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Signature SDEs from an affine and polynomial perspective
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Signature stochastic differential equations (SDEs) constitute a large class of stochastic processes, here driven by Brownian motions, whose characteristics are linear maps of their own signature, i.e. of iterated integrals of the process with itself, and therefore allow for a generic path dependence. We show that their prolongation with the corresponding signature is an affine and polynomial process taking values in the set of group-like elements of the extended tensor algebra. By relying on duality theory for affine or polynomial processes, we obtain explicit formulas in terms of converging power series for the Fourier-Laplace transform and the expected value of entire functions of the signature process' marginals. The coefficients of these power series are solutions of Riccati and linear ordinary differential equations (ODEs) with values in the extended tensor algebra, respectively, whose vector fields can be expressed in terms of the characteristics of the corresponding SDEs. We thus construct a class of stochastic processes that is universal (in a sense specified in the introduction) within It\^o-diffusions with path-dependent characteristics and allows for an explicit characterization of the Fourier-Laplace transform and hence the full law on path space. The practical applicability of this affine and polynomial approach is illustrated by several numerical examples.
Forward citations
Cited by 8 Pith papers
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Exponentially Fading Memory Signature
Defines the EFM-signature as a stationarized, mean-reverting analogue of the path signature with algebraic properties, Markovian evolution as a group-valued OU process, and explicit formulas for expectations.
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A General Theory of Paths: Signatures, Jump Lifts, and Expected Signatures of Self-Exciting Processes
Develops path signature theory with Geometricity-Defect Theorem and Hopf Square, then shows finite-dimensional linear closures for truncated expected signatures of affine/exponential Hawkes processes after state augmentation.
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The Singular Values of L\'evy's Area Matrix
Explicit density for singular values of Lévy's area matrix, determinantal point process characterization, and d to infinity asymptotics including absolute Cauchy limit.
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On the Structural Foundations of Signature Volatility Models: Existence, Arbitrage, Completeness, and the Hedging-Error Decomposition
Establishes existence, uniqueness, NFLVR, completeness via signature span density, and hedging-error decomposition for signature SDEs under summability and exponential-integrability conditions.
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A Call to Lagrangian Action: Learning Population Mechanics from Temporal Snapshots
Wasserstein Lagrangian Mechanics learns second-order population dynamics from observed marginals without specifying the Lagrangian and outperforms gradient flow methods on periodic dynamics like vortex motion and flocking.
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A Call to Lagrangian Action: Learning Population Mechanics from Temporal Snapshots
Wasserstein Lagrangian Mechanics formalizes second-order dynamics in Wasserstein space and provides an algorithm to learn them from observed marginals without specifying the Lagrangian, outperforming gradient flows on...
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A Call to Lagrangian Action: Learning Population Mechanics from Temporal Snapshots
Wasserstein Lagrangian Mechanics learns second-order population dynamics from observed marginal snapshots without specifying the Lagrangian and outperforms gradient flow methods on tasks like vortex dynamics and embry...
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Signature McKean-Vlasov stochastic differential equations
Introduces signature McKean-Vlasov SDEs driven by expected rough path signatures, proves strong well-posedness, approximation of path-dependent equations, and propagation of chaos.
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