Matrix Exponential Moment Method

Updated 24 November 2025

Matrix Exponential Moment Method is a framework that generalizes the classical matrix exponential by incorporating moment differential operators for both deterministic and stochastic analyses.
It provides closed-form transient and stationary moment formulas through explicit matrix-exponential representations and handles intricate nested matrix structures.
Applications include solving moment ODEs in Markov processes, deriving sharp exponential bounds for random matrix martingales, and enabling efficient numerical algorithms for high-dimensional problems.

The matrix exponential moment method is a framework that leverages generalized matrix exponentials to provide explicit representations and estimates for the moments of solutions to linear systems and for the norms of random matrix martingales, as well as closed-form formulas for the transient and stationary moments in a broad class of Markov processes. The method generalizes classical exponential matrix approaches by accommodating moment differential operators and intricate nested matrix structures, yielding powerful tools for both deterministic and stochastic systems analysis (Lastra et al., 2023, Daw et al., 2019, Formica et al., 24 Jan 2024).

1. Generalized Matrix Exponentials and Moment Differential Operators

Central to the matrix exponential moment method is the generalization of the exponential matrix to accommodate solutions of systems governed by moment differential operators. Given a strongly regular sequence $m=(m_p)_{p\ge0}$ (satisfying log-convexity, moderate growth, and non-quasianalyticity), the moment differential operator $D_m$ acts on the Taylor coefficients of a formal power series $\hat f(z) = \sum_{p=0}^{\infty} a_p z^p$ via

$D_m(\hat f) = \sum_{p=0}^{\infty} \frac{a_{p+1}}{m_p} z^p,$

with the associated m-exponential of a matrix $A \in \mathbb{C}^{n \times n}$ defined as

$E_m(A; t) = \sum_{p=0}^{\infty} m_p A^p t^p.$

Special cases recover the classical exponential ( $m_p = 1/p!$ ), the fractional exponential ( $m_p = \Gamma(1 + p/s)$ ), and the $q$ -exponential ( $m_p = [p]_q!$ ) (Lastra et al., 2023).

2. Algebraic and Analytic Structure of Generalized Exponentials

The generalized matrix exponential $E_m(A; t)$ exhibits important properties:

Formally, $D_m[E_m(A\,z)] = A E_m(A\,z)$ .
If $AB=BA$ , then $E_m(A)E_m(B)=E_m(B)E_m(A)$ .
If $A = CBC^{-1}$ , $E_m(A) = C E_m(B) C^{-1}$ .
For $m$ with sufficiently rapid growth ( $\liminf_{p\to\infty} m_p^{1/p} = +\infty$ ), $E_m(A; t)$ converges for all $t$ in $\mathbb{C}$ and is entire, with the estimate $\|E_m(A; t)\| \le E_m(\|A\|\,|t|)$ .
$E_m(A; t)$ is everywhere invertible under these growth conditions; the inverse admits a recursive series representation.
In general, $E_m((A+B)t) \ne E_m(A\,t)E_m(B\,t)$ unless $m$ is the Gevrey-1 sequence (Lastra et al., 2023).

3. Matrix Exponential Representation of Linear Systems and Markov Processes

Moment ODEs for Markovian Systems

For Markov processes $X(t)$ where the generator $\mathcal{L}$ maps monomials to lower-degree polynomials,

$\mathcal{L}[x^k] = \sum_{j=0}^k a_{k,j} x^j,$

the vector of moments $\mathbf{m}(t) = (m_0(t), m_1(t), \ldots, m_n(t))^T$ with $m_k(t) = \mathbb{E}[X(t)^k]$ satisfies the linear ODE

$\frac{d}{dt} \mathbf{m}(t) = A \mathbf{m}(t),$

where $A$ is lower-triangular with entries $A_{k,j} = a_{k,j}$ for $j \leq k$ (Daw et al., 2019).

Explicit Matrix-Exponential Solution

The solution for the moment vector is given by

$\mathbf{m}(t) = e^{A t} \mathbf{m}(0).$

If the ODE includes a constant shift vector $c$ , then

$\mathbf{m}(t) = e^{A t}\mathbf{m}(0) + A^{-1}(e^{A t} - I)c$

yields closed-form transient and, for $\lim_{t\to\infty}$ , stationary expressions. For nested ("Matryoshkhan") block lower-triangular matrices, the structure supports recursive computation of the exponential and eigen-decomposition (Daw et al., 2019).

4. Exponential Moment Bounds for Random Matrix Martingales

For a matrix martingale $\{M_k\}$ with differences $\Delta_k$ and operator norm $\|\cdot\|$ , sharp Lebesgue-Riesz and Grand Lebesgue Space (GLS) norm estimates are established for $\|M_n\|$ , in terms of $\ell_p$ -norms of the differences, and the entropic dimension $K$ of the set of extremal test pairs

$Z = \{(x, y) \in S_0(\mathbb{R}^d) \times S_0((\mathbb{R}^d)^*): \|x\|=1, \|y\|=1\}.$

The main result gives

$\|\,\|M_n\|\,\|_p \le \sqrt{n}\, B(p),\qquad B(p) = v(p) \frac{p}{p-K} 10^{(p-K)/(p-K)},$

where $v(p)$ depends on maximal $p$ -norms of entries $\Delta_k(i,j)$ and the Osekowski–Burkholder constant (Formica et al., 24 Jan 2024).

Embedding $n^{-1/2}\|M_n\|$ in a suitable GLS yields exponential tail inequalities:

$\mathbb{P}(n^{-1/2}\|M_n\| > t) \le \exp\left[-h^*(\ln t)\right],$

where $h^*(t)$ denotes the Young–Fenchel transform of $h(p) = p \ln B(p)$ (Formica et al., 24 Jan 2024).

5. Practical Computation and Illustrative Examples

Matrix-Exponential Moment Method for Markovian Models

The method provides explicit transient and steady-state formulas for broad Markovian models, such as:

Markovian Hawkes processes: yields, for the first time, closed-form transient expressions for all orders of the intensity process.
Shot noise models, growth-collapse processes, affine Itô diffusions, and ephemerally self-exciting birth–death–immigration systems.

The methodology is summarized:

Construct the generator-derived lower-triangular matrix $A$ up to the required moment order.
Initialize the moments vector at $t=0$ .
Compute $e^{A t}$ , using Padé/Schur methods or the Matryoshkhan recursion.
Obtain the full moment vector by a single matrix multiplication.

Closed-form expressions for both the finite-time and stationary moments emerge directly from this computation, bypassing iterative numerical integration (Daw et al., 2019).

Solution of Linear Moment Differential Systems

For generalized linear systems $D_m x(t) = A x(t) + f(t)$ , the system admits the solution

$x(t) = E_m(A\,t) x_0 + \int_0^t E_m(A\,(t-s)) f(s) ds,$

directly paralleling the classical variation-of-constants formula but in the context of the generalized operator and exponential (Lastra et al., 2023).

Explicit Representation with Jordan–Block Decomposition

For any matrix $A$ , reduction to Jordan form enables computation of $E_m(A\,t)$ via block-diagonalization and closed-form sums over the sequence $m_p$ , with explicit handling of nilpotent Jordan blocks.

6. Examples and Sharpness Analysis

Sharpness and optimality of the moment and tail bounds are established for both polynomial-growth differences and heavy-tailed difference cases in matrix martingale settings:

For differences with $\max_k H_k[p] \leq C p^A \ln p$ , the tail bound on $n^{-1/2}\|M_n\|$ decays as $\exp(-C_1 t^{1/(A+1)})$ .
For heavy-tailed entries, polynomial (possibly logarithmic) tails are shown, with the derived rate being optimal in the scalar case (Formica et al., 24 Jan 2024).

7. Numerical and Algorithmic Remarks

Efficiency and stability benefit from the nested block structure of Matryoshkhan matrices and the controlled growth of $m_p$ in practical cases (e.g., Gevrey, fractional, $q$ -extensions):

The block structure allows recursive computation for large moment systems, with each principal submatrix governing lower-order moments.
For fractional and $q$ -exponentials, classical numerical techniques for matrix exponentials, such as scaling and squaring or Padé approximants, extend naturally due to the factorial-type growth of $m_p$ (Lastra et al., 2023).
The methodology converts infinite-dimensional moment closure problems into finite-dimensional linear algebra, achieving substantial gains over traditional forward-integration schemes (Daw et al., 2019).