Deep-MKV-TS: Path-Dependent McKean--Vlasov Control for Financial Time Series Generation
Samer El Boustany, Théo Basseras, Samy Mekkaoui, Alexandre Alouadi, Yadh Hafsi, Huyên Pham
q-fin.CP, cs.CE, cs.LG, math.OC
2026-08-20
Deep-MKV-TS keeps the reference drift and corrects volatility with path-dependent McKean-Vlasov control. On Heston, RV W1 falls 0.089 to 0.014; held-out futures CRPS also improves.
Financial scenario generators are judged on paths, not one-step returns. Drawdown length, volatility clustering, and the coupling from early losses to later uncertainty all change risk and execution decisions. GANs, neural SDEs, and diffusion can learn a full path law, but they rebuild a high-dimensional distribution from finite samples and can overwrite dynamics that were already calibrated. After the fact it is hard to say why volatility moved at a given step.
A complementary move: keep an interpretable reference process and repair the path and volatility features it misses.
The reference is a path-dependent volatility model with short- and long-memory trend and activity signals, calibrated by Gaussian negative log-likelihood and then frozen. The controlled process keeps that drift. A controller supplies the volatility matrix at each step. The cost has two distributional terms, MMD on paths and MMD on volatility features (absolute returns, realized variance, rolling vol, lag correlations), plus a running specific-entropy penalty that charges deviations from reference volatility.
Pontryagin's stochastic maximum principle turns the control into a forward-backward system. A GRU with a 96-dimensional hidden state approximates the conditional expectation in the backward signal, summarizing the path so far. The loop samples forward, backprops a pathwise adjoint, fits a ridge proxy for the conditional expectation, and trains the GRU to that proxy. At each time one can read how far controlled volatility moved from the reference and which history triggered it.
Direct training parametrizes the same network and minimizes the same cost without the SMP loop. SBTS, LS4, and CSDI mark the quality band of flexible generators. They are context, not the claim of replacement.
On an enumerable GARCH tree (N=6, 729 paths), the sampled procedure recovers 96.4% of the exact solver's objective gain.
Heston, variance hidden from the model, four-seed medians, lower is better:
| Method | RV W1 | r | ACF | Early-future | MDD W1 | |
| Reference | 0.089 | 0.068 | 0.214 | 0.059 | ||
| Deep-MKV-TS | 0.014 | 0.016 | 0.018 | 0.022 | ||
| SBTS | 0.005 | 0.004 | 0.012 | 0.039 |
On an eight-regime Heston mixture, reference RV W1 0.479 and regime TVD 0.629 fall to 0.095 and 0.331; MDD W1 goes 0.309 to 0.009. Delayed-volatility tasks stretch memory across 128/256/512 steps. The correction removes 31.6%, 39.7%, and 62.2% of the reference's distributional error. At N=512, direct training finishes 2 of 4 seeds and temporal dependence gets worse than the reference. Deep-MKV-TS finishes all four.
Live data: ES, NQ, and YM index futures, 496/144/123 sessions in time order. Condition on the first 65 log-prices, retrieve 256 nearest of 8,192 generated histories, predict the next 32 steps. On ES, CRPS on cumulative return falls 1.045 to 1.003, increments 0.276 to 0.267, realized vol 0.897 to 0.634, with the same three-way lift on NQ and YM. Under a 1% loss budget and a 10% violation target, mean exposure rises 1.85 to 1.99 at an 8.13% violation rate.
Desks often already own a calibrated reference. What they lack is path-dependent volatility detail. This method corrects rather than retraining a black box, and the correction is traceable along the path. On delayed volatility, direct training is brittle while the structured signal keeps working, which is evidence that the backward process carries long memory more reliably than smashing the same cost end to end.
It does not beat SBTS or LS4 on every unconditional distance. The claim is to lift the reference into that neighborhood while keeping the skeleton.
Main experiments are one-dimensional, with the observation grid equal to the simulation grid. Frozen drift cannot repair a biased mean. The MMD kernel and volatility features are design choices; change them and the correction target changes. Conditional evaluation on futures uses path-shadowing neighbors, which is itself an inductive bias. The downstream test is a drawdown constraint on ES, not trading P&L. The paper has no standalone limitations section. The discussion treats flexible generators as a waterline and does not claim universal dominance.