a Hilbert space), the exponential rate is dimension-free; the long-maturity or stationary covariance regime justifies replacing a time-inhomogeneous process by its invariant law with explicit Riccati formulas for the Laplace transform.

7. Applications and Computational Techniques

Time-inhomogeneous affine processes are essential for:

• Interest rate modeling: affine LIBOR and inflation market models, where calibration to observed term structures or volatility surfaces is made more flexible by time-inhomogeneity (Waldenberger, 2015).
• Stochastic filtering: efficient filters for affine signals observed with noise, where generalized Riccati (often stochastic) equations exploit affine tractability for high-dimensional or non-Gaussian inference (Gonon et al., 2018).
• Structural modeling in environmental sciences: modeling of river discharge and water quality via superpositions of interacting affine and CEV processes, using exact discretization and Riccati-based computation (Yoshioka, 18 Apr 2025).

For numerical solution, exact discretization methods are developed to consistently simulate square-root diffusions and jump processes. Magnus expansion and splitting methods for the operator evolution, as well as measure-Riccati or Volterra-integral formulations, are implemented for practical pricing, estimation, and simulation in the presence of pronounced time-inhomogeneity.

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Summary Table: Common Structures in Time-Inhomogeneous Affine Processes

Structure Type	Key Equations	Context/Implications
Affine Transform with Riccati ODE/Integral	$\phi_{s,t}, \psi_{s,t}$	Option pricing, risk measures, filtering
Time-Change / Superposition	$X_t = \sum Z^{(i)}(\theta^{(i)}_t)$	Modeling memory, pathwise simulation
Semimartingale with Affine Characteristics	$B_t, C_t, \nu$ in affine form	Treatment of predictable/inaccessible jumps
Volterra / Non-Markovian Extensions	Kernel $K(t, s)$, Riccati-Volterra	Rough volatility, long-memory effects
Markov-Modulated (MMAP)	Joint generator, Riccati+Cauchy prob.	Regime switching, environmental dependencies

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Time-inhomogeneous affine processes generalize, unify, and extend the tractability of the affine paradigm to embrace temporal nonstationarity, environmental dependence, predictable jump events, and non-Markovian structure—while retaining computational and analytic transparency via exponential-affine transform formulas and Riccati-type equations. These properties are central both for rigorous probabilistic analysis and real-world implementations across quantitative finance, filtering, and applied stochastic modeling.