Hu-Meyer Formulas in Stochastic Calculus

• Hu-Meyer formulas are decompositional identities that express multiple stochastic integrals as explicit combinations of Itô and Stratonovich integrals with correction terms.
• They employ series expansions on complete orthonormal systems to achieve high-order approximations, ensuring strong convergence in numerical methods for stochastic differential equations.
• The formulas extend to modular forms and signal processing, providing insights into Dedekind sums, Fourier eigenmeasure characterizations, and analytic signal decompositions.

The Hu-Meyer formulas are a class of identities that relate multiple stochastic integrals in the Itô and Stratonovich frameworks, and more generally, establish connections between analytic and algebraic structures appearing in wavelet theory, modular forms, arithmetic zeta functions, and the theory of stochastic differential equations. Although their original context was in stochastic calculus, subsequent generalizations have linked them to eigenmeasure characterization in harmonic analysis, Dedekind/arithmetic sum decompositions in number theory, and analytic signal representations in time-frequency analysis.

1. Fundamental Structure of the Hu-Meyer Formulas

The canonical Hu-Meyer formula expresses a multiple Wiener (Itô) stochastic integral as an explicit combination of multiple Stratonovich integrals, and vice versa, typically in the setting of a multidimensional Wiener process. For a kernel $\Phi$ and indexing over $k$ variables, a prototypical formulation for a multiple Stratonovich integral $J^S[\Phi]_{T,t}^{(i_1\ldots i_k)}$ is: $$J^S[\Phi]_{T,t}^{(i_1\ldots i_k)} = J'[\Phi]_{T,t}^{(i_1\ldots i_k)} + \sum_{r=1}^{[k/2]}\sum_{\text{admissible partitions}} \prod_{s=1}^r 1_{\{i_{g_{2s-1}} = i_{g_{2s}} \neq 0\}} J'[T^{(k-2r)}_{g_1,\ldots,g_{2r}} \Phi]_{T,t}^{(i_{q_1},\ldots,i_{q_{k-2r}})}$$ with $J'[\,\cdot\,]$ denoting multiple Itô integrals and $T^{(k-2r)}_{g_1,\ldots,g_{2r}} \Phi$ a limiting trace in the Volterra kernel $\Phi$ after identification of paired indices. The inverse formula uses alternating signs and additional traces to express Wiener in terms of Stratonovich integrals.

This algebraic decomposition systematically captures corrections due to stochastic calculus conventions—noncommutativity, Itô–Stratonovich correction terms, symmetrisation, and trace-contributions when integrating over potentially repeated indices.

2. Series Expansion and Limiting Traces

A critical methodology underpinning recent generalizations of Hu-Meyer formulas comprises the series expansion of iterated stochastic integrals on an arbitrary complete orthonormal system (CONS) in $L_2([t,T])$. For a Volterra-type kernel,

$$\Phi(t_1,\ldots, t_k) = \begin{cases} \psi_1(t_1)\cdots\psi_k(t_k), & t_1 < \cdots < t_k;\ 0, & \text{otherwise}, \end{cases}$$

the multiple Stratonovich integral admits the mean-square convergent representation: $$J^S[\Phi]_{T,t}^{(i_1\ldots i_k)} = \lim_{p \to \infty} \sum_{j_1, \ldots, j_k=0}^p C_{j_k \ldots j_1} \prod_{l=1}^k \zeta_{j_l}^{(i_l)},$$ where each $C_{j_k \ldots j_1}$ is a $k$-dimensional Fourier coefficient and $$\zeta_j^{(i)} = \int_t^T \phi_j(\tau) d w_\tau^{(i)}$$ is the stochastic projection onto the $j$-th basis function. The decomposition and identification of corrections relies upon explicit construction and verification of limiting traces $T_{g_1, ..., g_{2r}}^{(k-2r)} \Phi$, capturing symmetrisations imposed by Stratonovich conversions.

These expansions enable the systematic organization of pathwise corrections, supporting both the analysis and numerical approximation of non-commutative Itô SDEs with high-order strong convergence.

3. Sufficient Conditions and Convergence Properties

Validity of the Hu-Meyer formulas in this generalized setting depends on mean-square convergence of the series expansion. This is guaranteed if, for each paired index partition,

$$\lim_{p \to \infty} \| F^{(p)}_{g_1, ..., g_{2r}} \|^2_{L_2([t, T]^{k-2r})} = 0,$$

where $F^{(p)}_{g_1, ..., g_{2r}}$ quantifies the deviation between truncated and fully symmetrized Fourier expansions. Sufficient conditions have been rigorously established for arbitrary bases for multiplicities up to six, and using the trigonometric Fourier and Legendre polynomial bases for orders seven and eight. These results ensure the practical applicability of Hu-Meyer formulas to high order iterated Stratonovich stochastic integrals.

The vanishing of remainder terms under these convergence conditions facilitates the rigorous design of high-order strong numerical schemes for complex Itô SDEs, particularly those with non-commutative stochastic noise terms.

4. Analytic Signal Decomposition and Time-Frequency Connections

In wavelet theory, an explicit connection has been drawn between the Hu-Meyer formula structure and time-frequency analytic signal decomposition. For instance, the Meyer wavelet $\psi_{\text{mey}}(t)$ admits a decomposition akin to Hu-Meyer representations,

$$\psi_{\text{mey}}(t) = \psi_c(t) \cos(\omega_0 t) + \psi_s(t) \sin(\omega_0 t),$$

in which $\psi_c$, $\psi_s$ are in-phase and quadrature components respectively, and $\omega_0$ is an appropriate carrier modulation frequency. This decomposition is equivalent to

$$\psi_{\text{mey}}(t) = \psi_{\text{mey}}(t) \cos(\omega_0 t) + \mathcal{H}\{\psi_{\text{mey}}\}(t) \sin(\omega_0 t),$$

with $\mathcal{H}$ denoting the Hilbert transform, obtaining a full analytic signal representation. In practice, this structure supports efficient synchronous detection and demodulation in communication systems and multiresolution analysis.

The analytic decomposition is a manifestation of Hu-Meyer style formulas in signal analysis, with direct implementation consequences for algorithm design in real-time fault detection and OFDM systems.

5. Arithmetic and Modular Extensions: Dedekind Sums and Poisson Summation

Extensions of Hu-Meyer formulas into number theory illuminate their applicability well beyond stochastic calculus. The higher Hickerson formula for generalized Dedekind sums $s_{i,j}(p,q)$, defined via Todd power series of lattice cones,

$$s_{i,j}(p,q) = s_{i,j}^{(I)}(p,q) + s_{i,j}^{(R)}(p,q),$$

introduces a natural decomposition into integral and rational parts, mirroring Hu-Meyer algebraic separation strategies. This decomposition is fundamental to expressing partial zeta values in Siegel's formula and generalizing Meyer’s formula to higher orders. The constructed formulas yield, under cancellation of the rational part in specific contexts, pure integral representations for special zeta values—a direct generalization of the classical Meyer result.

In harmonic analysis and the theory of modular forms, summation identities reminiscent of Hu-Meyer formulas are recognized in the explicit construction of Fourier eigenmeasures and weighted Poisson-type summation formulas. The central result is an equivariant isomorphism between $k$-spherical measures and modular-type Fourier series,

$$\text{Meas}_k \leftrightarrow FS_k,\quad \theta_{\mu}(-1/\tau) = \sqrt{-i\tau}^{\, k}\,\theta_{\mu}(\tau),$$

producing eigenmeasure identities and connecting the sums to concrete modular transformation properties (Alfes et al., 24 May 2024).

6. Computational and Practical Implications

The established representations of Hu-Meyer formulas serve as a foundation for the numerical solution of complex Itô SDEs, especially via Taylor-Stratonovich expansions in non-commutative systems. Series expansions enable conversion between Itô and Stratonovich interpretations, quantification of remainder terms, and facilitate high-order strong approximation schemes with well-characterized convergence. The requirement for arbitrarily chosen orthonormal bases broadens the scope of computational methods, accommodating a range of numerical environments including those utilizing trigonometric and Legendre polynomial bases.

In signal processing, the explicit time-domain representations deriving from Hu-Meyer style decompositions (see Meyer wavelet expressions) eliminate the need for costly numerical integration and enable more efficient designs in communications, image processing, and mathematical physics.

7. Historical Extensions and Terminological Notes

The term “Hu-Meyer formula” is variably used, with precise scope depending on disciplinary context. Originally tied to stochastic integral identities, its usage has broadened to encompass algebraic decompositions in number theory (Dedekind sums), series expansions in harmonic analysis (Poisson-type summation), and analytic signal structures in time-frequency analysis. Recent research has extended the definitional reach, generalizing both algebraic and analytic decompositions and connecting them to modular symmetries, eigenmeasure characterizations, and advanced numerical approximation of stochastic systems.

The unifying algebraic feature remains the explicit decomposition of complex integrals/sums into combinatorially structured terms—whether indexed by paired partitions (as in stochastic calculus), continued fraction expansions (arithmetic sums), or modular form coefficients (harmonic analysis).

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The Hu-Meyer formulas therefore delineate a broad spectrum of decompositional identities, each facilitating precise analytic, algebraic, and numeric treatments of high-complexity phenomena in stochastic processes, arithmetic geometry, harmonic analysis, and signal processing, with implementations validated across diverse bases, domains, and convergence regimes (Kuznetsov, 8 Oct 2025, Lee et al., 2016, Vermehren et al., 2015, Alfes et al., 24 May 2024).