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

The paper solves path-dependent PDEs by representing the solution in the reproducing-kernel Hilbert space of the signature kernel and collocating the PPDE there, turning a non-local functional PDE into a kernel regression problem with convergence guarantees. We read it as the numerical complement to the rough-PPDE theory: a mesh-free solver for equations whose state is an entire path.

u(·) is approximated in the RKHS of the signature kernel k_sig(x, y) = ⟨𝕊(x), 𝕊(y)⟩; PPDE residuals are collocated at sampled paths, yielding a linear system in the kernel coefficients.

Signature kernels make the infinite-dimensional path state computationally addressable without truncating the signature explicitly, which is the recurring bottleneck. It is a genuine numerical enabler for path-dependent pricing.

Makes path-dependent valuation problems the desk would otherwise avoid at least approachable; relevant for exotic, memory-dependent event exposures where Markovian solvers do not apply.

Kernel-collocation cost scales with the number of sampled paths; conditioning of the kernel system and the choice of collocation measure materially affect accuracy.