DNN Expression Rate Analysis of High-dimensional PDEs: Application to Option Pricing

1.1 Motivation

• Empirical work suggests deep learning avoids the curse of dimensionality in option-pricing PDEs, but the introduction notes that few theoretical results support this claim.The cited theoretical results address other PDE settings or algorithms.Source: §1.1 · p. 1

1.2 Contributions and Main Result

• The paper shows that low-rank functions built from tensor products can be approximated by deep ReLU networks, extending the approach to high-dimensional problems with implicit low-rank structure.The tensor-product construction has logarithmic connectivity scaling in ε^(-1) when its univariate factors admit logarithmic scaling.Source: §1.2 · p. 2
• The option-pricing application assumes uncorrelated asset log-returns, while the authors present their approximation tools as applicable to broader settings with implicit low-rank structure.They compare their result with [20], which treats a more general problem but gives weaker approximation guarantees.Source: §1.2 · pp. 2–3
• O(d^(2+1/n)ε^(-1/n)) network size gives pointwise ε-accuracy for the basket option on [a,b]^d, with fixed n and constants depending on n.The authors describe this spectral approximation rate as avoiding the curse of dimensionality.Source: §1.2 · p. 3

1.3 Outline

• The paper develops a semi-explicit Black–Scholes option-price formula, regularity estimates, quadrature methods, and tensor-product network approximation before proving the main rate.The introduction outlines these steps across Sections 2–7.Source: §1.3 · p. 4

2 High-dimensional derivative pricing

• The model is a linear parabolic Black–Scholes equation for a European maximum option on d stocks whose log-returns are mutually uncorrelated.The equation is endowed with the option's terminal condition.Source: §2 · p. 4
• Proposition 2.1 gives a solution representation, and its proof identifies the first equality as following from the Feynman–Kac formula.The proposition concerns the solution evaluated at a point x in the positive orthant.Source: §2 · p. 4
• The proof uses a complementary distribution-function formula to establish the second equality in the representation.Combining that formula with Proposition 2.1 completes the proof.Source: §2 · pp. 4–5

3 Regularity of the Cumulative Normal Distribution

• Regularity estimates for the integrand are needed to approximate the multivariate integral by quadrature and to apply neural-network approximation results to the solution.The factors are compositions involving the cumulative normal distribution and the natural logarithm.Source: §3 · p. 5
• All derivatives of the auxiliary function f are bounded on (0,∞), although their suprema scale with derivative order.The passage notes that the supremum of f^(n) scales like e^(cn^2) for some positive c.Source: §3 · p. 7
• The section also analyzes derivatives of the transformed univariate factors and their tensor product for the later quadrature approximation.The stated estimates establish infinite differentiability for the functions f and h.Source: §3 · pp. 9–10

4 Quadrature

• The integral is truncated after showing that its integrand Fx decays sufficiently fast as c tends to infinity.The remaining definite integral is then approximated numerically.Source: §4 · p. 12
• Composite Gaussian quadrature provides a weighted-sum approximation with an error bound explicit in the stepsize and quadrature order.The section recalls this bound as a tool for approximating the finite integral.Source: §4 · p. 12
• The weighted sum uses values Fx(cj), which the paper proposes realizing efficiently with neural networks because of their tensor-product structure.The resulting network approximation targets the integral in the option-price formula.Source: §4 · p. 12

• A network’s architecture and weights define its realization, but the same function can have multiple nontrivially different network representations.The realization is a function on R^d; the network representation makes properties such as depth and size meaningful.Source: §5 · p. 16
• Network size is measured by its nonzero entries M(Φ) and depth L(Φ); for nondegenerate networks, total weights are at most M(Φ)^2 + M(Φ).Thus the qualitative polynomial dependence on dimension also holds when counting all weights.Source: §5 · p. 17
• Composition constructs a network whose realization is the composition of the realizations of its two constituent networks.The paper uses this construction to relate a derived network’s complexity to its component networks.Source: §5 · p. 17
• Parallelization combines networks on distinct input-coordinate blocks, supporting linear combinations or tensor products of functions that are already approximated.The construction is presented for ReLU networks of differing depths by extending them to a common depth without changing their realizations.Source: §5 · pp. 17, 19

• ReLU networks approximate x² through sawtooth-based linear interpolation, and an algebraic identity then yields approximate multiplication.The identity expresses xy through evaluations of the square function.Source: §6 · pp. 19–20
• Products of multiple factors are approximated by hierarchical pairwise multiplication.The paper notes that the product approximation’s error bound includes an unavoidable B^(m−1) term from multiplying m inaccurate values; this is not an issue when B = 1 or factor networks have logarithmic size in ε.Source: §6 · pp. 20, 23
• Direct multivariate Taylor approximation yields networks with size exponential in the function dimension, so the paper treats univariate smooth functions for its polynomial-dimension goal.The univariate approximation is intended for use with high-dimensional functions having tensor-product structure.Source: §6 · p. 25

7 DNN Expression Rates for High-Dimensional Basket prices

• The construction uses Proposition 6.4 to approximate the required tensor product.The section first establishes general expression-rate results and then applies them to the specific problem.Source: §7 · pp. 26–27
• Quadrature estimates are added to obtain networks whose size depends polynomially on the problem dimension.Theorem 7.3 provides the network construction for the functions Fd.Source: §7 · p. 30
• Theorem 7.3 concerns functions Fd defined on (0, ∞) × [a, b]^d and supplies networks Γd,ε for ε ∈ (0, 1].The theorem’s detailed assumptions and bounds are not present in these excerpts.Source: §7 · p. 30

8 Discussion

• Theorem 7.3 formally treats one specific high-dimensional PDE; extending the approach depends on solutions having a similar low-rank structure.With a suitably well-behaved low-rank structure, Section 6 might yield connectivity growing logarithmically in inverse approximation error.Source: §8 · p. 33
• The constructive result is a step toward higher-order numerical solution of high-dimensional PDEs, which classical discretization and randomized approaches find troublesome.The passage also cautions that approximation existence alone does not establish whether or how such networks can be found.Source: §8 · p. 33
• Prior work reports dimension-free Monte Carlo sample requirements under assumptions and empirical evidence of efficient PDE learning, but formal analysis of training is not yet available in the cited discussion.The passage cites for the sample result and [2] [23] [41] for empirical results.Source: §8 · p. 33
• The ReLU networks are not adapted to the available semi-explicit formula, while a PDE-specific spline dictionary might attain similar rates but would lack the stated universality.The passage points to approximation results for piecewise smooth functions [36], affine and Gabor systems [12], and fractal structures [9].Source: §8 · p. 33

A.1 Technical Lemma

• Lemma A.1 addresses t ∈ (0, exp(−2r²)] for r > 0, and its proof uses the corresponding range x ∈ [exp(2r²), ∞).The supplied excerpts do not include the lemma’s bound itself.Source: §A.1 · p. 36

A.2 Proof of Lemma 6.1

• The proof follows [44] and supplies constants in the depth and width bounds, including their dependence on scaling parameter B.The passage states this as the reason for providing the proof.Source: §A.2 · p. 36
• The function fm is the affine linear interpolant of x² at the grid points k/2^m on [0, 1].The proof uses this interpolant in the network approximation argument.Source: §A.2 · p. 38
• The networks ϕm have depth m and at most 15m nonzero weights.This is the construction’s stated depth and size bound.Source: §A.2 · p. 40

A.3 Proof of Lemma 6.2

• The proof combines networks from Lemma 6.1 with auxiliary networks α and Σ to establish bounds for ε ∈ (0, B^2).It records L(α) = 2 and L(Σ) = 1, and gives [R̺(Σ)](x, y, z) = 2B^2x − 2B^2y − 2B^2z.Source: §A.3 · pp. 40–41
• The constructed networks satisfy L(µε) = 1 and M(µε) = 0 for ε ∈ (B^2, ∞), with M1(µε) = 8 and M_L(µε)(µε) = 3 for every ε > 0.The proof identifies these properties using Lemmas 5.3 and 5.4.Source: §A.3 · p. 41

A.4 Proof of Theorem 6.5

• The functions hN,j have local support, and adjacent functions sum to 1 on their shared partition interval.The proof uses this partition structure in combining the local approximations.Source: §A.4 · p. 42
• The networks ϕf,ε,N are assembled from λN, the product network P applied to local networks ψf,ε,N,j, and ∇1,0,2N+2.The proof introduces these networks to realize the local Taylor approximation.Source: §A.4 · pp. 42–43
• The proof concludes that the neural networks (Φf,ε) satisfy the properties required by Theorem 6.5.The proof assembles approximation and network bounds before concluding the theorem.Source: §A.4 · pp. 45–47

A.5 Proof of Corollary 6.6

• Theorem 6.5 therefore establishes that there exist neural networks (Φg,η)g∈Bn; the cited passage does not specify additional properties for those networks.Source: §A.5 · p. 48