Time-Inhomogeneous SDEs with Jumps

• Time-Inhomogeneous SDEs with jumps are stochastic processes defined by time-dependent drift, diffusion, and jump components that capture non-stationary dynamics.
• They require rigorous structural conditions, such as Lipschitz continuity and moment integrability, to ensure well-posedness and control phenomena like explosion or extinction.
• Advanced numerical methods, including ε-Euler–Maruyama schemes, balance the treatment of small diffusive fluctuations with large jump impacts to achieve convergent approximations.

A time-inhomogeneous stochastic differential equation with jumps (time-inhomogeneous SDE with jumps) is a stochastic process governed by coefficients that depend explicitly on time, state, and potentially exhibit discontinuous evolution due to random jump events. These SDEs generalize classical diffusion models by incorporating Lévy-type or more general jump mechanisms with non-stationary features, and they play a central role in contemporary probability theory, stochastic analysis, and a range of applications including finance, neuroscience, and statistical mechanics.

1. Mathematical Formulation and Model Classes

Time-inhomogeneous jump SDEs are formulated as càdlàg processes $X=(X_t)_{t\ge0}$ on a filtered probability space, involving both continuous diffusive and discontinuous jump-driven dynamics. Their general Itô form is

$$dX_t = b(t,X_{t^-})\,dt + \sigma(t,X_{t^-})\,dW_t + \int_{E} c(t,z,X_{t^-})\,N(dt,dz) + \int_{E} c(t,z,X_{t^-})\,\tilde{N}(dt,dz)$$

where:

• $b(t,x)$: time-/state-dependent drift,
• $\sigma(t,x)$: time-/state-dependent diffusion coefficient,
• $W_t$: multi-dimensional Brownian motion,
• $N(dt,dz)$: (possibly state/time-inhomogeneous) Poisson random measure with compensator dependent on $t,x$,
• $c(t,z,x)$: jump size function,
• $E$: mark space for the jumps, $\tilde N = N-\nu$.

Alternative formulations use a triplet $(b,\sigma,\nu)$ with random measures parametrized by time and state (Bossy et al., 17 Jan 2024, Qiu et al., 2021, Löcherbach, 2017, Chen et al., 21 Nov 2025). The generator incorporates both continuous and jump components, often expressed as

$$L_t f(x) = \sum_{i=1}^d \partial_{x_i}f(x)\,b^i(t,x) + \tfrac 12 \sum_{i,j=1}^d \partial_{x_ix_j}^2 f(x)\,a^{ij}(t,x) + \int_{E} [f(x+c(t,z,x)) - f(x)]\gamma(t,z,x)\mu(dz)$$

where $\gamma(t,z,x)$ is a possibly state/time-dependent jump intensity and $a^{ij}(t,x)$ the local covariance.

Process classes include:

• Multi-regime jump diffusions with fast diffusion-like small jumps, drift-producing intermediate jumps, and non-vanishing (slow) large jumps (Löcherbach, 2017).
• Time/state-inhomogeneous compound Poisson or general Levy-driven processes (Qiu et al., 2021, Bossy et al., 17 Jan 2024).
• Population and mean-field models, where the coefficients depend on the law of the process (Chen et al., 21 Nov 2025).

2. Structural Assumptions and Regimes

Establishing the well-posedness and qualitative properties of time-inhomogeneous SDEs with jumps requires structural hypotheses:

• Lipschitz and Growth Conditions: Local or global Lipschitz continuity and linear growth bounds on $b$, $\sigma$, $c$, and $\gamma$ (Löcherbach, 2017, Bossy et al., 17 Jan 2024, Chen et al., 21 Nov 2025).
• Jump Partitioning: The mark space $E$ is often partitioned into regimes $E_1$ (rapidly vanishing jumps for diffusive scaling), $E_2$ (moderate jumps yielding deterministic drift), and $E_3$ (persistent big jumps) (Löcherbach, 2017).
• Moment and Integrability: Finite moments up to prescribed orders for the compensated jump measure; Blumenthal–Getoor index ($\beta$) quantifies jump activity and tail distribution (Bossy et al., 17 Jan 2024).
• Nondegeneracy and Ergodicity: Nonvanishing diffusion or irreducible jump component for exponential contractivity and controllability (Chen et al., 21 Nov 2025).

Under such conditions, existence/uniqueness of strong solutions, Markov property, and control of explosion/extinction probabilities can be rigorously established (Chen et al., 21 Nov 2025).

3. Long-Time Behavior: Equilibrium and Ergodicity

Time-inhomogeneous jump SDEs exhibit intricate convergence properties determined by the competition between time dependence and stochasticity:

• Homogenization and Limiting Dynamics: If coefficients stabilize as $t\to\infty$, $L_t\to L$, then $X_t$ converges in distribution to a time-homogeneous SDE with jumps, for which explicit equilibrium characterization is possible. Semigroup convergence in operator norm and the concept of “asymptotic pseudo-trajectory” of the limit semigroup are central (Löcherbach, 2017).
• Regeneration via Big Jumps: For limit processes, coupling arguments built on regeneration at large jumps allow one to construct explicit minorization and recurrence, leading to exponential ergodicity and quantitative total variation bounds (Löcherbach, 2017).
• Ergodicity and Exponential Mixing: For Markov processes with a Lyapunov function and sufficient spread in the jump/diffusion component, one obtains explicit exponential contractivity rates in weighted total variation or Wasserstein distance (Chen et al., 21 Nov 2025).

Convergence results also extend to path-dependent and mean-field SDEs when the coefficients have suitable structure (Löcherbach, 2017, Chen et al., 21 Nov 2025).

4. Numerical Methods and Approximation Schemes

Numerical approximation of time-inhomogeneous SDEs with jumps is complex due to discontinuities and time-dependent coefficients.

• $\varepsilon$-Euler–Maruyama Schemes: A two-parameter discretization is introduced, combining classical Euler–Maruyama for drift/diffusion with truncation at scale $\varepsilon$ for jumps: jumps smaller than $\varepsilon$ are replaced by a Gaussian increment (Asmussen–Rosiński compensation) or neglected (Rubenthaler truncation) (Bossy et al., 17 Jan 2024).
• Convergence Rates:
  • Strong rate (in $L^p$-sup norm): order $1/p$ under regularity and moment assumptions, with refinement by the Blumenthal–Getoor index and jump compensation method.
  • Weak rate: $O(1/n + \varepsilon^{3-\beta})$ for the compensated scheme and $O(1/n + \varepsilon^{2-\beta})$ for simple truncation (Bossy et al., 17 Jan 2024).
• Recursive/Picard Representations: Jump decoupling leads to dimension-free sequences of PDEs/PIDEs, facilitating both probabilistic and deterministic solvers (Qiu et al., 2021).

Scheme	Strong Rate ($L^p$)	Weak Rate
Asmussen–Rosiński (comp.)	$O(n^{-1/p})$	$O(1/n + \varepsilon^{3-\beta})$
Rubenthaler (truncation)	$O(n^{-1/p})$ with small $\varepsilon$	$O(1/n + \varepsilon^{2-\beta})$

These algorithms underlie modern Monte Carlo, finite element, and deep PDE-based solvers for SDEs with jumps (Qiu et al., 2021, Bossy et al., 17 Jan 2024).

5. Boundary Phenomena: Extinction, Explosion, and Contraction

Time-inhomogeneous SDEs with jumps can display extinction (absorption at zero), explosion (blow-up), or exhibit uniform contractivity.

• Criteria for Extinction/Explosion: Quantitative criteria in terms of the drift $\gamma_0$, volatility/jump intensities, and jump measure $\mu$ govern almost sure extinction or explosion probabilities, often via test function martingale arguments and fine Lyapunov analysis (Chen et al., 21 Nov 2025).
• Exponential Contraction: Weighted total variation (or Wasserstein) contraction arises when the generator satisfies a Lyapunov dissipation condition and the noise components are suitably nondegenerate, yielding exponential ergodicity (Chen et al., 21 Nov 2025).
• Mean-Field and Population Models: These criteria extend to non-linear (McKean–Vlasov) and killed particle systems, as in mean-field SDEs appearing in population dynamics and neuroscience (Chen et al., 21 Nov 2025, Löcherbach, 2017).

6. Representative Examples and Applications

Several canonical models illustrate the scope and flexibility of time-inhomogeneous jump SDEs:

• Triple-Regime Jump Diffusion: Decomposition of jumps into rapidly vanishing (Gaussian), intermediate (drift), and persistent (large jump) regimes yields Cox–Ingersoll–Ross–type SDEs (Löcherbach, 2017).
• Ruin Probabilities and Survival Models: Pure-jump models with exit/ruin probabilities, for which closed-form and recursive solutions exist (Qiu et al., 2021).
• Hawkes Networks and Renewal Processes: Mean-field and network systems where stochastic memory, variable-length dependence, and time-varying intensities emerge naturally (Löcherbach, 2017).
• Anomalous Diffusion: Jump-driven models for turbulent rod-tumbling and other anomalous transport phenomena where time-inhomogeneous Lévy flights replace classical Brownian motion (Bossy et al., 17 Jan 2024).

7. Research Directions and Open Problems

Emerging research includes:

• Generalization to infinite-activity and heavy-tailed jump measures, with truncation and compensation (Qiu et al., 2021, Bossy et al., 17 Jan 2024).
• Adaptive thinning and backward SDEs with jumps.
• Quantitative large deviation theory and sharp pre-asymptotic error bounds in numerical schemes (Qiu et al., 2021).
• Extension to non-Markovian frameworks and systems with memory or functional dependence.

Comprehensive theoretical and practical analyses of time-inhomogeneous SDEs with jumps continue to deepen understanding of stochastic processes with non-stationary structure and discontinuous evolution (Löcherbach, 2017, Qiu et al., 2021, Bossy et al., 17 Jan 2024, Chen et al., 21 Nov 2025).