Rough Path Signatures: Learning Neural RDEs for Portfolio Optimization

TL;DR

The paper tackles high-dimensional, path-dependent valuation and control by proposing a deep BSDE/2BSDE solver that fuses truncated log-signatures with a Neural RDE backbone to approximate $(Y_t,Z_t,\boldsymbol{\Gamma}_t)$. It introduces a CVaR-tilted terminal objective and an optional 2BSDE head to capture second-order effects, enabling risk-sensitive pricing and nonlinear control. Across Asian options, barrier features, and stochastic-volatility portfolio control up to dimension $d=200$, the approach achieves superior tail calibration (e.g., CVaR$_{0.99}$ improvements) and lower HJB residuals compared to strong baselines, while maintaining training stability and reasonable compute cost. The work demonstrates a productive two-way dialogue between stochastic analysis and deep learning, where path-aware representations and continuous-time dynamics expand the class of solvable PPDE/BSDE problems at scale, and suggests concrete extensions for adaptive signatures, mean-field settings, and distributionally robust drivers.

Abstract

We tackle high-dimensional, path-dependent valuation and control and introduce a deep BSDE/2BSDE solver that couples truncated log-signatures with a neural rough differential equation (RDE) backbone. The architecture aligns stochastic analysis with sequence-to-path learning: a CVaR-tilted terminal objective targets left-tail risk, while an optional second-order (2BSDE) head supplies curvature estimates for risk-sensitive control. Under matched compute and parameter budgets, the method improves accuracy, tail fidelity, and training stability across Asian and barrier option pricing and portfolio control: at d=200 it achieves CVaR(0.99)=9.80% versus 12.00-13.10% for strong baselines, attains the lowest HJB residual (0.011), and yields the lowest RMSEs for Z and Gamma. Ablations over truncation depth, local windows, and tilt parameters confirm complementary gains from the sequence-to-path representation and the 2BSDE head. Taken together, the results highlight a bidirectional dialogue between stochastic analysis and modern deep learning: stochastic tools inform representations and objectives, while sequence-to-path models expand the class of solvable financial models at scale.

Figures (4)

• Figure 1: Scaling with dimension $d$ (T1). RPE vs. $d$ at fixed param count; time/epoch inset.
• Figure 2: Tail calibration under CVaR tilt: $\mathrm{CVaR}_q$ vs. $q\in[0.90,0.99]$ (T1, $d{=}100$).
• Figure 3: Ablations on T1 ($d{=}100$): signature depth $m$, RDE width $p$, window $K$, and Malliavin.
• Figure 4: Performance–diagnostics diagram (T1/T3): RPE vs. time/epoch with NaN contours (log-scale).