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A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations

Philipp Grohs, Fabian Hornung, Arnulf Jentzen, Philippe von Wurstemberger

arXiv:1809.02362v3math.NAcs.LGmath.PRq-fin.MF

TL;DR

Rigorous results explaining ANNs’ apparent ability to approximate high-dimensional functions remain limited, including for PDEs. This paper proves that ANNs approximate Black-Scholes PDE solutions with parameter counts growing at most polynomially in dimension and reciprocal accuracy, thereby overcoming the curse of dimensionality.

  • Problem

    Only a few rigorous results explain ANNs’ empirical success in approximating high-dimensional functions, including solutions of PDEs.

    Source: §1 · p. 3

  • Method

    The paper constructs a suitable realization on an artificial probability space and establishes approximation bounds for fully connected ANNs.

    Source: §3.1 · p. 47, §3.6 · p. 78

  • Results

    ANN parameter counts grow at most polynomially in the PDE dimension d and reciprocal approximation accuracy ε^-1 when approximating Black-Scholes PDE solutions.

    Source: §1 · pp. 4–6

  • Takeaways & Limitations

    The results prove that ANNs overcome the curse of dimensionality for numerical approximation of Black-Scholes PDEs.

    Source: §1 · pp. 4, 6

    The result assumes affine linear drift and diffusion functions and stated growth conditions on the network output.

    Source: §1 · p. 4, §3.5 · p. 67

Abstract

from arXiv · show

Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems ranging from image classification/image recognition, speech recognition, time series analysis, game intelligence, and computational advertising to numerical approximations of partial differential equations (PDEs). Such numerical simulations suggest that ANNs have the capacity to very efficiently approximate high-dimensional functions and, especially, indicate that ANNs seem to admit the fundamental power to overcome the curse of dimensionality when approximating the high-dimensional functions appearing in the above named computational problems. There are a series of rigorous mathematical approximation results for ANNs in the scientific literature. Some of them prove convergence without convergence rates and some even rigorously establish convergence rates but there are only a few special cases where mathematical results can rigorously explain the empirical success of ANNs when approximating high-dimensional functions. The key contribution of this article is to disclose that ANNs can efficiently approximate high-dimensional functions in the case of numerical approximations of Black-Scholes PDEs. More precisely, this work reveals that the number of required parameters of an ANN to approximate the solution of the Black-Scholes PDE grows at most polynomially in both the reciprocal of the prescribed approximation accuracy $\varepsilon > 0$ and the PDE dimension $d \in \mathbb{N}$. We thereby prove, for the first time, that ANNs do indeed overcome the curse of dimensionality in the numerical approximation of Black-Scholes PDEs.

1 Introduction

The paper proves that ANNs can approximate solutions of Black-Scholes PDEs without the curse of dimensionality: required parameters grow polynomially in dimension and reciprocal accuracy. The result provides a theoretical explanation for efficient high-dimensional approximation on the entire unit cube.

Contribution

  • The main result proves that ANN parameter counts for approximating Black-Scholes PDE solutions grow at most polynomially in d and ε^-1.This establishes, for the first time in the paper’s stated scope, that ANNs overcome the curse of dimensionality for numerical Black-Scholes PDE approximation.Source: §1 · p. 4

Main theorem

  • Theorem 1.1 guarantees networks with P(ψ_d,ε) ≤ C d^C ε^-C that approximate the relevant PDE solutions under the stated assumptions.The theorem concerns Kolmogorov PDEs with affine linear drift and diffusion functions and prescribes continuous ANN realizations.Source: §1 · pp. 4–5

Scope

  • The approximation applies on the entire d-dimensional unit cube [0,1]^d, rather than only at a single spatial point.This distinguishes the result from the cited Monte Carlo methods, which overcome the curse of dimensionality at one fixed spatial point.Source: §1 · p. 6

Significance

  • The result theoretically supports deep-learning algorithms for Black-Scholes PDE approximation, including the numerical approach proposed by Beck et al..The introduction frames the theorem as a partial theoretical justification of observed algorithmic performance.Source: §1 · p. 6

Proof strategy

  • The proof combines the Feynman-Kac formula, Monte Carlo expectation approximations, affine dependence of SDE solutions on initial values, and probabilistic ANN construction.These ingredients construct suitable ANN approximations for viscosity solutions of the relevant Kolmogorov PDEs.Source: §1 · p. 6
  • A specially constructed elementary event provides the crucial bridge from the proof’s probabilistic arguments to its deterministic approximation conclusion.The paper identifies this event-existence argument as the crucial ingredient connecting the probabilistic construction with the deterministic ANN guarantee.Source: §1 · pp. 6–7

2 Probabilistic and analytic preliminaries

This section develops probabilistic and analytic preliminaries for Monte Carlo approximations, stochastic fields, affine-coefficient stochastic differential equations, and associated partial differential equations. It establishes error estimates, regularity, adaptation, continuity, well-posedness, and probabilistic representation results.

2 Probabilistic and analytic preliminaries

  • The preliminaries cover Monte Carlo approximations, affine functions, and SDEs, supplying tools used in later analysis.The section introduces these auxiliary results before the viscosity-solution material handled elsewhere.Source: §2 · p. 8

2.1 Monte Carlo approximations

  • The Monte Carlo subsection derives an Lp-Monte Carlo estimate from Proposition 2.3 and Lemma 2.4 using Kahane-Khintchine-type estimates.The results are presented as Corollary 2.6, with supporting propositions and lemmas included for completeness.Source: §2.1 · pp. 8, 10

2.2 Properties of affine functions

  • The affine-function subsection characterizes affine maps and establishes their linear-growth and Lipschitz bounds.Corollaries 2.9–2.11 provide these properties, including corresponding bounds for matrix-valued affine functions.Source: §2.2 · pp. 12–14

2.3 A priori estimates for solutions of stochastic differential equations

  • For SDEs with at most linearly growing coefficients, Proposition 2.15 establishes an elementary a priori estimate.The proof uses Gronwall, Minkowski-type, and Burkholder-Davis-Gundy inequalities.Source: §2.3 · pp. 15, 16, 21

2.4 Stochastic differential equations with affine coefficient functions

  • For SDEs with affine coefficients, Proposition 2.21 establishes elementary regularity properties using moment estimates and a version of the Kolmogorov-Chentsov theorem.The subsection also uses modification and adaptedness results for stochastic processes.Source: §2.4 · pp. 21–22
  • Pointwise almost-surely equal random fields are indistinguishable when the fields satisfy the stated separability and continuity assumptions.The proof reduces equality over the full index set to equality on a countable dense subset and concludes probability one for simultaneous equality.Source: §2.4 · pp. 23, 25
  • A modification of an adapted process remains adapted when it is pointwise almost surely equal at every time.The proof uses the usual conditions on the filtration to show measurability of the modification’s time-indexed events.Source: §2.4 · p. 27
  • The Kolmogorov-Chentsov-type lemma supplies a continuous random-field modification that agrees pointwise almost surely with the original field.The construction proceeds from local moment bounds to Hölder-continuous versions on expanding domains, which are then patched consistently.Source: §2.4 · pp. 27–30
  • Unique continuous adapted solutions exist for the affine-coefficient SDE family, up to indistinguishability.The result is obtained under the stated filtered-probability-space, Brownian-motion, and coefficient assumptions.Source: §2.4 · pp. 31–32
  • The affine SDE solution family inherits the required joint regularity in initial conditions and time for the subsequent analysis.The proof combines moment estimates, continuity of sample paths, and the modification and indistinguishability lemmas.Source: §2.4 · pp. 32, 34, 35, 36
  • For affine-coefficient SDEs, the paper proves existence and uniqueness of stochastic processes Z^x satisfying the stated equations.These processes are adapted and have continuous sample paths.Source: §2.4 · pp. 38, 40

2.5 Viscosity solutions for partial differential equations

  • Under coercivity-type conditions, the Lyapunov function V(x) = 1 + ∥x∥^p satisfies the required differential inequality for viscosity-solution analysis.The lemma assumes p ∈ [4, ∞) and provides a constant ρ ensuring the bound.Source: §2.5 · pp. 40–41
  • Proposition 2.23 establishes a continuous viscosity solution with polynomial growth, uniqueness on finite time intervals, and a stochastic representation via the Feynman-Kac formula.The result applies to continuous, at most polynomially growing initial data under the stated coefficient assumptions.Source: §2.5 · pp. 42, 43, 45, 46
  • For affine drift and diffusion coefficients, Corollary 2.24 specializes these results to a unique continuous viscosity solution with polynomial growth and the corresponding SDE-based representation.The affine identities for µ and σ are the key additional assumption.Source: §2.5 · pp. 46–47

3 Artificial neural network approximations

This section develops the ANN approximation framework for Black–Scholes PDEs, including representation properties, constructions, and cost estimates for continuous activations. It establishes approximation results with parameter counts growing polynomially in the PDE dimension and inverse approximation accuracy.

3.1 Construction of a realization on the artificial probability space

  • Theorem 3.14 establishes that the ANN parameter count grows at most polynomially in both the reciprocal accuracy ε^-1 and PDE dimension d.The proof uses an artificial probability space to show that a realization with the required approximation properties exists.Source: §3.1 · p. 47

3.2 Approximation error estimates

  • Under polynomial-growth and regularity assumptions, Proposition 3.4 and Corollary 3.5 provide approximation error estimates for ANN-based representations of the PDE solution.They assume a unique continuous viscosity solution with polynomial growth and establish the existence of finite neural-network parameters satisfying the approximation requirements.Source: §3.2 · pp. 49, 51, 57

3.3 Cost estimates

  • Under its stated assumptions, Proposition 3.6 considers n ∈ N with n at least c^2C^2ε^-2 and states existence of a unique continuous function plus matrices and vectors.Source: §3.3 · p. 61

3.4 Representation properties for artificial neural networks

  • Lemma 3.8 shows how to construct a fully connected feedforward ANN representing combinations of n copies of a given network while controlling the resulting parameter count.The constructed network preserves the input-output dimension and provides explicit parameter-count bounds in terms of n and the original network size.Source: §3.4 · p. 63
  • The representation argument proves Lemma 3.8 by inductively establishing the required layerwise identities for the constructed artificial neural network.The proof uses the base case and induction step to complete the network representation.Source: §3.4 · pp. 65, 67

3.5 Cost estimates for artificial neural networks

  • P(ψε) ≤C a bz c4 ε−4−z and P(ψε) ≤C a bz c2 ε−2−z, while R(ψε) ∈Md, under polynomial-growth and linear-growth assumptions.Proposition 3.10 constructs approximating networks for every ε ∈(0, R] and propagates the complexity of the initial payoff approximation.Source: §3.5 · p. 70
  • For every dimension d, Corollary 3.13 provides unique continuous PDE solutions and networks (ψd,ε)d∈N, ε∈(0,R] with dimension- and accuracy-dependent complexity bounds.The result applies uniformly across the family of dimensions under the stated growth and approximation assumptions.Source: §3.5 · p. 77

3.6 Artificial neural networks with continuous activation functions

  • Theorem 3.14 establishes that fully-connected ANNs overcome the curse of dimensionality for Black-Scholes PDE approximation measured in Lp-norms under general probability measures νd.The theorem is specialized later to continuous uniform distributions on [0,1]^d.Source: §3.6 · p. 78
  • P(ψd,ε) ≤C d(5+z)θ+z+wz+4v ε−4−z and P(ψd,ε) ≤ C d(5+z)θ+z+wz+2v ε−2−z, with R(ψd,ε) ∈C(Rd, R).These networks use continuous activation functions and exist for all d ∈N and ε ∈(0, R].Source: §3.6 · pp. 79–80
  • The construction uses neural networks with continuous activation functions whose realizations belong to C(R^d,R), under the stated assumptions on the PDE data and activation-network complexity.The assumptions include continuous coefficient and initial-data functions and polynomial parameter bounds for the approximating activation networks.Source: §3.6 · pp. 81–83
  • The associated functions u_d: [0,T] × R^d → R exist uniquely, are continuous, and satisfy the initial condition u_d(0,x) = ϕ_d(x).The result applies simultaneously for all d ∈ N.Source: §3.6 · p. 82
  • For every dimension d and accuracy ε ∈ (0,1], an ANN ψ_d,ε has P(ψ_d,ε) ≤ C d^C ε^-C and continuous realization R(ψ_d,ε).The constant C is independent of d and ε.Source: §3.6 · p. 83

4 Artificial neural network approximations for Black-Scholes partial differential equations

This section develops ANN approximation results for Black–Scholes PDEs with basket call, basket put, call-on-max, and call-on-min payoffs. It establishes viscosity-solution constructions with parameter bounds polynomial in the dimension and reciprocal approximation accuracy.

4.1 Elementary properties of the Black-Scholes model

  • Under bounded coefficients, the Black-Scholes model uses componentwise linear drift and diagonal multiplicative volatility, with the payoff map given by coordinatewise positive parts.The setting assumes uniformly bounded α_d,i and β_d,i and defines μ_d(x) = (α_d,1x_1, ..., α_d,dx_d) and σ_d(x) = diag(β_d,1x_1, ..., β_d,dx_d)B_d.Source: §4.1 · pp. 84–85

4.2 Transformations of viscosity solutions

  • Time reversal preserves viscosity-solution structure, and Corollary 4.5 makes terminal-value and initial-value formulations equivalent under the stated continuity, uniqueness, and growth conditions.The transformation maps u(t, x) to U(t, x) = u(a + b − t, x) and establishes the corresponding transformed PDE solution.Source: §4.2 · pp. 88, 90, 91, 92, 93

4.3 Artificial neural network approximations for basket call options

  • The basket-call payoff max{c_d,1x_1 + ... + c_d,dx_d − K_d, 0} is represented exactly by a three-layer feedforward ANN using at most 4d parameters.Lemma 4.6 gives P(φ_d) = d + 3 ≤ 4d.Source: §4.3 · p. 94
  • For ε ∈ (0, 1], continuous approximations satisfy parameter-count bounds P(ψd,ε) ≤ C d5θ+1 ε−4 and P(ψd,ε) ≤ C d5θ+1 ε−2.The associated functions ud are unique and continuous, with terminal payoff ud(T, x) = max{cd,1x1 + cd,2x2 + . . . + cd,dxd − Kd, 0}.Source: §4.3 · pp. 94–95
  • Basket call options admit ANN approximations with parameter and computational bounds C d^5θ+1 ε^-4 and C d^5θ+1 ε^-2.The associated Black–Scholes solution is unique and continuous, with a continuous ANN realization for every dimension d and accuracy ε.Source: §4.3 · pp. 96–97

4.4 Artificial neural network approximations for basket put options

  • For basket put options, ANN approximations overcome the curse of dimensionality, with payoff-network size P(φd) ≤ 4d and bounds C d5θ+1 ε−4 and C d5θ+1 ε−2.Source: §4.4 · pp. 97, 98, 100

4.5 Artificial neural network approximations for call on max options

  • For call-on-max payoffs, the cited constructions use fully connected feedforward networks with d + 2 layers, at most 6d^3 parameters, and bounds C d^5θ+3 ε^-4 and C d^5θ+3 ε^-2.Source: §4.5 · pp. 102, 106, 107
  • The passages state that u_d restricted to (0,T) × R^d is a viscosity solution for every d ∈ N, and that an additional condition holds for every ε ∈ (0,1].Source: §4.5 · p. 108

4.6 Artificial neural network approximations for call on min options

  • The call-on-min payoff admits a fully connected feedforward ANN with d + 2 layers and at most 6d^3 parameters.Its realization is continuous and represents the payoff exactly before the PDE approximation step.Source: §4.6 · pp. 109, 113
  • For call-on-min options, Proposition 4.15 establishes unique continuous viscosity solutions and ANN approximations with parameter bounds C d^5θ+3 ε^-4 and C d^5θ+3 ε^-2.The terminal payoff is max{min{cd,1x1, cd,2x2, . . . , cd,dxd} −Kd, 0}, under uniformly bounded coefficients cd,i.Source: §4.6 · pp. 113–116

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