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Orthogonal polynomials on path-space

Ilya Chevyrev, Emilio Ferrucci, Darrick Lee, Terry Lyons, Harald Oberhauser, Nikolas Tapia

TL;DR

This work develops an analogue of classical orthogonal polynomials on path-space by combining signature expansions with an orthogonalisation procedure under an infinite radius of convergence, enabling $L^2$-convergent representations of square-integrable functionals on rough paths. It recasts the shuffle algebra as a graded commutative polynomial algebra over the free Lie algebra, establishing a Favard-type theory and a recurrence framework with multi-term structure, and analyzes when inner products arise from measures on the dual and on path space. The Brownian case is studied in depth, showing that time augmentation yields a natural, dimension-free Itô orthogonal polynomial system, while the non-time-augmented case does not admit such a natural basis; numerical experiments demonstrate practical use for path-functional approximation and regression on Wiener measure. The results offer a cohesive framework for spectral, probabilistic, and computational aspects of orthogonal functionals on path-space with potential applications to numerical schemes and data-driven path analysis.

Abstract

We consider the orthogonalisation of the signature of a stochastic process as the analogue of orthogonal polynomials on path-space. Under an infinite radius of convergence assumption, we prove density of linear functions on the signature in $L^p$ functions on grouplike elements, making it possible to represent a square-integrable function on (rough) paths as an $L^2$-convergent series. By viewing the shuffle algebra as commutative polynomials on the free Lie algebra, we revisit much of the theory of classical orthogonal polynomials in several variables, such as the recurrence relation and Favard's theorem. Finally, we restrict our attention to the case of Brownian motion with and without drift, and prove that dimension-independent orthogonal signature exists with drift but not without. We end with numerical examples of how orthogonal signature polynomials of Brownian motion can be applied for the approximation of functions on paths sampled from the Wiener measure.

Orthogonal polynomials on path-space

TL;DR

This work develops an analogue of classical orthogonal polynomials on path-space by combining signature expansions with an orthogonalisation procedure under an infinite radius of convergence, enabling -convergent representations of square-integrable functionals on rough paths. It recasts the shuffle algebra as a graded commutative polynomial algebra over the free Lie algebra, establishing a Favard-type theory and a recurrence framework with multi-term structure, and analyzes when inner products arise from measures on the dual and on path space. The Brownian case is studied in depth, showing that time augmentation yields a natural, dimension-free Itô orthogonal polynomial system, while the non-time-augmented case does not admit such a natural basis; numerical experiments demonstrate practical use for path-functional approximation and regression on Wiener measure. The results offer a cohesive framework for spectral, probabilistic, and computational aspects of orthogonal functionals on path-space with potential applications to numerical schemes and data-driven path analysis.

Abstract

We consider the orthogonalisation of the signature of a stochastic process as the analogue of orthogonal polynomials on path-space. Under an infinite radius of convergence assumption, we prove density of linear functions on the signature in functions on grouplike elements, making it possible to represent a square-integrable function on (rough) paths as an -convergent series. By viewing the shuffle algebra as commutative polynomials on the free Lie algebra, we revisit much of the theory of classical orthogonal polynomials in several variables, such as the recurrence relation and Favard's theorem. Finally, we restrict our attention to the case of Brownian motion with and without drift, and prove that dimension-independent orthogonal signature exists with drift but not without. We end with numerical examples of how orthogonal signature polynomials of Brownian motion can be applied for the approximation of functions on paths sampled from the Wiener measure.
Paper Structure (21 sections, 28 theorems, 143 equations, 4 figures)

This paper contains 21 sections, 28 theorems, 143 equations, 4 figures.

Table of Contents

  1. Introduction
  2. L² orthogonal signature expansions
  3. Signature of a stochastic process
  4. Orthogonalisation of signature features
  5. Density of linear functions on the signature in Lᵖ
  6. General properties of orthogonal shuffle polynomials
  7. Shuffle polynomials as graded commutative polynomials
  8. Recurrence relation
  9. Rank condition
  10. Favard's theorem
  11. Jacobi matrices and measures on Ŵ
  12. Background on spectral theory
  13. Commutativity and cyclic vectors
  14. Bounded case
  15. Unbounded case
  16. ...and 6 more sections

Key Result

corollary 1

Let $Y = F(X) \in L^2(\mathcal{G})$ and $\{(X^i, Y^i = F(X^i) + \varepsilon^i\}_{i = 1}^M$ be an i.i.d. sample of input-output pairs, where $\varepsilon^i$ are i.i.d. errors independent of $X$. Let $\Phi \in \mathbb R^{M \times D}$ be the data matrix with with $n \leq N$ and $D = D(d, N) = \frac{d^{N+1} - 1}{d - 1}$ the dimension of $T_N(\mathbb R^d)$. Then the estimator for ordinary least square

Figures (4)

  • Figure 1: Taylor and $L^2$ approximations (computed with Legendre orthogonal polynomials) of $f(x) = \frac{1}{1 + x^2}$. Since $f$ has poles at $\pm i$, the Taylor approximations do not converge uniformly on $(-1,1)$.
  • Figure 2: Comparison of correlation heatmaps for flattened signatures, on the subspace making $( \, \cdot \,, \, \cdot \,)_{\mathbin{\widehat{\shuffle}}}$ is non-degenerate \ref{['eq:nullspace']}, computed over $100$k $2$-dimensional time-augmented sample Brownian paths, with $T = 1$ and $1$k grid points. The Stratonovich signature features are far from orthogonal, the Itô ones are much sparser but still not orthogonal (because of residual correlations inside each Wiener chaos), and finally their Gram-Schmidt orthogonalisation is verified to be fully orthogonal (modulo numerical errors). This and similar checks can be performed with the notebook orthsig.
  • Figure 3: Comparison of Taylor and Orth models: out-of-sample L$^2$ error (left) and coefficient of determination (R$^2$, right) across different dataset sizes. The Brownian motion is taken to have dimension $d = 2$, $Y$ is scalar, and errors/R$^2$ are averaged over $10$ random choices of the matrix $A$ normalised to have Euclidean norm $1$.
  • Figure 4: Comparison of coefficients of determination for OLS regression (Regr) on the truncated signature (with non-orthogonal coordinates, out-of-sample) and orthogonal signature expansion (Orth). The scalar Black Scholes model has parameters $\sigma = 0.2$, $\mu = 0$, $S_0 = 1$, and the call option is struck at $K = 1$ at time $T = 1$. We observed worse performance for both models for OTM options, and for ITM options Orth was performing worse than Regr.

Theorems & Definitions (62)

  • remark 2.1: $\mathcal{G} = \mathcal{F}$
  • corollary 1: Learning $F(X)$ by linear regression
  • corollary 2: Series expansion of an $L^2$ function on paths
  • proof
  • theorem 1
  • lemma 1
  • proof
  • lemma 2
  • proof
  • proof : Proof of \ref{['thm:density']}
  • ...and 52 more