Markov-Additive Process (MAP) Overview

• MAP is a bivariate process where an additive component evolves under the influence of a modulating Markov process, generalizing Lévy processes.
• The matrix-exponential and scale matrix formulations enable explicit analysis of transition probabilities, first-passage events, and fluctuation identities.
• MAPs are applied in queuing theory, self-similar process modeling, and regime-switching financial models while supporting strong limit theorems and ergodicity.

A Markov-additive process (MAP) is a strong Markov process on a product state space, conventionally $\mathbb{R} \times E$, in which the "ordinate" coordinate evolves additively, conditional on the path of a modulating Markov process ("modulator") on $E$. The transition mechanism is such that, given the modulator state at a stopping time, the future increments of the ordinate and the modulator depend only on the current state, and the law of future increments is independent of the path prior to the stopping time. This structure generalizes Lévy processes by allowing the increments of the additive component to be modulated by a Markovian environment, leading to models that capture versatile regime-switching, phase modulation, or multi-type behaviors in both theory and applications.

1. Formal Definition and Structure

A MAP is a bivariate process $(\xi_t, J_t)_{t\geq 0}$ with modulator $J_t$ living on a space $E$ (often finite or Polish) and additive component $\xi_t\in\mathbb{R}$ (or $\mathbb{R}^d$). The defining properties are:

• Markov property: $(\xi,J)$ is a Markov process on $\mathbb{R} \times E$ (Yaran et al., 8 Dec 2025, Yaran et al., 2024).
• Additivity: For any $t\ge0$, conditional on $E$0,

$E$1

has the same law as $E$2 under $E$3 (Kyprianou et al., 2015, Yaran et al., 8 Dec 2025, Yaran et al., 2024).

• Regime-dependent increments: Conditional on the modulator $E$4, $E$5 increments follow a Lévy process of type $E$6, and at jumps of $E$7 from $E$8 to $E$9, $(\xi_t, J_t)_{t\geq 0}$0 incurs an additional (possibly random) jump, independent from the path (Kyprianou et al., 2015, D'Auria et al., 2010, Behme et al., 2020).
• Right-process generality: For the broadest MAPs, $(\xi_t, J_t)_{t\geq 0}$1 is a right process on $(\xi_t, J_t)_{t\geq 0}$2 with the additivity property for measurable $(\xi_t, J_t)_{t\geq 0}$3,

$(\xi_t, J_t)_{t\geq 0}$4

(Noba, 31 Mar 2026).

This structure includes:

• Piecewise Lévy description: In each modulator state, $(\xi_t, J_t)_{t\geq 0}$5 behaves as a Lévy process (possibly with killing), and upon each jump of $(\xi_t, J_t)_{t\geq 0}$6 an independent jump (whose law depends on the transition) is added to $(\xi_t, J_t)_{t\geq 0}$7 (Döring et al., 2021, Kyprianou et al., 2015).
• Finite-state or general modulator: $(\xi_t, J_t)_{t\geq 0}$8 can be finite, countable, or more general Polish space (Kyprianou et al., 2015, Yaran et al., 2024, Kyprianou et al., 16 May 2025).

2. Matrix Exponent and Semigroup Representation

The law of a MAP with finite-state modulator is characterized by a matrix-exponential functional. For a process on $(\xi_t, J_t)_{t\geq 0}$9:

$J_t$0

where:

• $J_t$1 is the generator of $J_t$2,
• $J_t$3 is the Laplace exponent of the Lévy process in state $J_t$4, and
• $J_t$5 is the (Laplace or Fourier) transform of the jump from $J_t$6 to $J_t$7 in $J_t$8 (Kyprianou et al., 2015, D'Auria et al., 2010, Döring et al., 2021, Ivanovs et al., 2011).

For $J_t$9,

$E$0

This semigroup encodes all transition probabilities and can integrate non-lattice, spectrally one-sided, or lattice increment structures through appropriate choices of $E$1 and $E$2 (Kyprianou et al., 2015, Ivanovs et al., 2024).

The generator for smooth test functions $E$3 is:

$E$4

(Ivanovs, 2015).

3. Fluctuation Theory and Scale Matrices

Fluctuation identities for MAPs generalize those for Lévy processes using matrix-analytic methods:

• First passage/scale matrices: For a spectrally negative MAP, define the scale matrix $E$5 as the unique solution of

$E$6

(D'Auria et al., 2010, Ivanovs et al., 2011, Ivanovs et al., 2024).

• Two-sided exit: The probability of exiting an interval $E$7 upward before downward is

$E$8

(Ivanovs et al., 2011, D'Auria et al., 2010, Ivanovs et al., 2024).

• Generalizations: The scale matrix formalism extends to processes with state-dependent killing or discount (omega-killing) via Volterra equations for the generalized scale matrix $E$9 (Czarna et al., 2018).

Ladder processes—the times and overshoots at new maxima/minima—are Markov additive and their exponents enter the Wiener–Hopf factorization (Döring et al., 2021, Yaran et al., 2024). These tools unify the analysis of first-passage, reflection/skorokhod problems, and queuing analysis (Ivanovs et al., 2011, Mandjes et al., 10 Feb 2026).

4. Conditioning, Time-Change, and Self-Similarity

MAPs underpin the Lamperti–Kiu representation of real self-similar Markov processes (rssMp). For index $\xi_t\in\mathbb{R}$0, the time-change

$\xi_t\in\mathbb{R}$1

gives

$\xi_t\in\mathbb{R}$2

(Kyprianou et al., 2015, Kyprianou et al., 27 Jun 2025), so that $\xi_t\in\mathbb{R}$3 has the self-similarity property $\xi_t\in\mathbb{R}$4.

The Cramér–Esscher (Doob $\xi_t\in\mathbb{R}$5-) transform at $\xi_t\in\mathbb{R}$6 with $\xi_t\in\mathbb{R}$7 leads to a new MAP with "reversed" drift. For self-similar processes, this gives the construction for conditioning to avoid or be absorbed at the origin (Kyprianou et al., 2015).

5. Limit Theorems and Long-Time Behavior

MAPs exhibit classical and anomalous scaling limits:

• Strong law of large numbers: Under light regularity, if the modulator is Harris recurrent with invariant measure $\xi_t\in\mathbb{R}$8 and $\xi_t\in\mathbb{R}$9 has finite first moment, $\mathbb{R}^d$0 a.s., where $\mathbb{R}^d$1 aggregates the drift, Lévy, and environment-switch jump means (Kyprianou et al., 16 May 2025, Yaran et al., 2024).
• Functional central limit theorem: Under additional Lindeberg and ergodic conditions, $\mathbb{R}^d$2 admits Brownian (potentially time-changed by a stable subordinator) limits (Kyprianou et al., 16 May 2025).
• Overshoot, ergodicity, and mixing: The overshoot process $\mathbb{R}^d$3 is itself a Markov process, and its mixing and ergodic rates are governed by moment conditions and Lyapunov methods (Döring et al., 2021).
• Trichotomy for occupation: Under mild assumptions, $\mathbb{R}^d$4 either drifts to $\mathbb{R}^d$5 or oscillates, with explicit criterion in terms of ladder