A desk already running particle-style local stochastic volatility calibration gets the same calibration formula after adding roughness. Two things still change. Conditional densities must come from a transformed pathwise PDE whose coefficients are rough and path-dependent. The common noise must also be simulated jointly as (Z^i, V^i).
The leverage function is the calibration target. Start with Dupire's local variance from the quoted surface, divide by the conditional expectation of the stochastic variance factor when spot sits at the strike, then take the square root. Brigo and Lucic write L̂²(t,K) = σ²_Dup(t,K)/Θ̂(t,K). Here Θ̂(t,K) is a Monte Carlo ratio of conditional densities weighted by f²(V_t). Roughness leaves the form of that ratio unchanged.
We could not reproduce the method. The paper supplies a conditional-density and calibration framework; it never specifies a trading rule that can be run as stated. A test would require us to build both a pathwise Fokker-Planck solver for local stochastic rough volatility and a Rao-Blackwellized calibration routine. The likely starting point would be pure rough Heston, where the density already has a closed form. Any output from such an exercise would be our own construction, not a result the authors reported.
The conditional-density SPDE
The model is dS_t = (r_t - d_t)S_t dt + S_t L(t,S_t) f(V_t) dB_t, while a Volterra kernel with Hurst parameter H in (0, 1/2) generates V_t. Its spot driver is dB_t = ρ dZ_t + √(1-ρ²) dζ_t, and Z itself drives volatility. The authors call Z and V together the common environment. The independent ζ supplies the idiosyncratic component. This model class is local stochastic rough volatility (LSRV), with the local factor L(t,S) multiplying rough or Volterra-driven stochastic variance.
Freeze the environment.
The conditional density p_t of S_t then follows an SPDE. Its first-order transport term is driven by dZ_t and has loading Γ_t(s) = ρ s L(t,s) f(V_t). Total instantaneous variance separates as Σ²_t = Γ²_t + Δ²_t, where Δ²_t(s) = (1-ρ²) s² L²(t,s) f²(V_t). Γ carries the common noise that the transformation removes. Δ preserves the residual idiosyncratic diffusion.
The transport disappears
Use the flow generated by the transport coefficient: dΦ_t(x) = Γ_t(Φ_t(x)) dZ_t. Its Jacobian J_t(x) = ∂_x Φ_t(x) is strictly positive almost surely, since a linear SDE with an exponential solution governs it. Set q_t(x) = p_t(Φ_t(x)) J_t(x), then apply the Itô-Wentzell formula (Kunita 1990, Theorem 3.3.1).
The cancellation comes in two stages. Following the density along the flow makes the dZ_t terms from the SPDE and dΦ_t eliminate each other, apart from -Γ'_t p_t dZ_t. The drift expansion contains ½(Γ²_t p)_ss + ½Γ²_t p_ss - Γ_t(Γ_t p)_ss. Nearly every term vanishes, leaving exactly (Γ'_t)² p_t. Multiplication by J_t adds the cross-variation d⟨Y,J⟩_t = -(Γ'_t)² q_t dt, which eliminates the remainder. The authors summarize it accurately: "The stochastic integral and the (Γ′_t)^2 drift corrections vanish completely."
The resulting divergence-form equation is ∂_t q_t = -∂_x(μ̃_t q_t) + ½∂_xx(Δ̃²_t q_t). Its transformed coefficients are Δ̃²_t(x) = Δ²_t(Φ_t(x))/J_t(x)² and μ̃_t(x) = μ_t(Φ_t(x))/J_t(x) - ½ Δ²_t(Φ_t(x)) H_t(x)/J_t(x)³. H_t denotes the second spatial derivative of the flow. The rough driver has disappeared from the equation.
Roughness causes no obstruction because the argument uses spatial regularity of Γ_t in s. Assumption 1(i) requires C³, with bounded derivatives of orders one through three. Temporal requirements extend only to local boundedness and Borel measurability in t under Assumption 1(ii). With the environment fixed, V_t enters as a locally bounded measurable forcing term.
Rough Heston closes the equation
Set L ≡ 1 and f = √·. The coefficient Γ_t(s) = ρ s √V_t is now affine, so the flow is geometric: Φ_t(x) = x M_t, with M_t = exp(ρ∫√V dZ - (ρ²/2)∫V ds). Consequently J_t(x) = M_t and H_t(x) = 0. The troublesome Δ² H/J³ drift correction disappears. The transformed drift becomes (r_t - d_t)x, while transformed variance becomes (1-ρ²)x²V_t.
For X_t = S_t/M_t, the dynamics are dX_t/X_t = (r_t - d_t)dt + √(1-ρ²)√V_t dζ_t. Its logarithm is Gaussian with variance A_t = (1-ρ²)∫₀ᵗ V_u du. This gives the conditional density in closed-form lognormal form. Uniqueness follows from the Fourier identity û_t(ξ) = exp(-½ξ²A_t) û_0(ξ), including when A_t has flat portions.
The paper's standing assumption that a conditional-density random field exists is verified here as well. General LSRV takes that existence as an assumption. Verification occurs only in this explicit specification, which requires no PDE solver. The authors make the same point: the lognormal representation avoids abstract parabolic theory entirely and provides a benchmark for the numerical step.
Is this cheaper to solve?
The numerical proposal is a five-step loop. It starts with joint simulation of (Z, V), followed by evolution of the flow and its two spatial derivatives. Next comes one deterministic Fokker-Planck solve per path. The density is then recovered through the inverse flow at x_K(t) = Φ_t^{-1}(K), before the ratio estimator is evaluated.
Step one requires a dedicated discretization for the nonlinear Volterra square-root equation. The authors identify a specific trap: Cholesky of the joint covariance can sample correlated Gaussians, yet by itself it does not discretize the rough Heston variance equation. Step three requires a PDE for every path, using a grid that the authors say must be fine. Since the coefficient Hölder exponent in time is bounded by H < 1/2, standard time-stepping can be expected to achieve a lower order. The paper supplies no rate and no grid size.
Step five adds a Picard iteration because p, q and Φ depend on the current L. There is no convergence proof or iteration count. The ratio estimator is self-normalized and therefore biased at finite N, as the authors concede. Their claim is consistency as N goes to infinity. They quantify no variance reduction factor against naive Monte Carlo.
No numerical experiment appears anywhere in the paper. Nor is there a comparison with the fractional Riccati and Fourier pricer discussed in the setup. The solvable case is their response to this gap. Rough Heston "provides a closed-form benchmark for the numerical implementation of Step 3", and the conclusion presents it as a useful check on the general transformation.
Yet this benchmark prices only the rough Heston leg, where a PDE solve is unnecessary. It gives no evidence on the expense of the general LSRV calculation that a desk would actually run. The abstract promises a pathwise formulation that connects naturally with Rao-Blackwellized calibration, and the paper delivers that structure. As a case for rewriting a working calibration engine, the result remains an existence statement with the cost side blank.
Degeneracy, then the excluded corner
For rough Heston, the transformed PDE has diffusion coefficient Δ̃²_t(x) = (1-ρ²)x²V_t. It vanishes whenever V_t = 0. The paper cites recent work showing that Volterra square-root variance can hit zero with positive probability before any fixed horizon. A Feller-type condition therefore offers no rescue, and global uniform ellipticity cannot be assumed.
The authors repair the issue for the case they need. With V₀ > 0 and continuity, ∫₀ᵗ V_s ds > 0, preserving the lognormal formula. This fix applies specifically to pure rough Heston.
General LSRV leaves four possible readings: degenerate parabolic theory; weak or distributional solutions; local uniform parabolicity on cylinders where diffusion remains bounded away from zero; or the explicit rough Heston solution. Under the weak reading, existence, uniqueness and regularity in the degenerate case need further model-specific coercivity, growth and boundary hypotheses beyond those imposed here.
The |ρ| = 1 corner stays excluded throughout. In that case Δ̃² is identically zero, parabolicity disappears, and the conditional law becomes a Dirac instead of a density. The paper names a measure-valued reformulation without developing it.
A practical implementation faces another restriction. Assumption 1(i) calls for C³ smoothness with bounded derivatives of Γ_t(s) = ρ s L(t,s) f(V_t), and Γ contains L. The paper verifies condition (i) only when L ≡ 1. It explicitly leaves condition (i') to the chosen L: local C² and C¹ regularity of the idiosyncratic variance Δ²_t and drift μ_t must be checked separately for the selected local-volatility function. In live calibration, L is an interpolated surface generated by the loop itself. The smoothness assumptions therefore apply to the object being solved for.
The conclusion also concedes another open end. Continuity of the filter with respect to the observation path, under a rough-path or Volterra-path topology, is not established here.
Evidence that would change the verdict
One worked example would do it: an LSRV leverage surface calibrated with N paths, with error measured against the rough Heston lognormal benchmark proposed by the authors, and wall clock compared with a Fourier pricer on the same surface. The cancellation itself is exact. The dZ_t coefficient is exactly zero in equation (26), and the (Γ'_t)² drift corrections cancel. This note leaves unanswered whether the transformed PDE deserves more compute than the transform methods a desk already owns.