Mesh-Free Solution of Path-Dependent PDEs in a Signature-Kernel RKHS

Pannier and Salvi develop a mesh-free numerical solver for path-dependent PDEs by representing the solution in the reproducing kernel Hilbert space (RKHS) generated by the signature kernel — the positive definite kernel on path space defined by k_sig(x, y) = ⟨𝕊(x), 𝕊(y)⟩_{T((R^d))} — and collocating the PPDE residual at a finite set of sampled paths, transforming the infinite-dimensional functional PDE into a finite-dimensional linear system in the kernel coefficients. The signature kernel, introduced by Salvi and collaborators in the context of statistical learning on sequential data, is the inner product between the signatures of two paths in the full (untrunacted) tensor algebra, and can be computed efficiently via a PDE on the two-parameter domain [0,T]² without explicit computation of the signature components; this makes it possible to work in an effectively infinite-dimensional signature space without incurring the curse of dimensionality that would arise from explicit signature truncation. The reproducing kernel Hilbert space framework for PPDE solution generalizes classical kernel collocation methods for PDEs (the Kansa method, radial basis function collocation) from finite-dimensional state spaces to the infinite-dimensional path space that is the natural domain of PPDEs. In the finite-dimensional case, the solution V(t, x) is approximated in the RKHS of a kernel k(x, x') on R^d, and the PDE is collocated at a finite set of state-space points; in the path-dependent case, the solution V(t, x_{·∧t}) is approximated in the RKHS of the signature kernel k_sig on the space of continuous paths, and the PPDE is collocated at a finite set of sampled paths. The convergence theory for kernel collocation in finite dimensions — which guarantees that the RKHS approximant converges to the true PDE solution as the number of collocation points grows and their fill distance shrinks — extends to the path-space setting under appropriate completeness and density conditions on the signature kernel RKHS. The computational workflow consists of three phases: (1) sampling a set of n collocation paths from a reference measure on path space (typically Monte Carlo paths from the underlying dynamics); (2) computing the kernel matrix K ∈ R^{n×n} with K_{ij} = k_sig(x^{(i)}, x^{(j)}) via the signature kernel PDE; and (3) solving the linear system (K + λ_reg I) α = F for the kernel coefficients α ∈ R^n, where F encodes the PPDE residual at each collocation path and λ_reg is a Tikhonov regularization parameter. The solution at a new path x is then V(t, x_{·∧t}) = Σ_{i=1}^n α_i k_sig(x^{(i)}, x_{·∧t}), computable in O(n) inner product evaluations.

The signature kernel on continuous paths of bounded p-variation is k_sig(x, y) = ⟨𝕊(x)_{0,T}, 𝕊(y)_{0,T}⟩_{T((R^d))} = Σ_{n=0}^∞ ⟨𝕊^n(x)_{0,T}, 𝕊^n(y)_{0,T}⟩_{(R^d)^{⊗n}} where 𝕊^n(x)_{0,T} = ∫_{0<t_1<...<t_n<T} dx_{t_1} ⊗ ... ⊗ dx_{t_n} is the n-th iterated integral. The kernel is well-defined as a convergent series for paths of bounded 2-variation (in particular for Brownian paths by Lyons' extension theorem) and is positive definite on the space of paths, so by the Mercer theorem the associated RKHS H_sig is a Hilbert space of path functionals in which evaluation and continuity are well-defined. The PPDE for the value function V(t, x_{·∧t}) is written as L_PPDE V = 0 where L_PPDE is the PPDE operator encoding the backward equations for the conditional expectation; the kernel collocation approximation is V_n(t, x_{·}) = Σ_{i=1}^n α_i k_sig(x^{(i)}_{·∧t}, x_{·∧t}) and α = (K + λI)^{-1} y where K_{ij} = k_sig(x^{(i)}, x^{(j)}), λ > 0 is the regularization parameter, and y_i = (L_PPDE V_n)(t_i, x^{(i)}_{·∧t_i}) evaluated at the collocation paths. The signature kernel PDE enabling efficient computation of K is ∂_{s,t} k_sig(x_{[0,s]}, y_{[0,t]}) = k_sig(x_{[0,s]}, y_{[0,t]}) · ⟨dx_s, dy_t⟩, initialized at k_sig(x_{[0,0]}, y_{[0,t]}) = k_sig(x_{[0,s]}, y_{[0,0]}) = 1.

The signature kernel PDE — the key computational innovation enabling efficient kernel matrix computation — is a first-order PDE on the two-parameter domain [0,T]² whose solution gives the inner product of two signatures evaluated at any pair (s, t) with O(n_s · n_t) operations for paths discretized on grids of size n_s and n_t respectively, rather than the O(N^{d^N}) cost that would arise from explicit computation of the signature components up to level N. This computational advantage is dramatic for high-dimensional path problems: for d = 5 and N = 5, explicit signature computation requires 5^5 = 3125 components per pair of paths, while the signature kernel PDE requires only n_s × n_t ≈ 250 × 250 = 62,500 operations at daily discretization over a 1-year path, comparable in cost but with the advantage of operating in the full infinite-dimensional RKHS rather than its level-N truncation. The kernel collocation framework for PPDEs addresses a fundamental limitation of classical numerical methods for path-dependent problems: grid-based PDE solvers (finite differences, finite elements) require discretization of the state space, and for path-dependent problems the state space is the space of continuous functions on [0,T], which is infinite-dimensional and cannot be discretized by any finite grid without projecting onto a finite-dimensional subspace. The signature kernel RKHS provides a principled infinite-dimensional function space in which the PPDE can be formulated and solved, with the collocation paths playing the role of grid points and the kernel matrix encoding the geometry of the function space. The regularization parameter λ in the kernel collocation system controls the trade-off between fidelity to the PPDE residual at collocation paths (λ small) and smoothness of the solution in the RKHS norm (λ large): small λ produces solutions that satisfy the PPDE precisely at the collocation paths but may oscillate wildly between them, while large λ produces solutions that are smooth in the signature kernel RKHS sense but may have significant PPDE residual. The optimal λ from the theory of kernel ridge regression depends on the true smoothness of V in the RKHS norm — a quantity not directly observable — and is estimated in practice by cross-validation on held-out paths, with the caveat that non-stationarity of the PPDE solution across different regions of path space may require spatially adaptive λ.

For derivatives with payoffs that depend on the full path of the underlying — Asian options, lookbacks, barrier options, realized variance contracts — the signature kernel PPDE solver provides a method for computing conditional expectations that is free of the state-space dimensionality assumptions required by classical grid methods. In particular, for payoffs that depend on a path at a set of discrete observation dates {t_1, ..., t_m}, the state space is R^m — potentially high-dimensional for large m — and classical grid methods face the curse of dimensionality, while the signature kernel method collocates on m-dimensional paths and the computational cost scales polynomially in n (the number of collocation paths) rather than exponentially in m. The ability to compute functional sensitivities D_x V via kernel differentiation — differentiating the kernel collocation representation V_n with respect to the endpoint x_{t} of the collocation path — provides a deterministic, variance-free greeks computation method for path-dependent options that complements the Monte Carlo bump-and-reprice approach. For the desk's structured derivatives book, maintaining a signature kernel PPDE solver calibrated to the current rough volatility model enables intraday greeks computation for path-dependent exotics at a speed and accuracy not achievable by Monte Carlo, and the mesh-free nature of the method avoids the step-size calibration issues that plague finite-difference greeks for discontinuous payoffs.

The conditioning of the kernel matrix K — whose eigenvalue spectrum determines the stability of the linear system (K + λI)α = y — is the primary numerical challenge: for large n and a fixed collection of collocation paths, the kernel matrix can be ill-conditioned (many small eigenvalues corresponding to near-identical paths) or poorly conditioned (eigenvalues spanning many orders of magnitude), and the regularized solution α = (K + λI)^{-1} y may be dominated by the regularization term rather than the PPDE residual, producing solutions that are smooth but inaccurate as PPDE solutions. Choosing collocation paths that span the relevant region of path space as diversely as possible — using quasi-Monte Carlo sequences, conditional sampling, or importance sampling — reduces kernel matrix conditioning but requires a priori knowledge of where the PPDE solution is most sensitive, information that is not available without first solving the problem. The convergence theory for kernel collocation in the path-space RKHS setting is not as well-developed as in finite dimensions: the key result from kernel collocation theory — that the approximation error decays as the fill distance of the collocation set shrinks — requires a concept of fill distance in path space that depends on the topology induced by the signature kernel, and characterizing this topology in terms of familiar path-space metrics (uniform, p-variation) requires deep results from the theory of reproducing kernel Hilbert spaces on non-compact metric spaces. The current convergence theory gives qualitative guarantees (the approximant converges as n → ∞ if the collocation paths are dense in the appropriate sense) but not quantitative rates relevant to practical computation, leaving the choice of n for a given accuracy target to empirical study rather than theoretical prescription.