Noise estimation of SDE from a single data trajectory

Battle Count: 81

Highly relevant for quantitative trading in several ways: (1) The Black-Scholes SDE is explicitly addressed as a non-ergodic, non-stationary model, which is the standard stock price model; (2) The method recovers the underlying physical Brownian motion from a single price path, enabling noise-aware modeling; (3) Model discovery from a single trajectory is directly applicable to calibrating stochastic models for derivatives pricing and risk management; (4) The Girsanov transformation framework connects to risk-neutral pricing; (5) However, the method is primarily about model identification rather than direct trading signal generation, and requires high-frequency data which may not always be available.

High complexity due to: (1) Computing quadratic variation on fine grids; (2) Implementing Girsanov transformations and constructing risk-neutral Brownian motion; (3) Computing iterated Stratonovich integrals (signatures) up to order 2-3; (4) Solving the ODE for drift recovery via Euler scheme; (5) Implementing the SSISDE sparse regression with elastic-net penalty and k-fold time-series cross-validation; (6) De-biasing via restricted least squares; (7) The 7-step pipeline requires careful numerical implementation. The signature/iterated integral computations are particularly non-trivial and may require specialized libraries.

The paper provides detailed algorithmic steps (7 steps), explicit formulas for Taylor expansions, Girsanov transformations, and the SSISDE optimization problem. Numerical experiments include specific parameter settings (N=100000, M=1000, k=7 folds, specific α and ρ grids, polynomial libraries). However, no code repository is mentioned, and the implementation of iterated Stratonovich integrals and signature computations is non-trivial. The Euler scheme for ODE solving and the sequential thresholding least-squares algorithm are referenced but not fully specified in code.