Non-Adversarial Neural SDE Calibration via Strictly Proper Signature Kernel Scoring Rules

Issa, Horvath, Lemercier, and Salvi address a fundamental methodological problem in the training of neural stochastic differential equations for financial market simulation: the instability and mode collapse endemic to generative adversarial network (GAN) approaches to path-space distribution matching. Neural SDEs — SDEs of the form dX_t = b_θ(t, X_t) dt + σ_θ(t, X_t) dW_t where the drift b_θ and diffusion σ_θ are neural networks parameterized by θ — are natural candidates for flexible market simulators because they combine the inductive bias of the SDE framework (continuous paths, no-arbitrage structure, interpretable dynamics) with the universal approximation capacity of neural networks. The standard training approach matches the distribution of simulated paths X under the neural SDE to the distribution of historical market paths via a GAN objective — a discriminator neural network D_ψ is trained to distinguish simulated from real paths while the generator (the neural SDE) is trained to fool D_ψ — but this min-max formulation is notoriously unstable in practice, suffering from discriminator collapse, gradient vanishing, and failure to converge to a unique equilibrium. The paper replaces the GAN objective with a strictly proper scoring rule on path space: a functional S(P, ω) such that E_{ω' ∼ P}[S(P, ω')] ≥ E_{ω ∼ Q}[S(P, ω)] for all P ≠ Q, with equality if and only if P = Q, where ω denotes a realized path. A strictly proper scoring rule defines a proper divergence D_S(P, Q) = E_Q[S(P, ω)] - E_P[S(P, ω)] ≥ 0 that is zero only when P = Q, meaning minimizing D_S(P_θ, P_data) over the neural SDE parameters θ achieves training by minimizing a well-defined loss function — not a min-max game — that has a unique global minimum when the neural SDE correctly captures the data distribution. The key innovation is the construction of a strictly proper scoring rule on the infinite-dimensional space of continuous paths using the signature kernel: the scoring rule S_{sig}(P, ω) = E_{X ∼ P}[k_{sig}(X, ω)] - (1/2) E_{X,X' ∼ P}[k_{sig}(X, X')] + const, where k_{sig}(X, Y) is the signature kernel (the inner product of the path signatures in the tensor algebra), is strictly proper because the signature kernel is a characteristic kernel on path space — it distinguishes all distinct path distributions — and the scoring rule S_{sig} is the kernel score associated to k_{sig}, which is strictly proper by the Gretton–Sriperumbudur–Fukumizu theorem for characteristic kernels. The computational advantage over GAN training is substantial: the training loss D_S(P_θ, P_data) = E_{X ∼ P_θ, ω ∼ P_data}[k_{sig}(X, ω)] - (1/2) E_{X,X' ∼ P_θ}[k_{sig}(X, X')] is a simple expected value that can be estimated by Monte Carlo without any discriminator network, the gradient ∂D_S/∂θ can be computed by differentiating through the SDE simulation (using adjoint sensitivity methods for the SDE) and through the signature kernel computation, and the optimization problem is a standard minimization rather than a min-max game, enabling stable gradient descent with well-understood convergence guarantees. The signature kernel k_{sig}(X, Y) = Σ_{n=0}^{∞} 〈X^{⊗n}, Y^{⊗n}〉 / n! can be computed exactly for polynomial kernels or via the kernel PDE (∂_{s,t} k_{sig}(X_{[0,s]}, Y_{[0,t]}) = k_{sig}(X_{[0,s]}, Y_{[0,t]}) · dX_s · dY_t), and the Gram matrix (k_{sig}(X_i, X_j))_{i,j} for a batch of simulated paths can be assembled on a GPU with matrix-parallel computation whose cost is comparable to the SDE simulation itself.

The neural SDE is dX_t = b_θ(t, X_t, Z_t) dt + σ_θ(t, X_t, Z_t) dW_t where (b_θ, σ_θ) are feed-forward neural networks and Z_t is a latent state satisfying dZ_t = f_θ(t, Z_t) dt + g_θ(t, Z_t) dB_t (a latent factor SDE with independent Brownian B). The path signature is 𝕊(X)_{s,t} = (1, ∫_s^t dX^{i_1}_{r_1}, ∫_s^t ∫_s^{r_2} dX^{i_1}_{r_1} dX^{i_2}_{r_2}, …) ∈ T(R^d) = ⊕_{n=0}^∞ (R^d)^{⊗n}, the sequence of iterated integrals of X over [s, t]. The signature kernel between two paths X : [0, T] → R^d and Y : [0, T] → R^d is k_{sig}(X, Y) = 〈𝕊(X)_{0,T}, 𝕊(Y)_{0,T}〉_{T(R^d)} = Σ_{n=0}^∞ Σ_{i_1,…,i_n} (∫_{0<t_1<…<t_n<T} dX^{i_1}⋯dX^{i_n})(∫_{0<t_1<…<t_n<T} dY^{i_1}⋯dY^{i_n}), computable via the kernel PDE ∂_{s,t} k(X_{[0,s]}, Y_{[0,t]}) = k(X_{[0,s]}, Y_{[0,t]}) · 〈dX_s, dY_t〉. The training objective is MMD²_{k_{sig}}(P_θ, P_{data}) = E[k_{sig}(X,X')] - 2E[k_{sig}(X,Y)] + E[k_{sig}(Y,Y')] where X,X' ∼ P_θ and Y,Y' ∼ P_{data}, which is zero if and only if P_θ = P_{data} by the characteristic property of k_{sig} on continuous path distributions (Chevyrev–Oberhauser). Gradients ∂MMD²/∂θ are computed via adjoint sensitivity: d(∂X/∂θ) = (∂b_θ/∂X · ∂X/∂θ + ∂b_θ/∂θ) dt + (∂σ_θ/∂X · ∂X/∂θ + ∂σ_θ/∂θ) dW.

The shift from GAN training to proper scoring rule minimization is conceptually significant beyond the practical stability benefits: in a GAN, the training objective is not a divergence between P_θ and P_data but rather a measure of the ability of the discriminator D_ψ to distinguish them, and the equilibrium of the training game is the Nash equilibrium of the generator–discriminator game rather than the global minimum of any fixed loss function. This means GAN training may converge to a measure P_θ that fools the discriminator without matching P_data — the discriminator may have insufficient capacity to detect the mismatch, leading to mode collapse. The proper scoring rule framework replaces this game with a fixed divergence D_S(P_θ, P_data) whose unique global minimum is P_θ = P_data, so any local minimum must at least approximately match the data distribution. The choice of the signature kernel k_{sig} as the reproducing kernel for the scoring rule has the specific virtue of being characteristic on the space of continuous paths equipped with the p-variation topology: by the Chevyrev–Oberhauser theorem, the expected signature E_{X ∼ P}[𝕊(X)_{0,T}] uniquely characterizes the law P of any continuous semimartingale whose iterated integrals are integrable, meaning that MMD²_{k_{sig}}(P_θ, P_data) = 0 iff E_{P_θ}[𝕊(X)] = E_{P_{data}}[𝕊(X)] iff P_θ = P_data. This characteristic property holds for all continuous path distributions including those with heavy tails, path-dependent volatility, and non-Markovian dynamics — precisely the distributions relevant for financial market simulation — making the signature kernel more expressive than kernel choices based on pointwise evaluations of the paths, which may fail to distinguish distributions with the same marginals but different temporal correlation structure. The practical implication for market simulation — particularly the simulation of joint equity, volatility, and rates paths for structured product pricing — is that the neural SDE trained via signature kernel scores captures the full distributional geometry of the historical paths, including non-Gaussian returns, volatility clustering, return–volatility correlation, and the term structure of realised correlation between assets, without any parametric assumptions about the functional form of the dynamics. The GAN approach, by contrast, requires careful discriminator architecture to ensure that these path statistics are included in the discriminator's feature map; if the discriminator uses only pointwise path features, temporal correlation structure will not be matched.

For the desk's simulation infrastructure, the non-adversarial neural SDE provides a stable and principled way to train a flexible market simulator that can replace parametric models (local volatility, rough Heston, SABR) for pricing structured products whose payoffs depend on the full distributional geometry of multi-asset paths — including barriers, autocallables, variance dispersion swaps, and worst-of products. The signature kernel scoring rule provides a training objective with a single effective hyperparameter (the signature truncation depth) versus GAN training which requires tuning discriminator architecture, learning rate schedules, gradient penalties, and the generator–discriminator update ratio. The stable gradient dynamics of the proper scoring rule objective also enable uncertainty quantification via ensemble training: training multiple neural SDEs with different random seeds and measuring the spread of their outputs provides an estimate of model uncertainty directly from the simulation infrastructure without a separate confidence interval calculation. The computational cost of evaluating the signature kernel Gram matrix — proportional to N² × L² for N paths of length L — means that stochastic gradient descent with mini-batches of size M has per-step cost O(M² · L²), which scales quadratically with the mini-batch size M. For typical market simulation problems with L = 252 (daily paths over one year) and M between 100 and 1000 paths per mini-batch, the per-step cost is between 10⁶ and 10⁸ floating-point operations, feasible on a GPU in milliseconds. The kernel PDE formulation of k_{sig} enables matrix-parallel GPU computation with negligible overhead compared to the SDE simulation itself, and the adjoint method for computing ∂MMD²/∂θ requires only one forward and one backward SDE solve per mini-batch, the same cost as standard backpropagation through an ODE.

The signature kernel MMD objective measures the divergence between P_θ and P_data in the topology induced by k_{sig}, which is the topology of convergence in distribution on path space with respect to the expected signature. However, the MMD objective does not distinguish whether a deviation of P_θ from P_data corresponds to a deviation in a high-signature-order statistic (which may be small in magnitude and unimportant for option pricing) or a low-order statistic (which may dominate option payoff moments). In practice, the signature kernel with truncation depth n compares only the first n levels of the signature, and for n = 3 or 4 (the typical depth used for computational tractability), the MMD is sensitive primarily to one-, two-, and three-point path statistics — marginal distributions, lagged correlations, and third-order cross-moments — but not to higher-order temporal dependencies that may matter for long-dated structured products. The desk should verify empirically that the trained neural SDE reproduces the path statistics relevant for the specific payoff being priced, using payoff-specific diagnostic statistics rather than relying on MMD smallness alone. A subtler issue is that the signature MMD measures distributional distance on path space without reference to the financial no-arbitrage structure: the neural SDE may converge to a measure P_θ that minimizes the MMD to P_data but is not a local martingale (the drift b_θ may be non-zero under the risk-neutral measure). The SDE formulation does not constrain the drift b_θ to be zero (risk-neutral) or equal to the risk-free rate (physical measure), so the trained neural SDE is a statistical model of the historical measure P_data rather than a risk-neutral pricing model. Using P_θ directly for option pricing requires an additional change of measure from P to Q, which requires estimating the market price of risk process λ_t and is subject to the estimation uncertainty inherent in any P-to-Q measure change, introducing an additional source of model risk that the signature kernel score training does not address.