---
title: Time-Inhomogeneous SDEs with Jumps
url: https://www.emergentmind.com/topics/time-inhomogeneous-sdes-with-jumps
type: topic
---

# Time-Inhomogeneous SDEs with Jumps

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 [2401.09338], [2105.13015], [1712.03507], [2511.16973]. 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 [1712.03507].
- Time/state-inhomogeneous compound Poisson or general Levy-driven processes [2105.13015], [2401.09338].
- Population and mean-field models, where the coefficients depend on the law of the process [2511.16973].

## 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 $\gamma$ [1712.03507], [2401.09338], [2511.16973].
- **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) [1712.03507].
- **Moment and Integrability:** Finite moments up to prescribed orders for the compensated jump measure; Blumenthal–Getoor index ($\beta$) quantifies jump activity and tail distribution [2401.09338].
- **Nondegeneracy and Ergodicity:** Nonvanishing diffusion or irreducible jump component for exponential contractivity and controllability [2511.16973].

Under such conditions, existence/uniqueness of strong solutions, Markov property, and control of explosion/extinction probabilities can be rigorously established [2511.16973].

## 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 [1712.03507].
- **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 [1712.03507].
- **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 [2511.16973].

Convergence results also extend to path-dependent and mean-field SDEs when the coefficients have suitable structure [1712.03507], [2511.16973].

## 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) [2401.09338].
- **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 [2401.09338].
- **Recursive/Picard Representations:** Jump decoupling leads to dimension-free sequences of PDEs/PIDEs, facilitating both probabilistic and deterministic solvers [2105.13015].

| 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 [2105.13015], [2401.09338].

## 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 [2511.16973].
- **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 [2511.16973].
- **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 [2511.16973], [1712.03507].

## 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 [1712.03507].
- **Ruin Probabilities and Survival Models:** Pure-jump models with exit/ruin probabilities, for which closed-form and recursive solutions exist [2105.13015].
- **Hawkes Networks and Renewal Processes:** Mean-field and network systems where stochastic memory, variable-length dependence, and time-varying intensities emerge naturally [1712.03507].
- **Anomalous Diffusion:** Jump-driven models for turbulent rod-tumbling and other anomalous transport phenomena where time-inhomogeneous Lévy flights replace classical Brownian motion [2401.09338].

## 7. Research Directions and Open Problems

Emerging research includes:
- Generalization to infinite-activity and heavy-tailed jump measures, with truncation and compensation [2105.13015], [2401.09338].
- Adaptive thinning and backward SDEs with jumps.
- Quantitative large deviation theory and sharp pre-asymptotic error bounds in numerical schemes [2105.13015].
- 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 [1712.03507], [2105.13015], [2401.09338], [2511.16973].

Source: https://www.emergentmind.com/topics/time-inhomogeneous-sdes-with-jumps