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Lévy Area and Iterated Stochastic Integrals

StochasticDiffEqLevyArea provides Fourier-series approximations of Lévy areas and iterated stochastic integrals. It supports direct random sampling, pre-generated coefficients for deterministic reuse, and Brownian-path reconstruction within a time step.

Algorithms

StochasticDiffEqLevyArea.Fourier — Type
Fourier()

Basic Fourier-series approximation of Lévy areas. Its truncation error has convergence order $1/2$ in the number of retained Fourier terms.

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StochasticDiffEqLevyArea.Milstein — Type
Milstein()

Fourier-series Lévy-area approximation with Milstein's Gaussian tail approximation. Its truncation error has convergence order $1/2$ in the number of retained Fourier terms.

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StochasticDiffEqLevyArea.Wiktorsson — Type
Wiktorsson()

Wiktorsson's Fourier-series Lévy-area approximation with a Gaussian tail-sum correction. Its truncation error has convergence order $1$ in the number of retained Fourier terms.

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StochasticDiffEqLevyArea.MronRoe — Type
MronRoe()

Mrongowius–Rößler Fourier-series Lévy-area approximation with an improved tail-sum correction. Its truncation error has convergence order $1$ in the number of retained Fourier terms.

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Error norms

StochasticDiffEqLevyArea.MaxL2 — Type
MaxL2

Error-norm type that takes the maximum of the entrywise $L^2$ norms of an iterated-integral approximation. Construct an instance with MaxL2().

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StochasticDiffEqLevyArea.FrobeniusL2 — Type
FrobeniusL2

Error-norm type that takes the Frobenius norm of the entrywise $L^2$ norms of an iterated-integral approximation. Construct an instance with FrobeniusL2().

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Coefficients and iterated integrals

StochasticDiffEqLevyArea.LevyAreaCoefficients — Type
LevyAreaCoefficients{T}

Stores the pre-generated random numbers (Fourier coefficients) needed by a Lévy area algorithm. Separating coefficient generation from area computation enables:

  • Deterministic Lévy area given the same coefficients
  • Path reconstruction from the Fourier expansion
  • Consistent iterated integrals across step-size changes
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StochasticDiffEqLevyArea.generate_coefficients — Function
generate_coefficients(m, n, alg, rng = default_rng(); T = Float64)

Generate Fourier coefficients for Lévy area computation. The layout of X and Y depends on the algorithm:

  • Fourier, Milstein, Wiktorsson: X is n×m, Y is m×n
  • MronRoe: X is m×n, Y is n×m
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StochasticDiffEqLevyArea.levyarea — Function
levyarea(W, n, alg; rng = default_rng())
levyarea(W, n, alg, coeffs)

Approximate the antisymmetric Lévy-area matrix for the normalized Wiener increment W using n Fourier terms and algorithm alg.

The keyword form draws the required random coefficients from rng. Passing pre-generated LevyAreaCoefficients makes the computation deterministic. Use iterated_integrals to obtain the complete matrix of iterated integrals over a time step of length h.

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StochasticDiffEqLevyArea.iterated_integrals — Function
iterated_integrals(W::AbstractVector, h, eps=h^(3/2); ...)

Simulate iterated stochastic integrals ∫₀ʰ∫₀ˢdWᵢ(t)dWⱼ(s) for all pairs i,j.

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iterated_integrals(W::AbstractVector, h, coeffs::LevyAreaCoefficients; ...)

Compute iterated integrals using pre-generated Fourier coefficients (deterministic).

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iterated_integrals(W::AbstractVector, q_12, h, eps; ...)

Q-Wiener process version.

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StochasticDiffEqLevyArea.reconstruct_path — Function
reconstruct_path(dW, h, coeffs, t_points)

Reconstruct the Brownian motion at specified time points within [0, h] using the Karhunen-Loève (Fourier) expansion:

W_j(t) = (t/h)*dW_j + Σ_{k=1}^n a_{j,k} * √(2h)/(kπ) * sin(kπt/h)

where a_{j,k} are the Fourier coefficients stored in coeffs.

Returns a vector of m-dimensional vectors, one per time point.

Note: The coefficient layout differs by algorithm. For Fourier/Milstein/Wiktorsson, coeffs.X[k,j] = a{j,k}. For MronRoe, coeffs.X[j,k] = a{j,k}.

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StochasticDiffEqLevyArea.iterated_integrals_subinterval — Function
iterated_integrals_subinterval(
    dW, h, coeffs, t_start, t_end;
    n_quadrature = 64, ito_correction = true
)

Compute iterated Stratonovich integrals over the sub-interval [tstart, tend] ⊂ [0, h] by reconstructing the Brownian path from Fourier coefficients and computing via Riemann sums.

Returns an m×m matrix J where:

  • With itocorrection=true: J{jk} = Ito iterated integral ∫∫ dWj dWk
  • With itocorrection=false: J{jk} = Stratonovich iterated integral

The accuracy improves with n_quadrature (number of evaluation points within the sub-interval) and with the truncation level n in the coefficients.

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