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Structured Neural SDEs for Functional Calibration

Francesco Piatti, Andrea Iannucci, Thomas Cass

arXiv:2609.34831v1cs.LGmath.PR

TL;DR

Functional calibration fits expectations of path functionals when trajectories are not observed. SLiSDE uses structured stochastic layers with gated in-flow coupling; across the reported benchmarks, it achieves the lowest or a statistically tied calibration loss on every benchmark.

  • Problem

    Functional calibration must fit prescribed expectations of path functionals even though the trajectories themselves are unobserved.

    Source: §1 · p. 1, §2.2 · p. 4

  • Method

    SLiSDE combines structured affine stochastic layers evaluated by parallel scans with gated in-flow coupling between layers.

    Source: §1 · p. 2, Contributions · p. 2

  • Results

    SLiSDE attains the lowest or a statistically tied calibration loss on every benchmark, trains faster than SLiCE in every tested setting, and is faster than Neural SDEs in all but two.

    Source: Contributions · p. 3

  • Takeaways & Limitations

    The associative scan reduces sequential dependence to O(log T) parallel depth, compared with O(T) for standard Neural SDE solvers.

    Source: Contributions · p. 2, §2.1 · p. 3

    Finitely many expectation constraints do not determine the path law, so the population calibration objective has no unique minimiser.

    Source: §2.2 · p. 4

Abstract

from arXiv · show

Neural Stochastic Differential Equations (Neural SDEs) provide flexible continuous-time generative models, but generic neural drift and diffusion networks are costly to simulate on long horizons and can give unstable gradients when the training signal is a path functional rather than a pointwise observation. We introduce SLiSDE, a family of Neural SDE models built from structured linear stochastic layers. Parallel-in-time simulation is obtained at the layer level, while expressivity is recovered by gated in-flow stacking: previous-layer paths modulate the next layer's latent flow through learned gates. For functional calibration tasks in which rare paths dominate the loss, we add an optional Girsanov tilt that acts as a learned importance sampler with an exact likelihood-ratio correction. We prove well-posedness, a discretisation error bound, validity of the change of measure, and a universality result: the terminal laws of the gated stack are dense in the space of square-integrable laws. Experiments on functional calibration benchmarks show that the structured model outperforms fully neural SDE baselines while retaining parallel-time simulation and stable importance weights.

1 Introduction

SLiSDE addresses functional calibration, where models must fit expectations of path functionals without observed trajectories, by combining scan-compatible structured stochastic layers with nonlinear coupling between layers. Its contributions include a last-layer Girsanov tilt for rare-event objectives and theoretical and empirical results.

Related work.

  • Functional calibration fits prescribed expectations of path functionals when trajectories are unobserved, unlike generative training that matches a model law to an empirical path distribution.The paper situates this setting alongside earlier parametric calibration families and structured sequence models.Source: §1 · p. 1, Related work · p. 2

Contributions.

  • Structured affine layers reduce parallel depth in sequence length to O(log T), compared with O(T) for standard Neural SDE solvers.The parallel associative scan composes affine recurrences; the stated complexity concerns parallel depth.Source: Contributions · p. 2, §1 · p. 2
  • Gated in-flow stacking lets previous-layer paths modulate the next layer’s affine transition and offset while each layer remains affine in its own state.This moves nonlinearity to cross-layer coupling while retaining scan compatibility.Source: Contributions · pp. 2–3, §1 · p. 2
  • For rare-event functional calibration, an optional last-layer Girsanov tilt uses a learned adapted controller as a self-normalised importance sampler with a closed-form likelihood-ratio correction.The contribution passage reports improved rare-event calibration performance with little added compute.Source: Contributions · p. 3
  • The gated stack’s terminal laws are dense in P2(Rp) under the 2-Wasserstein distance, so structural restrictions do not limit which terminal laws can be approximated.This is the paper’s stated universality result.Source: Contributions · p. 3
  • SLiSDE attains the lowest or a statistically tied calibration loss on every benchmark, trains faster than SLiCE in every tested setting, and is faster than Neural SDEs in all but two.The benchmarks cover nonlinear path-functional calibration tasks and real option-surface benchmarks.Source: Contributions · p. 3

2 Mathematical Background

The mathematical background shows how discretised affine SDE recurrences can be evaluated by an associative scan and defines calibration as matching model expectations to prescribed functional values. Because finitely many expectation constraints do not determine a unique path law, the objective need not have a unique minimiser.

2.1 Linear SDEs and affine flows

  • A discretised affine linear SDE has an affine recurrence whose transition and offset are built from the coefficients and Brownian increments.The paper uses either matrix-exponential or Euler–Maruyama transitions, with Euler–Maruyama used in its experiments.Source: §2.1 · p. 3
  • Affine maps compose associatively, allowing all recurrence prefixes to be computed with a balanced scan of O(log T) parallel depth while exactly matching the sequential recursion.The scan reduces sequential dependence but does not reduce total arithmetic work across time steps.Source: §2.1 · pp. 3–4

2.2 Functional calibration objective

  • Functional calibration minimises discrepancies between prescribed expectations of path functionals and expectations estimated from fresh model paths, without pairing model paths to data paths.The objective is a function of the model law rather than a samplewise path error.Source: §2.2 · p. 4
  • Finitely many expectation constraints do not determine the path law, so the population calibration objective has no unique minimiser and fitted models are validated on functionals not used for fitting.Where reference dynamics are available, a relative-entropy penalty can serve as a selection rule.Source: §2.2 · p. 4

3 The SLiSDE Model

SLiSDE composes structured linear stochastic layers, coupling them through gated in-flow modulation and a learned linear readout. The layerwise affine structure supports parallel scans, while an optional final-layer tilt targets rare path-functional events.

3 The SLiSDE Model

  • A SLiSDE composes structured linear stochastic layers and maps the final latent path to observations with a learned linear readout.The Girsanov component is described separately from the core model.Source: §3 · p. 4, §3.4 · p. 6

3.1 Structured linear base layer

  • Under coefficient bounds, each structured layer has a unique strong solution with moment bounds independent of the time grid.The matrices are selected from diagonal, block-diagonal, or dense families, which remain closed under multiplication and preserve scan compatibility.Source: §3.1 · pp. 4–5

3.2 Residual stacking (sequence-model baseline)

  • Residual stacking is unsuitable because driving a layer with the previous Brownian path would add depth without injecting fresh randomness per layer.The model instead couples layers through their flows.Source: §3.2 · p. 5

3.3 Gated in-flow stacking

  • Gated in-flow stacking uses previous-layer paths to introduce nonlinearity while keeping each layer affine in its own state and therefore scan-compatible.The previous path modulates the next layer’s affine flow through learned gates.Source: §3.3 · p. 5, Appendix C.4 · p. 5

Appendix C.4.

  • The Appendix C.4 recurrence modifies diagonal transition entries by scaling and adds an offset correction, retaining affine dependence on the current state.Its dependence on the previous layer’s path supplies the nonlinearity, and the recurrence remains exactly evaluable by the parallel associative scan.Source: Appendix C.4 · p. 5

3.4 Linear decoder

  • A learned linear readout maps the final layer’s latent path to observation space, with an optional shift when the initial observation is prescribed.Source: §3.4 · p. 6

3.5 Discretisation and scan

  • Layerwise Euler–Maruyama discretisation is evaluated by parallel associative scans, with strong error of order 1/2 for every fixed depth.At each time step, the layer forms an affine pair from coefficients frozen at that grid time.Source: §3.5 · p. 6

4 Last-Layer Girsanov Tilt

  • For rare-event objectives, the optional tilt changes the final layer’s innovation under a learned adapted control and applies a closed-form likelihood-ratio correction.The tilt also shifts the initial latent state; its correction is exact and normalised, and model samples are drawn under the original measure without the training-time tilt.Source: §4 · p. 6

5 Theoretical Results

The theoretical results establish well-posedness and depth-independent moment bounds for the gated stack, validate the last-layer change of measure, and show terminal-law density under specified architectural assumptions. They also give exact parallel-scan and fixed-depth discretisation guarantees.

5 Theoretical Results

  • The parallel associative scan returns the sequential iterates exactly and evaluates a depth-L stack in O(L log T) parallel depth.These results apply to layers whether gated or not.Source: §5 · p. 7
  • For every fixed depth, the layerwise Euler–Maruyama scheme with gates evaluated on the discrete previous layer has strong order 1/2.Source: §5 · p. 7

Standing assumptions.

  • Under Assumptions B.1–B.3, the gated stack has a unique strong solution, with moment bounds using one constant independent of layer and stack depth.The bounds hold for any correlation structure of the layer drivers.Source: Standing assumptions · p. 7
  • With a progressively measurable last-layer controller satisfying Novikov’s condition, the change of measure defines an equivalent Q and yields an exact likelihood ratio for integrable functionals.The ratio depends only on the final-layer control, initial-state shift, and shifted Brownian innovation.Source: Standing assumptions · pp. 7–8
  • Under the stated RMS-normalisation and time-feature conditions, terminal laws are dense in P2(R^p) for any square-integrable initial condition, including one with a degenerate law.One ungated layer followed by one gated layer suffices.Source: Standing assumptions · p. 8

6 Experiments

Across toy and option-surface benchmarks, SLiSDE improves calibration relative to the fully neural SDE baseline while using scan-compatible structured dynamics. The experiments also assess gated stacking, rare-event importance sampling, and runtime across batch sizes and horizons.

6 Experiments

  • Affine composition enables scan depth O(log T) rather than O(T), but dense transitions can offset practical gains and were excluded from the main experimental analysis.The experiments compare two- and three-layer models against a fully connected Neural SDE and SLiCE; the toy task uses nonlinear long-horizon path statistics and no tilt.Source: §6 · p. 8, §6.1 · p. 9

6.1 Path-dependent toy benchmark (TOY)

  • The gated in-flow variant achieves the lowest total loss on the toy path-functional benchmark at both depths, with fewer parameters than the Neural SDE and SLiCE baselines.SLiSDE also remains faster than the Neural SDE; the benchmark compares models without the Girsanov tilt.Source: §6.1 · p. 9

6.2 Option-surface benchmark (DAX)

  • On SPX, under the same protocol, SLiSDE matches SLiCE’s two-layer held-out loss and attains the lowest held-out and far-tail loss with three layers.The three-layer model uses 6–8× fewer parameters and has 3.5–4× faster epochs; both structured models are far ahead of the Neural SDE.Source: §6.2 · pp. 9–10
  • On DAX, gated SLiSDE has the lowest held-out loss at both depths and the lowest far-tail loss with three layers, while using fewer parameters than SLiCE and about half its epoch time.The held-out evaluation masks 25% of threshold levels for interpolation; the optional tilt reduces mean far-call held-out error to 0.64× the vanilla model at fixed path budget.Source: §6.2 · pp. 9–10

Appendix E.

  • At equal parameter count, sequential SLiSDE steps are 6–7× faster than Neural SDE at B = 64 and 1.3–1.4× faster at B = 1024, but slower at B = 4096.At B = 4096, Neural SDE takes 179 ms against SLiSDE’s 242 ms.Source: Appendix E · p. 10

7 Conclusion

  • SLiSDE attains lower calibration loss than the Neural SDE baseline on every benchmark and at both depths, and trains faster in all but two tested settings.Against the structured-CDE baseline, it is clearly better on toy and has lower or statistically tied losses on the option surfaces, with 1.4–8× fewer parameters and 2–4× faster epochs.Source: §7 · pp. 10–11

A.1 Girsanov change of measure

  • Under the Girsanov change of measure, the inverse density supplies the importance weight that converts expectations under the tilted measure back to reference-measure expectations.The change-of-measure statement assumes a progressively measurable controller satisfying Novikov’s condition.Source: §A.1 · p. 14

A.2 Calibration objective: population limit, non-uniqueness and model selection

The population calibration objective matches prescribed functional expectations without rewarding collapse to a mean, but finitely many constraints leave the path law non-unique. Reference-based penalties can guide selection among calibrated models.

A.2 Calibration objective: population limit, non-uniqueness and model selection

  • The Monte Carlo objective converges almost surely to the population objective, with its variance term scaled by 1/n and vanishing as n grows.The population objective is a weighted sum of squared errors between model and target functional expectations.Source: §A.2 · p. 14
  • Collapse onto the data mean is not incentivized by the population objective, whose Monte Carlo variance contribution vanishes as 1/n.The text also states that collapse is incompatible with the population objective when the target marginal has the specified strictly convex call-price map.Source: §A.2 · pp. 14–15
  • Finitely many functional constraints do not determine the path law, so the population objective can have infinitely many calibrated diffusion models.Positive-part expectations across every strike determine a marginal law, but a finite strike grid and additional path-dependent functionals do not identify the entire path law.Source: §A.2 · p. 15
  • When a reference dynamics or other prior information is available, an additive penalty can select among calibrated models, including through a relative-entropy criterion.The Girsanov controller uses a KL(Q∥P) anchor that penalises tilts moving too far from the untilted reference model.Source: §A.2 · p. 15

B Theoretical Results: Assumptions, Auxiliary Results and Proofs

The appendix sets out assumptions and proofs for the paper’s theoretical results, including depth-uniform bounds for gated layers. The standing assumptions specify structured layers, gates, and driver arrangements under which those results are established.

B.1 Standing assumptions

  • The admissible configuration stacks a finite number of common-width layers, with each layer driven by Brownian motion adapted to a common filtration.Drivers may be shared, independent, or correlated across layers.Source: §B.1 · p. 17
  • The appendix distinguishes discrete and continuous gating conventions, relating them through diagonal drift modulation and grid-realizable transition scaling.The discrete offset gate has no explicit Δt factor, while the continuous model gates coefficients; the stated correspondence includes an exact identity and a fixed-grid realization.Source: §B.1 · p. 17
  • Gates are causal and measurable, and the implemented gates satisfy the boundedness and Lipschitz properties required by the assumptions.The gates use previous-layer paths and deterministic time features; the stated implementation disables the optional causal convolution.Source: §B.1 · pp. 17–19

B Theoretical Results: Assumptions, Auxiliary Results and Proofs

  • The appendix proves that gated layers inherit coefficient bounds with constants independent of layer depth and without requiring a moment bound on the previous layer.Its proof roadmap also includes well-posedness, exact scan evaluation, and stacked discretisation error.Source: B Theoretical Results: Assumptions,… · p. 16, §B.1 · p. 19

B.2 Well-posedness: proof of Theorem 5.1

Under the stated coefficient and gate conditions, the stacked SDE has unique strong solutions with moment bounds uniform in layer depth. The affine recurrence within each layer also admits an exact associative scan, enabling logarithmic parallel depth in time.

B.2 Well-posedness: proof of Theorem 5.1

  • Each layer has a unique strong solution, and the moment bounds are uniform in layer index and stack depth.The result holds for any correlation structure among the layer drivers.Source: §B.2 · p. 23
  • Normalising the gate input keeps the offset forcing bounded, which makes the moment-bound argument uniform in stack depth.Without normalisation, the bound could grow geometrically with depth unless an additional weight constraint holds.Source: §B.2 · p. 23

B.3 Exactness of the parallel associative scan

  • Associative composition of affine pairs returns the same sequential iterates under any scan evaluation order, with O(log T) parallel depth.The scan reduces sequential dependence, not total arithmetic work.Source: §B.3 · pp. 23–24
  • Gated in-flow stacking preserves exact scans because gates depend on the other layer, not the current state, leaving each layer affine.The layers run sequentially, giving L scans of depth O(log T) each.Source: §B.3 · p. 24

B.4 Strong discretisation error

The layerwise Euler–Maruyama scheme has strong order 1/2 for each fixed stack depth under the stated assumptions. Its error estimates allow arbitrary correlation structures among layer drivers, while the error constant is not claimed to be uniform in depth.

B.4 Strong discretisation error

  • The strong-error estimates hold for any correlation structure of the layer drivers.The proof uses no conditioning on other layers.Source: §B.4 · p. 25
  • The layerwise Euler–Maruyama scheme has strong order 1/2 for every fixed depth under the stated assumptions.The error constant may depend on depth, although it does not depend on the time step.Source: §B.4 · p. 29
  • The rate is not accompanied by a depth-uniform error constant: the constant depends on depth through growing moment orders and gate-Lipschitz factors.The authors state that order 1 would require iterated Itô integrals and Lévy areas, which the scheme does not simulate; they claim only order 1/2 without assuming commuting diffusion matrices.Source: §B.4 · p. 31

B.5 Proof of Theorem 5.2

The proof establishes a valid change of measure for the tilted process and shows that the discrete likelihood ratio is exact and normalized. Its ratio depends only on the final-layer control, initial-state shift, and shifted innovation.

B.5 Proof of Theorem 5.2

  • The initial-state shift changes the initial law to N(µθ, Id) and contributes its own likelihood factor; when µθ = 0, this contribution vanishes.The initial-state factor has expectation one and is independent of the innovation weight.Source: §B.5 · p. 32
  • The adapted control satisfies Novikov’s condition, making the stochastic exponential a strictly positive martingale and defining an equivalent probability measure under which the shifted innovation is Brownian.The prefix driver retains its law and remains independent of the shifted innovation.Source: §B.5 · p. 32
  • On the simulation grid, multiplying conditional Gaussian density ratios gives the exact likelihood ratio for the full increment sequence, and its expectation is one.The discrete weight is an exact density ratio, not an approximation of the continuous-time expression.Source: §B.5 · pp. 32–33
  • The full ratio is local to the final-layer control, initial-state shift, and shifted innovation because the prefix computation is unchanged under the two measures apart from its shifted starting point.The change-of-measure identity applies to every terminal-measurable functional integrable under the reference measure.Source: §B.5 · pp. 32–33

B.6 Proof of Theorem 5.3

The proof constructs gated-layer terminal outputs that approximate arbitrary square-integrable target laws. It does so by building a continuous scalar feature, approximating finite-support laws, and controlling discretisation error.

B.6 Proof of Theorem 5.3

  • A first-layer scalar feature is atomless, enabling a continuous distribution function and interval-based encoding of the finite-support target.The feature has continuous paths and is bounded.Source: §B.6 · pp. 33–34
  • Since finitely supported probability measures are dense in P2(Rp), the proof reduces approximation of an arbitrary target law to a finite-support target.The target law is chosen within an arbitrary tolerance η.Source: §B.6 · pp. 33–34
  • The proof approximates the finite-support encoding with sigmoid features and realizes those features in a second gated layer with zero diffusion.The second layer uses a constant coordinate and shifted sigmoid coordinates.Source: §B.6 · pp. 34–35
  • The gated construction can make the terminal output arbitrarily close to the target law, establishing density of the gated stack’s terminal laws in the square-integrable law space.The conclusion follows because the target law and approximation tolerance were arbitrary.Source: §B.6 · pp. 35–36
  • For fixed construction parameters, the discretised model converges in L2 as the grid is refined, so the density conclusion also holds for the discretised architecture when the mesh may vary.The first layer’s Euler–Maruyama iterates are exact, and the second layer’s explicit Euler scheme converges pathwise.Source: §B.6 · p. 37

C Implementation Details

The implementation combines structured linear layers with gated in-flow conditioning and an optional final-layer Girsanov tilt. It specifies discretisation, controller training, and estimator choices to keep simulation and rare-event calibration practical.

C Implementation Details

  • SLiSDE uses gated in-flow stacking with structured block-diagonal base layers; the optional Girsanov tilt is applied only to the final layer.Reported results also use RMS gate-input normalisation, optional causal convolutions and time features, and no residual stacking.Source: C Implementation Details · p. 38

C.1 Discretisation of a linear layer

  • The linear-layer Euler–Maruyama transition preserves the affine recurrence structure and is used throughout the experiments; the exponential flow is exact only when the coefficient matrices commute pairwise.Otherwise, the exponential is an approximation and the affine term remains first-order.Source: §C.1 · p. 38

C.2 Time-dependent coefficient map

  • A small deterministic decoder can make structured coefficients time-dependent while respecting each layer’s diagonal, block-diagonal, or dense matrix structure.Zero-initialised decoders start from a time-homogeneous structured SDE and learn time variation only when useful.Source: §C.2 · p. 38

C.3 Feature conditioning in gated layers

  • Gated layers condition on the previous layer at matching grid times, with normalisation controlling gate-input scale but not bounding the paths.Optional causal convolution uses only states up to the current time, preserving adaptation; deterministic time features can also be appended after normalisation.Source: §C.3 · p. 39

C.4 Gated in-flow implementation

  • The in-flow gate modulates the next layer’s structured drift through a bounded diagonal scale and affine residual offset, while the shared initial state is reused across layers.The gate begins near zero, so the stack starts close to an unmodulated structured SDE and learns cross-layer coupling gradually.Source: §C.4 · p. 39

C.5 Last-layer Girsanov tilt: design details

  • For rare-event calibration, the backbone uses self-normalised importance sampling on tilted batches, while an adapted controller learns a Girsanov tilt that shifts both the initial state and final-layer innovation.The initial-state shift matters because innovation-only controllers could not reach rare regions in the reported toy experiments.Source: §C.5 · p. 40

C.6 Girsanov controller

  • The controller is a diagonal state-space recurrence whose adapted control depends only on prior innovation increments and can itself be scanned in parallel.Its recurrence rates are constrained to (0, 1); an unconstrained rate above one can cause geometric growth and non-finite weights.Source: §C.6 · pp. 40–41

C.7 Self-normalised importance sampling and ESS

  • The SNIS estimator is biased at finite sample size but consistent, and the effective sample size tracks weight concentration; for rare-event probabilities, the paper instead reports the vanilla estimator.With an initial-state shift, non-event tilted paths can carry large weights that affect only the SNIS denominator, making it markedly worse in the reported comparison.Source: §C.7 · p. 41, Appendix C.8 · p. 41

D Experimental Setup and Hyperparameter Grids

The experiments evaluate path-functional calibration on a controlled synthetic benchmark and two historical option-surface datasets. Training and search choices use held-out strikes, multiple seeds, and structured model configurations.

toy Dataset

  • The toy benchmark separates path fitting from functional calibration using a nonlinear SDE and targets that include tail-sensitive path functionals.Threshold-crossing probabilities and high-level positive-part targets test rare excursions or extrema that typical-path statistics may miss.Source: toy Dataset · p. 43

DAX Dataset

  • DAX calibration uses historical European option quotes as expectations of terminal-value functionals, with far-tail targets posing the hardest calibration challenge.Evaluation holds out a non-contiguous quarter of strikes to test interpolation across the observed surface rather than extrapolation.Source: DAX Dataset · pp. 44–45

SPX Dataset

  • SPX provides an out-of-sample option-surface check, using the same target groups, normalisation, grid, and non-contiguous 25% strike hold-out as DAX.Two controllers are used for the far tails when the Girsanov overlay is enabled.Source: SPX Dataset · p. 45

Training protocol

  • Models use a common AdamW training protocol with seven seeds per configuration, and reported values are means with standard errors over those seeds.Models are simulated on 512 time steps; Girsanov runs use 1,024 reference paths and two tilted batches of 512, at 1.3× the epoch time of the budget-matched untilted arm.Source: Training protocol · p. 45

Search grids

  • The search varies block size, noise type, time-embedding dimension, tilt type, controller objective, and far-tail estimator; DAX ablations use one-factor sweeps around a two-layer configuration.The reported block-size and noise-type sweeps use general-noise and block-size-8 settings, respectively.Source: Search grids · pp. 45–46

E Additional Experimental Results

Additional experiments compare SLiSDE with baselines on option-surface calibration, structural and timing ablations, toy-law recovery, and Girsanov-tilt studies. Results show trade-offs across batch sizes and estimators, alongside gains in selected calibration and rare-event settings.

spx option-surface dataset

  • At three layers, SLiSDE attains the lowest held-out loss and far-put error on spx, with 6–8× fewer parameters and 3.5–4× faster epochs.At two layers, its held-out loss matches SLiCE (6.44 against 6.43).Source: Table 4 · p. 47, spx option-surface dataset · p. 47

Noise type and block size

  • Noise types are within one standard error on total loss; additive noise is cheaper, while multiplicative-noise variants fit far puts better.Total loss is flat across block sizes; blocks of 4 or more improve far puts over the diagonal model, at increasing epoch time.Source: Table 5 · p. 47, Noise type and block size · p. 47

Timing study

  • SLiSDE training steps are 6–7× faster than Neural SDE at B = 64, 3.2× at B = 256, and 1.3–1.4× at B = 1024.At B = 4096, the Neural SDE step is 1.4× faster; SLiCE runs out of memory beyond B = 1024 at T = 512.Source: Table 6 · p. 48, Timing study · p. 48
  • The chunked scan shortens the forward pass by 1.4× at B = 64, T = 512, but is otherwise on par with or slower than sequential evaluation.Its advantage at these sizes is parallel depth rather than wall-clock time.Source: Table 7 · p. 48, Timing study · p. 48

Distributional recovery on toy

  • On toy, SLiSDE and Neural SDE recover the terminal marginal comparably, while SLiSDE is markedly better on the running-maximum law.SLiSDE is better in terminal-value KS distance, while Neural SDE is better in W1 distance.Source: Distributional recovery on toy · p. 48, Table 8 · p. 49

Girsanov tilt: calibration on dax and rare-event estimation on toy

  • On dax, the combined tilted variant lowers far-call error to 0.64× of vanilla and total held-out loss by 15%; the tilted-batch-only variant halves far-put error and lowers total loss by 24%.Against the budget-matched untilted arm, the tilted arms are never worse and are better on average on far tails, though those differences are not significant at seven seeds.Source: Table 9 · p. 49, Girsanov tilt: calibration on dax and… · p. 49
  • On toy, the tilt's relative-error gain grows from 2× at p = 10^-2 to 4× at p = 10^-4, where plain Monte Carlo returns zero in 90% of replications.Across budgets, the gain grows with event rarity, but heavy-tailed weights produce large standard errors and some cells lose to plain Monte Carlo.Source: Girsanov tilt: calibration on dax and… · p. 50, Table 11 · p. 51

Effect of the controller objective, the KL wall and the far-tail estimate

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