Path-Dependent PDEs for Volterra Models and the Weak Error of Discretization

The paper proves that conditional expectations in stochastic Volterra (rough volatility) models are the unique classical solutions of associated path-dependent PDEs, and uses this to establish weak convergence rates for discretized integrals of smooth functionals of Riemann–Liouville fractional Brownian motion with H ∈ (0, 1/2). We read it as the rigorous backbone connecting rough models to a PDE description and to controlled discretization error.

Value functionals u(t, X_{·∧t}) solve a PPDE with a non-local (path) derivative; the weak error of Euler-type schemes for ∫ f(W^H) is shown to decay at a rate governed by the Hurst parameter H.

The practical payoff is the separation of two error sources — model roughness and numerical scheme — with the weak rate quantifying the second. It gives the desk a defensible statement about how much of a rough-model price is discretization artefact.

Discretization-sensitive memo inputs derived from rough simulations should report scheme order against the H-dependent weak rate rather than assume convergence; coarse grids carry a quantifiable bias.

Smoothness assumptions on the functionals exclude some payoffs (barriers, digitals), and the constants in the rate are not always sharp enough for a priori grid selection.