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.CPAug 19, 2026

Deep-MKV-TS: Path-Dependent McKean--Vlasov Control for Financial Time Series Generation

Authors: Samer El Boustany, Théo Basseras, Samy Mekkaoui, Alexandre Alouadi, Yadh Hafsi, Huyên Pham

Organizations: Murex SAS · École Polytechnique · BNP-PAR and Ecole Polytechnique

Abstract

We introduce Deep-MKV-TS, a path-dependent McKean-Vlasov framework for financial scenario generation. The stochastic dynamics are chosen by matching selected path and volatility features of generated scenarios to those observed in the data. Starting from an interpretable reference model, Deep-MKV-TS preserves the reference drift and adjusts its volatility, while a regularization penalty limits unnecessary departures from the calibrated dynamics. We solve the resulting control problem using a neural, sample-based implementation of the stochastic maximum principle. We validate the method against an exactly computable oracle. On Heston and Heston-mixture models, Deep-MKV-TS substantially reduces path-dependent and volatility-related deficiencies of the reference model. In delayed-volatility experiments, the correction remains effective as the forecasting horizon increases, while direct training becomes less reliable. On held-out intraday equity-index futures, the corrected model improves conditional forecasts relative to the reference and reaches a level of performance comparable to flexible generative and historical baselines. The resulting scenarios also support greater exposure than the reference under a fixed drawdown-risk target. These results show that path-dependent McKean-Vlasov control can enrich an interpretable reference model without replacing it.

Figures & tables

Method	SWD	RV $W_{1}$	$\|r\|$ ACF	Early–future	MDD $W_{1}$
Reference	0.060	0.089	0.068	0.214	0.059
Deep-MKV-TS	0.062	0.014	0.016	0.018	0.022
SBTS	0.089	0.005	0.004	0.012	0.039
LS4	0.039	0.026	0.023	0.037	0.021
CSDI	0.104	0.204	0.011	0.080	0.118

Table 1. Heston: medians over four seeds, lower is better.

Figure 1. Comparison of volatility statistics for the target data, the Heston reference model, Deep MKV-TS, and SBTS. Left: Empirical cumulative distribution of annualized realized volatility. Right: ACF of absolute returns across different lags, measuring volatility persistence. Comparison of the distribution of realized volatility and the persistence of absolute returns for the target data, the Heston reference model, Deep MKV, and SBTS.

Method	SWD	RV $W_{1}$	$\|r\|$ ACF	Early–future	Regime TVD	MDD $W_{1}$
Reference	0.347	0.479	0.322	0.765	0.629	0.309
Deep-MKV-TS	0.237	0.095	0.046	0.046	0.331	0.009
SBTS	0.407	0.493	0.094	0.05	0.563	0.478

Table 2. Mixture medians over four seeds, lower is better.

Figure 2. Quantile comparison of annualized realized volatility between the target data and the generated paths. The dotted line represents the ideal quantile matching, while the curves correspond to the Heston reference model, Deep MKV, and SBTS. The reported $W_{1}$ values quantify the discrepancy between the generated and target realized-volatility distributions. Quantile comparison of annualized realized volatility for the target data, the Heston reference model, Deep MKV, and SBTS.

Figure 3. Delayed-volatility experiment. Left: Difference in rolling volatility between paths that cross and do not cross the early drawdown threshold, showing the delayed volatility response in the target and generated scenarios. Right: Deep-MKV-TS volatility relative to the reference, conditioned on the same two path groups. The learned correction responds differently after the early-path event, making the timing and magnitude of the path-dependent intervention directly traceable. Comparison of the delayed-volatility response and recovered temporal dependence for the target, reference model, Deep MKV, and SBTS.

$N$	Deep dist.	Deep dep.	Direct dist.	Direct dep.
128	31.6%	4.8%	15.7%	$-6.7$ %
256	39.7%	6.9%	14.1%	$-3.4$ %
512	62.2%	3.0%	52.6%	$-4.5$ %
Method	RV $W_{1}$	$\|r\|$ ACF	Lagged-response	Past–future error
Reference	0.208	0.032	0.194	0.672
SBTS	0.088	0.023	0.205	0.818

Table 3. Lagged volatility: horizon scaling (top) and detailed errors at

N=512

(bottom). All configurations complete

4/4

seeds except direct training at

N=512

(

2/4

). Statistics are computed over completed runs.

Index	Method	Cum. return	Increment	RV
ES	Reference	$1.045\pm 0.004$	$0.276\pm 0.0002$	$0.897\pm 0.005$
	Deep MKV	$1.003\pm 0.006$	$0.267\pm 0.0003$	$0.634\pm 0.013$
	SBTS	$1.040\pm 0.010$	$0.279\pm 0.0003$	$0.623\pm 0.003$
	Block bootstrap	$1.042$	$0.272$	$0.771$
	Session bootstrap	$1.066$	$0.282$	$0.620$
NQ	Reference	$1.383\pm 0.011$	$0.361\pm 0.0005$	$1.092\pm 0.002$

Table 4. Conditional-forecast CRPS (

\times 1000

; lower is better) on the 123-session chronological held-out test. The primary comparison is between Deep MKV and its matched fitted-drift reference. The remaining methods provide external benchmark levels. Deep MKV, its reference, and SBTS are reported as four-seed mean

\pm

sample standard deviation.

Method	Average exposure	Violations
Deep-MKV-TS	$\mathbf{1.99\pm 0.06}$	$8.13\%\pm 0.66\%$
Fitted reference	$1.85\pm 0.09$	$7.11\%\pm 1.02\%$
SBTS	1.87	6.50%
Block bootstrap	1.84	8.13%
Session bootstrap	1.95	8.13%

Table 5. Frozen ES drawdown-risk decision: mean

\pm

std over four seeds where applicable. T he prescribed violation limit is 10%.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

$N$	128	128	128/256/512	128
	Heston	Mixture	Delayed vol.	Futures
$\Delta t$	0.004	0.004	0.004/0.002/0.001	one session
$(w_{\text{path}},w_{\text{vol}})$	(50,100)	(25,50)	(25,50)	(25,50)
$K$	2,500	3,000	3,000/2,500/2,500	400
Learning rate	0.002	0.0015	0.002	0.00025
Drift rule	fitted	zero	zero	zero

Appendix

Table 6. Locked Deep-MKV-TS settings.

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