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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
TL;DR
The paper addresses limited rigorous evidence that ANNs overcome the curse of dimensionality for high-dimensional function approximation. It proves that ANNs approximate Black-Scholes PDE solutions with parameter counts growing polynomially in dimension and reciprocal accuracy, under stated coefficient-growth assumptions.
Problem
Only a few mathematical results rigorously explain ANNs' empirical success in approximating high-dimensional functions, including solutions of Black-Scholes PDEs.
Method
The paper proves approximation results for fully connected ANNs using stochastic differential-equation estimates and an artificial probability space to construct a suitable realization.
Results
The required ANN parameters grow at most polynomially in PDE dimension d and reciprocal approximation accuracy ε^-1, establishing that ANNs overcome the curse of dimensionality for Black-Scholes PDEs.
Takeaways & Limitations
The theorem provides a partial theoretical justification for deep-learning algorithms that approximate solutions of Black-Scholes PDEs.
Takeaways & Limitations
The result applies under stated growth conditions on the coefficient functions and related assumptions.
Abstract
from arXiv · showhide
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 addresses the limited rigorous evidence explaining ANN success on high-dimensional functions by proving efficient approximation for Black-Scholes PDE solutions. Its main result shows that required ANN parameters grow polynomially with dimension and reciprocal accuracy, thereby overcoming the curse of dimensionality.
- ANNs have been successfully applied across computational problems, including numerical approximations of partial differential equations.
- Existing ANN theory establishes convergence, sometimes with rates, but only rarely explains empirical success for high-dimensional functions.
- The main contribution proves that ANN parameter counts for approximating Black-Scholes PDE solutions grow at most polynomially in dimension d and reciprocal accuracy ε.The result applies more generally to Kolmogorov PDEs with affine linear drift and diffusion functions.
- P(ψd,ε) ≤ C d^C ε^-C expresses the polynomial bound on the number of ANN parameters for dimension d and accuracy ε.Here P(Φ) denotes the number of parameters describing the network.
- Theorem 1.1 states that Black-Scholes PDE solutions can be approximated on the unit cube by ANNs without the curse of dimensionality.The paper presents this as a partial theoretical justification of deep-learning algorithms for these PDEs.
- The proof combines the Feynman-Kac formula, Monte Carlo expected-value approximations, affine SDE dependence, and a probabilistic existence argument.The existence of a suitable realization on an artificial probability space bridges the probabilistic proof and deterministic conclusion.
2 Probabilistic and analytic preliminaries
This section develops auxiliary probabilistic and analytic results used for the paper’s PDE approximation arguments, covering Monte Carlo estimates, affine functions, SDEs, and viscosity solutions.
- The section assembles preliminary results on Monte Carlo approximations, affine functions, stochastic differential equations, and viscosity solutions for PDEs.
- Monte Carlo approximations: Kahane–Khintchine-type estimates are used to present an Lp Monte Carlo estimate for i.i.d. random variables in Euclidean spaces.
- Affine functions: Affine-function results characterize affine maps and establish linear-growth and Lipschitz bounds for them.
- Stochastic differential equations: An a priori estimate is established for SDE solutions whose coefficient functions grow at most linearly.
- Stochastic differential equations with affine coefficient functions: The section establishes regularity properties for SDEs with affine coefficient functions using moment estimates, stochastic-process results, and a version of the Kolmogorov–Chentsov theorem.
3 Artificial neural network approximations
The paper constructs ANN approximations for solutions of high-dimensional Black-Scholes-related PDEs and proves that their parameter complexity grows only polynomially with dimension and inverse accuracy.
- Construction of a realization: The proof uses an artificial probability space to establish a realization with the desired ANN approximation properties.This realization argument is identified as an important ingredient in proving the main approximation theorem.
- Complexity bounds: The ANN parameter bounds are polynomial in the reciprocal accuracy ε and the PDE dimension d.The specialized theorem gives a bound of the form P(ψd,ε) ≤ C d^C ε^-C for ε ∈ (0,1].
- PDE solutions: Unique continuous viscosity solutions are established for the considered PDEs under polynomial-growth conditions on the data and affine conditions on drift and diffusion.The results include existence, uniqueness, continuity, initial-value satisfaction, and viscosity-solution properties.
- Main approximation result: The main theorem provides dimension- and accuracy-dependent parameter estimates for ANN approximations measured in Lp-norms with respect to general probability measures.A specialized corollary treats the continuous uniform distribution on the unit cube [0,1]^d.
- Conclusion: The result proves that fully connected ANNs overcome the curse of dimensionality for numerical approximation of Black-Scholes PDEs.The paper presents this as a partial theoretical justification for the performance of deep-learning algorithms applied to such PDEs.
4 Artificial neural network approximations for Black-Scholes partial differential equations
The section establishes ANN approximation results for Black-Scholes PDEs under bounded coefficient settings, covering basket, max, and min option payoffs. The resulting parameter bounds grow polynomially with dimension and reciprocal accuracy, demonstrating that these cases overcome the curse of dimensionality.
- 4 Artificial neural network approximations for Black-Scholes partial differential equations: The Black-Scholes setting uses coordinatewise linear drift and diagonal multiplicative volatility with uniformly bounded coefficients.The activation map is coordinatewise ReLU, A_d(x)=(max{x_1,0},…,max{x_d,0}).
- 4 Artificial neural network approximations for Black-Scholes partial differential equations: The results establish polynomial dependence on dimension and reciprocal approximation accuracy for ANN approximations of the covered Black-Scholes option classes.The propositions separately cover basket calls, basket puts, calls on maxima, and calls on minima.
- 4 Artificial neural network approximations for Black-Scholes partial differential equations: Basket call and put option solutions admit ANN approximations with parameter bounds C d^(5θ+1) ε^-4 and C d^(5θ+1) ε^-2.The constructions also produce continuous realizations R(ψ_d,ε)∈C(R^d,R).
- 4 Artificial neural network approximations for Black-Scholes partial differential equations: The option-value functions are established as unique continuous functions satisfying the relevant terminal conditions and viscosity-solution formulation of the Black-Scholes PDE.The call-on-min construction explicitly identifies the terminal payoff as ϕ_d(x).
- 4 Artificial neural network approximations for Black-Scholes partial differential equations: Call-on-max and call-on-min option solutions admit ANN approximations with parameter bounds C d^(5θ+3) ε^-4 and C d^(5θ+3) ε^-2.For call-on-min options, the terminal payoff is max{min{c_d,1x_1,…,c_d,dx_d}−K_d,0}.