Jump-Time Averaged State in Stochastic Processes

• Jump-time averaged state is defined as an effective state computed by averaging fast jump dynamics against invariant measures, occupation laws, or event samples.
• It is applied in slow–fast jump-diffusions, regime-switching systems, and non-autonomous SPDEs to simplify complex dynamics into effective drift terms.
• The concept integrates diverse approaches such as invariant-measure averaging, empirical jump-event distributions, and hazard-weighted post-jump beliefs to capture essential dynamics.

“Jump-time averaged state” is not a standard term with a single accepted definition across stochastic-process theory. In the averaging-principle literature, the closest rigorous object is usually the averaged slow dynamics obtained by freezing the slow variable and averaging the slow drift against the invariant law, or more generally the occupation law, of a fast jump-driven subsystem. In jump-location and empirical-process literature, the nearest object is instead the distribution of the process evaluated at jump epochs. In hazard-based and filtering formulations, the same phrase points to hazard-weighted jump-event distributions or to posterior beliefs after jump innovations (Zhang et al., 2017, Mao et al., 2022, Xu et al., 2018, Miles et al., 2017).

1. Terminological scope and principal meanings

In the terminology of "Weak order in averaging principle for stochastic differential equations with jumps" (Zhang et al., 2017), the object corresponding to a jump-time averaged state is the averaged dynamics or averaged slow motion of a slow–fast jump-diffusion. There, the averaging is performed by freezing the slow variable, constructing the invariant measure of the fast jump-diffusion, and replacing the slow drift by its invariant-measure average.

Several later formulations make the same conceptual point in different settings. For two time-scale regime-switching processes on a countably infinite state space, the effective state is obtained by averaging the slow drift against the invariant distribution $\pi^x$ of the fast jump chain conditional on frozen slow state $x$, and the paper explicitly emphasizes that this is not an average over explicit jump times (Mao et al., 2022). For non-autonomous reaction–diffusion equations with Poisson random measures, the averaged object is defined through a time-dependent evolution family of measures $\mu_t^x$ and a long-time Bohr mean, again not through averaging over jump epochs (Xu et al., 2018).

A distinct usage appears when the process is sampled at jump times themselves. "Jump Locations of Jump-Diffusion Processes with State-Dependent Rates" (Miles et al., 2017) makes the jump-epoch viewpoint primary: the central object is the sequence of jump locations $X_i=X_{t_i}$ and its stationary distribution $p_\star$. Large-deviation formulations sharpen this distinction by separating the ordinary time-averaged density $p(x)$ from the empirical distributions $Q^{-}(x)/n$ and $Q^{+}(x)/n$ of pre-jump and post-jump states per jump event (Monthus, 2021).

The resulting terminology is therefore context-dependent. A “jump-time averaged state” may mean an invariant-measure average over a fast jump mechanism, an event-sampled jump-location distribution, a hazard-weighted jump-event state law, or a posterior belief updated by jump observations.

2. Averaged slow state in slow–fast jump-diffusions

The canonical averaging-principle formulation is the two-time-scale jump-diffusion system

$$\begin{aligned} dX_t^\epsilon &= a(X_t^\epsilon,Y_t^\epsilon)\,dt+b(X_t^\epsilon)\,dB_t+c(X_{t-}^\epsilon)\,dP_t,\qquad X_0^\epsilon=x, \ dY_t^\epsilon &= \frac1\epsilon f(X_t^\epsilon,Y_t^\epsilon)\,dt+\frac1{\sqrt\epsilon}g(X_t^\epsilon,Y_t^\epsilon)\,dW_t+h(X_{t-}^\epsilon,Y_{t-}^\epsilon)\,dN_t^\epsilon,\qquad Y_0^\epsilon=y, \end{aligned}$$

where $P_t$ has intensity $x$0 and $x$1 has intensity $x$2 (Zhang et al., 2017). The slow variable $x$3 carries drift, diffusion, and jumps, while the fast variable $x$4 is accelerated in drift, diffusion, and jump intensity.

The standing assumptions are $x$5-regularity with bounded first and second derivatives, boundedness of $x$6, nondegeneracy of the fast diffusion, and a dissipativity condition that includes the jump coefficient $x$7. Under these hypotheses, the frozen fast motion

$x$8

has a unique invariant measure $x$9, exponential contraction,

$\mu_t^x$0

and ergodicity (Zhang et al., 2017).

The averaged drift is then

$\mu_t^x$1

This is the precise mathematical content of the jump-time averaged state in this setting: the slow drift is averaged against the equilibrium law of the fast jump-diffusion. The corresponding effective slow equation is

$\mu_t^x$2

Two structural points are essential. First, only the slow drift is averaged, because in this model $\mu_t^x$3 and $\mu_t^x$4 do not depend on the fast variable. Second, the two jump mechanisms play different roles: the fast jump term $\mu_t^x$5 changes the invariant measure $\mu_t^x$6 and therefore changes $\mu_t^x$7, whereas the slow jump term $\mu_t^x$8 survives explicitly in the limit equation (Zhang et al., 2017).

3. Generator formulation, Poisson equation, and weak order $\mu_t^x$9

The weak approximation theory is formulated through infinitesimal generators. For a slow test function $X_i=X_{t_i}$0,

$X_i=X_{t_i}$1

while the frozen fast generator is

$X_i=X_{t_i}$2

Hence the full generator is $X_i=X_{t_i}$3, and the averaged generator is

$X_i=X_{t_i}$4

For $X_i=X_{t_i}$5 with $X_i=X_{t_i}$6, the asymptotic expansion

$X_i=X_{t_i}$7

leads to

$X_i=X_{t_i}$8

Since $X_i=X_{t_i}$9, the leading term does not depend on $p_\star$0, and after integrating against $p_\star$1 one obtains

$p_\star$2

Thus $p_\star$3, where $p_\star$4.

The first correction solves the Poisson equation

$p_\star$5

with representation

$p_\star$6

This is the first-order weak correction generated by the fast jump-diffusion semigroup.

The main theorem is that for any $p_\star$7 and $p_\star$8,

$p_\star$9

The weak order in the averaging principle is therefore $p(x)$0, while the paper explicitly compares this with strong averaging order $p(x)$1, so the weak rate is essentially twice the strong one (Zhang et al., 2017).

4. Generalizations: fast pure-jump switching and non-autonomous jump-driven SPDEs

For fully coupled two-time-scale regime-switching systems, the slow component is a diffusion in $p(x)$2 and the fast component is a pure jump process on a countably infinite state space $p(x)$3. With frozen slow state $p(x)$4, the fast chain has generator $p(x)$5 and invariant law $p(x)$6, and the averaged drift is

$p(x)$7

The limit system is the deterministic ODE

$p(x)$8

because the diffusion coefficient is multiplied by $p(x)$9 and vanishes in the limit (Mao et al., 2022).

In that framework, the decisive issue is the regularity of $Q^{-}(x)/n$0. Under uniform strong ergodicity,

$Q^{-}(x)/n$1

so the averaged drift is Lipschitz and the slow process converges in $Q^{-}(x)/n$2 to the unique averaged limit. Under weaker ergodicity,

$Q^{-}(x)/n$3

the averaged drift is only Hölder continuous; the limit ODE still admits a solution, but uniqueness may fail, and only weak convergence is obtained in general (Mao et al., 2022). This makes explicit that the averaged state may exist while the averaged trajectory is not uniquely determined.

For non-autonomous stochastic reaction–diffusion equations with Poisson random measures, the autonomous invariant-measure picture is replaced by a time-dependent evolution family of measures $Q^{-}(x)/n$4 associated with the frozen fast equation. The basic averaged coefficient is

$Q^{-}(x)/n$5

and the effective drift is the Bohr mean

$Q^{-}(x)/n$6

The averaged equation keeps the slow Wiener and Poisson terms explicitly: $Q^{-}(x)/n$7 Only the coupling through the fast variable is averaged out (Xu et al., 2018).

5. Event-sampled states at jump epochs

A different and literal meaning arises when the state is sampled at jump times. In the framework of jump-diffusions with state-dependent jump intensity $Q^{-}(x)/n$8, jump times are $Q^{-}(x)/n$9 and jump locations are

$Q^{+}(x)/n$0

Their densities $Q^{+}(x)/n$1 evolve through an absorbing-process construction, and the stationary jump-location distribution $Q^{+}(x)/n$2 is characterized by

$Q^{+}(x)/n$3

with normalization

$Q^{+}(x)/n$4

If $Q^{+}(x)/n$5 is the stationary density of the full reinjected process, then

$Q^{+}(x)/n$6

Hence the stationary distribution of jump locations is the same as the stationary distribution of the full process if and only if $Q^{+}(x)/n$7 is constant (Miles et al., 2017). This is the basic event-bias relation for jump-time sampling.

Large-deviation theory makes the same distinction at the empirical level. The ordinary time-averaged density is

$Q^{+}(x)/n$8

whereas the empirical jump-flow is

$Q^{+}(x)/n$9

Its marginals

$$\begin{aligned} dX_t^\epsilon &= a(X_t^\epsilon,Y_t^\epsilon)\,dt+b(X_t^\epsilon)\,dB_t+c(X_{t-}^\epsilon)\,dP_t,\qquad X_0^\epsilon=x, \ dY_t^\epsilon &= \frac1\epsilon f(X_t^\epsilon,Y_t^\epsilon)\,dt+\frac1{\sqrt\epsilon}g(X_t^\epsilon,Y_t^\epsilon)\,dW_t+h(X_{t-}^\epsilon,Y_{t-}^\epsilon)\,dN_t^\epsilon,\qquad Y_0^\epsilon=y, \end{aligned}$$0

are the empirical densities of pre-jump and post-jump states per unit physical time, and

$$\begin{aligned} dX_t^\epsilon &= a(X_t^\epsilon,Y_t^\epsilon)\,dt+b(X_t^\epsilon)\,dB_t+c(X_{t-}^\epsilon)\,dP_t,\qquad X_0^\epsilon=x, \ dY_t^\epsilon &= \frac1\epsilon f(X_t^\epsilon,Y_t^\epsilon)\,dt+\frac1{\sqrt\epsilon}g(X_t^\epsilon,Y_t^\epsilon)\,dW_t+h(X_{t-}^\epsilon,Y_{t-}^\epsilon)\,dN_t^\epsilon,\qquad Y_0^\epsilon=y, \end{aligned}$$1

is the jump density. The normalized jump-epoch distributions are therefore $$\begin{aligned} dX_t^\epsilon &= a(X_t^\epsilon,Y_t^\epsilon)\,dt+b(X_t^\epsilon)\,dB_t+c(X_{t-}^\epsilon)\,dP_t,\qquad X_0^\epsilon=x, \ dY_t^\epsilon &= \frac1\epsilon f(X_t^\epsilon,Y_t^\epsilon)\,dt+\frac1{\sqrt\epsilon}g(X_t^\epsilon,Y_t^\epsilon)\,dW_t+h(X_{t-}^\epsilon,Y_{t-}^\epsilon)\,dN_t^\epsilon,\qquad Y_0^\epsilon=y, \end{aligned}$$2 and $$\begin{aligned} dX_t^\epsilon &= a(X_t^\epsilon,Y_t^\epsilon)\,dt+b(X_t^\epsilon)\,dB_t+c(X_{t-}^\epsilon)\,dP_t,\qquad X_0^\epsilon=x, \ dY_t^\epsilon &= \frac1\epsilon f(X_t^\epsilon,Y_t^\epsilon)\,dt+\frac1{\sqrt\epsilon}g(X_t^\epsilon,Y_t^\epsilon)\,dW_t+h(X_{t-}^\epsilon,Y_{t-}^\epsilon)\,dN_t^\epsilon,\qquad Y_0^\epsilon=y, \end{aligned}$$3 (Monthus, 2021). In this sense, a jump-time averaged state is not $$\begin{aligned} dX_t^\epsilon &= a(X_t^\epsilon,Y_t^\epsilon)\,dt+b(X_t^\epsilon)\,dB_t+c(X_{t-}^\epsilon)\,dP_t,\qquad X_0^\epsilon=x, \ dY_t^\epsilon &= \frac1\epsilon f(X_t^\epsilon,Y_t^\epsilon)\,dt+\frac1{\sqrt\epsilon}g(X_t^\epsilon,Y_t^\epsilon)\,dW_t+h(X_{t-}^\epsilon,Y_{t-}^\epsilon)\,dN_t^\epsilon,\qquad Y_0^\epsilon=y, \end{aligned}$$4 but the appropriate marginal of $$\begin{aligned} dX_t^\epsilon &= a(X_t^\epsilon,Y_t^\epsilon)\,dt+b(X_t^\epsilon)\,dB_t+c(X_{t-}^\epsilon)\,dP_t,\qquad X_0^\epsilon=x, \ dY_t^\epsilon &= \frac1\epsilon f(X_t^\epsilon,Y_t^\epsilon)\,dt+\frac1{\sqrt\epsilon}g(X_t^\epsilon,Y_t^\epsilon)\,dW_t+h(X_{t-}^\epsilon,Y_{t-}^\epsilon)\,dN_t^\epsilon,\qquad Y_0^\epsilon=y, \end{aligned}$$5.

The same structure appears in linear-response theory for Markov jump processes. The paper on martingale-based linear response treats jump-additive functionals

$$\begin{aligned} dX_t^\epsilon &= a(X_t^\epsilon,Y_t^\epsilon)\,dt+b(X_t^\epsilon)\,dB_t+c(X_{t-}^\epsilon)\,dP_t,\qquad X_0^\epsilon=x, \ dY_t^\epsilon &= \frac1\epsilon f(X_t^\epsilon,Y_t^\epsilon)\,dt+\frac1{\sqrt\epsilon}g(X_t^\epsilon,Y_t^\epsilon)\,dW_t+h(X_{t-}^\epsilon,Y_{t-}^\epsilon)\,dN_t^\epsilon,\qquad Y_0^\epsilon=y, \end{aligned}$$6

and the jump count $$\begin{aligned} dX_t^\epsilon &= a(X_t^\epsilon,Y_t^\epsilon)\,dt+b(X_t^\epsilon)\,dB_t+c(X_{t-}^\epsilon)\,dP_t,\qquad X_0^\epsilon=x, \ dY_t^\epsilon &= \frac1\epsilon f(X_t^\epsilon,Y_t^\epsilon)\,dt+\frac1{\sqrt\epsilon}g(X_t^\epsilon,Y_t^\epsilon)\,dW_t+h(X_{t-}^\epsilon,Y_{t-}^\epsilon)\,dN_t^\epsilon,\qquad Y_0^\epsilon=y, \end{aligned}$$7 is obtained by taking $$\begin{aligned} dX_t^\epsilon &= a(X_t^\epsilon,Y_t^\epsilon)\,dt+b(X_t^\epsilon)\,dB_t+c(X_{t-}^\epsilon)\,dP_t,\qquad X_0^\epsilon=x, \ dY_t^\epsilon &= \frac1\epsilon f(X_t^\epsilon,Y_t^\epsilon)\,dt+\frac1{\sqrt\epsilon}g(X_t^\epsilon,Y_t^\epsilon)\,dW_t+h(X_{t-}^\epsilon,Y_{t-}^\epsilon)\,dN_t^\epsilon,\qquad Y_0^\epsilon=y, \end{aligned}$$8. This provides the numerator and denominator needed for jump-count-normalized empirical state averages, even though the normalized ratio itself is not the primary object of that work (Faggionato et al., 2022).

6. Hazard-weighted jump-event states and filtered post-jump beliefs

In state-dependent hazard models, the natural jump-time state law is neither an invariant-measure average nor a stationary jump-flow marginal. For the Ornstein–Uhlenbeck internal-state model

$$\begin{aligned} dX_t^\epsilon &= a(X_t^\epsilon,Y_t^\epsilon)\,dt+b(X_t^\epsilon)\,dB_t+c(X_{t-}^\epsilon)\,dP_t,\qquad X_0^\epsilon=x, \ dY_t^\epsilon &= \frac1\epsilon f(X_t^\epsilon,Y_t^\epsilon)\,dt+\frac1{\sqrt\epsilon}g(X_t^\epsilon,Y_t^\epsilon)\,dW_t+h(X_{t-}^\epsilon,Y_{t-}^\epsilon)\,dN_t^\epsilon,\qquad Y_0^\epsilon=y, \end{aligned}$$9

with jump time

$P_t$0

the density $P_t$1 of still-alive particles satisfies

$P_t$2

The waiting-time density is

$P_t$3

so the joint jump-event density in state–time space is

$P_t$4

Consequently,

$P_t$5

This is the natural jump-time averaged state in a hazard-driven setting (Xue et al., 2023).

In partially observed jump-diffusion systems, the nearest analogue is not a state value but a belief distribution. The filtered conditional law is

$P_t$6

and the low-dimensional state summary exposed to the forecast model is

$P_t$7

The encoder performs a Strang-splitting update

$P_t$8

where $P_t$9 is the jump innovation. In that framework, the closest rigorous object to a jump-time averaged state is the posterior belief, or the belief feature $x$00, after the jump-innovation update (Leng et al., 23 May 2026).

A persistent misconception is therefore that jump-time averaging always means averaging over discrete jump epochs. The literature shows four non-equivalent constructions: invariant-measure averaging of a fast jump mechanism, occupation-measure averaging for fast switching, empirical state distributions at jump events, and hazard- or observation-weighted post-jump laws. What they share is not a common formula, but a common structural role: they replace the full jump-resolved dynamics by an effective state description adapted either to fast jump clocks or to event-conditioned sampling.