Solving the Kolmogorov PDE by means of deep learning

• SDEs and their associated Kolmogorov PDEs are widely used in engineering, finance, and natural sciences, including approximative financial-derivative pricing, but typically cannot be solved explicitly.Source: §1 · p. 2
• Deterministic PDE methods work efficiently in one or two dimensions but suffer from the curse of dimensionality, while Monte Carlo methods typically approximate only one fixed space-time point.Source: §1 · p. 3
• The proposed method aims to approximate the Kolmogorov PDE across [a, b]^d without suffering from the curse of dimensionality, and numerical results indicate high-dimensional effectiveness in accuracy and speed.Source: §1 · p. 3

• Section 2 derives the proposed approximation algorithm and describes it in special and general cases.Source: §2 · p. 4
• The section derives and specifies a numerical scheme for the approximation problem, alongside related derivations and schemes.Source: §2 · p. 4

• The PDE setup assumes Lipschitz continuous drift and diffusion functions and a solution with at most polynomially growing partial derivatives.Source: §2.1 · p. 4
• The approximation target is the function x ↦ u(T, x) on [a, b]^d.Source: §2.1 · p. 4

• The section recalls the classical Feynman–Kac formula as a probabilistic representation for solutions of the PDE.Source: §2.2 · p. 4
• The setup introduces a filtered probability space, Brownian motion, and adapted continuous-path processes indexed by initial state x.Source: §2.2 · p. 4
• The Feynman–Kac formula together with the PDE yields a relation that holds for every x.Source: §2.2 · p. 4

2.3 Formulation as minimization problem

• The proposed minimization problem has the function x ↦ u(T, x) as its unique global minimizer on [a, b]^d.Proposition 2.7 applies the random-field minimization result in the context of the Feynman–Kac representation.Source: §2.3 · pp. 5, 17
• The L2-minimization property for real-valued random variables is extended to random fields, yielding a unique continuous function equal to their pointwise expectations.Proposition 2.2 assumes finite second moments and continuity of the expectation function.Source: §2.3 · pp. 5–7
• The proof of Proposition 2.7 relies on auxiliary results in Lemmas 2.3–2.6, including joint measurability and stochastic-process estimates.The section also includes the proofs of those auxiliary lemmas.Source: §2.3 · pp. 5, 9, 11
• The unique continuous function U on [a, b]^d satisfies U(x)=u(T, x)=E[ϕ(X_T^x)].The equality to the stochastic expectation follows from the Feynman–Kac formula.Source: §2.3 · pp. 18–20
• The function V(x)=u(T, x) is the unique global minimizer of E[|ϕ(X_T)−V(ξ)|²].Here ξ is continuously uniformly distributed on [a, b]^d.Source: §2.3 · pp. 20–21
• The subsequent approximation discretizes X and uses a deep neural network to approximate x↦u(T, x).These steps are developed in Subsections 2.4 and 2.5, respectively.Source: §2.3 · p. 21

2.4 Discretization of the stochastic differential equation

• Euler–Maruyama is used to temporally discretize the SDE solution process.The scheme is defined on a time grid from 0 to T.Source: §2.4 · p. 21
• Theorem 2.8 gives a strong-convergence rate result for the Euler–Maruyama scheme under its stated assumptions.The preceding bound is stated to hold with a constant independent of N.Source: §2.4 · pp. 21–22
• The discretized process is specified by a stepwise stochastic update beginning from the initial random variable.The section relates this update to the SDE over each time interval.Source: §2.4 · pp. 21–22

2.5 Deep artificial neural network approximations

• A deep neural network is proposed to approximate u(T, x) for suitable parameters and inputs in [a, b]^d.The output is treated as an approximation of the PDE solution at time T.Source: §2.5 · p. 22
• The construction composes affine maps with a multidimensional logistic function across the network layers.The section gives the logistic function and parameterized affine-map construction.Source: §2.5 · pp. 22–23
• The described network uses logistic activation functions and has an input layer, hidden layers, and an output layer.The stated architecture has d neurons in the input and output layers and d neurons in each hidden layer.Source: §2.5 · p. 23

2.6 Stochastic gradient descent-type minimization

• The algorithm seeks parameters that approximately minimize the objective defining the network approximation to u(T, x).The network output is intended to approximate the terminal-time PDE solution across [a, b]^d.Source: §2.6 · pp. 23–24
• A stochastic gradient descent-type algorithm uses Monte Carlo to approximate the derivative of the objective.The objective is the function introduced in (107).Source: §2.6 · p. 23
• For sufficiently large iteration indices, the parameters are regarded as approximations of local minima and the resulting network as an approximation of u(T, x).The statement is conditional on appropriate hypotheses.Source: §2.6 · p. 24

2.7 Description of the algorithm in a special case

• The special case chooses a particular neural-network approximation and plain-vanilla stochastic gradient descent with a constant learning rate.The section refers readers to Subsection 2.8 for the more general description.Source: §2.7 · p. 24
• The algorithm samples independent initial points uniformly from [a,b]^d and uses independent Brownian motions to define its stochastic processes.The framework specifies a time grid from 0 to T.Source: §2.7 · p. 24
• The network composes affine maps Aθ,v with Ld: Rd→Rd, the componentwise logistic map that sends each coordinate x_i to exp(x_i)/(exp(x_i)+1).Source: §2.7 · pp. 24–25
• For sufficiently large m, U(Θm,x) is presented as an approximation of u(T,x) for every x in [a,b]^d, under appropriate hypotheses.The solution u is assumed to have at most polynomially growing partial derivatives.Source: §2.7 · p. 25

2.8 Description of the algorithm in the general case

• The framework covers the methods derived earlier and allows minimization algorithms beyond plain-vanilla stochastic gradient descent.The paper describes it as an extension of E et al., Han et al. [24], and Beck et al. [3] for linear Kolmogorov PDEs.Source: §2.8 · p. 25
• Framework 2.10 includes a time grid, stochastic processes X, and parameter and auxiliary processes Θ, S, and Ξ.It defines update functions for the parameter and auxiliary processes.Source: §2.8 · pp. 25–26
• For sufficiently large m, UΘm,1,Sm(x) is presented as an approximation of u(T,x) for every x in [a,b]^d, under appropriate hypotheses.The solution is assumed to have at most polynomially growing partial derivatives, and the coefficients µ and σ are sufficiently regular.Source: §2.8 · p. 26

3.1 Setting

• The shared implementation uses a fully connected network with two hidden layers, batch normalization, tanh activations, Xavier initialization, and Adam with mini-batches of 8192.Computations use single precision (float32).Source: §3 · pp. 26–27

3.2 Heat equation

• For the 100-dimensional heat equation, the section reports relative L1, L2, and L∞ errors, along with errors averaged over randomness, across training iterations.The heat-equation errors are estimated using Monte Carlo approximations; the cited passages describe the tables and plots but provide no numerical error values.Source: §3.2 · p. 28, Table 1 · p. 29, Table 2 · p. 31

3.3 Geometric Brownian motions

• For a Black–Scholes PDE with independent geometric Brownian motions, the experiments report relative L1, L2, and L∞ approximation errors across training iterations.The setup uses dimension 100 and Monte Carlo approximations; the supplied passages do not include the error values.Source: §3.3 · p. 36, Table 3 · p. 37

3.4 Black-Scholes model with correlated noise

• For a 100-dimensional Black–Scholes PDE with correlated noise, the experiments report relative L1, L2, and L∞ approximation errors across training iterations.The described noise correlation matrix has diagonal entries 1 and off-diagonal entries 1/2; the supplied passages do not include error values.Source: §3.4 · pp. 37–38, Table 4 · p. 39

3.5 Stochastic Lorenz equations

• The section applies the approximation algorithm to a stochastic Lorenz equation with dimension 3 and 100 time steps, using drift parameters α1 = 10, α2 = 14, and α3 = 8.The supplied passages describe the setup but provide no Lorenz-result values.Source: §3.5 · p. 39
• For the stochastic Lorenz equation, Table 5 presents relative L1, L2, and L∞ approximation errors across m values from 0 to 750000.The exact PDE solution was approximated using Monte Carlo with 104857 samples and 100 time steps; error estimates used Monte Carlo approximations with 20480 samples.Source: §3.5 · pp. 39–40, Table 5 · p. 40

3.6 Heston model

• For the Heston model in dimension 50, Table 6 presents relative L1, L2, and L∞ approximation errors across m values from 0 to 750000.The exact PDE solution was approximated using Monte Carlo with 1048576 samples and 100 time steps; error estimates used Monte Carlo approximations with 10240 samples.Source: §3.6 · pp. 40–42, Table 6 · p. 42

4.1 Python source code for the algorithm

• The shared implementation defines a batch-normalized neural network and a training/testing routine that periodically estimates errors by Monte Carlo rounds.The routine tracks absolute and relative L1, L2, and L-infinity errors.Source: §4.1 · pp. 43–45

4.2 A Python source code associated to the numerical simulations in Subsection 3.2

• The Subsection 3.2 example script imports the shared training/testing routine and calls it with example-specific inputs and training settings.It writes results to example3_1.csv.Source: §4.2 · pp. 45–46

4.3 A Python source code associated to the numerical simulations in Subsection 3.3

• The Subsection 3.3 script sets up a discounted maximum-based payoff, simulates terminal asset values, and passes the resulting inputs to the shared routine.The payoff uses the maximum across coordinates and strike K.Source: §4.3 · pp. 46–47

4.4 A Python source code associated to the numerical simulations in Subsection 3.4

• The Subsection 3.4 script calls kolmogorov_train_and_test with the SDE input, payoff, reference solution, training parameters, and example3_3.csv output.The supplied excerpt shows the call and output filename, but not the preceding simulation implementation.Source: §4.4 · p. 48

4.5 A Python source code associated to the numerical simulations in Subsection 3.5

• The Subsection 3.5 script sets Lorenz parameters alpha_1=10, alpha_2=14, alpha_3=8/3 and simulates paths before invoking the shared training-and-testing routine.The code defines the drift components and advances the process with sampled Gaussian increments; the training call writes example3_4.csv.Source: §4.5 · pp. 48–50

4.6 A Python source code associated to the numerical simulations in Subsection 3.6

• The Subsection 3.6 script simulates coupled state and variance processes with correlated Brownian increments, then passes the simulated state to kolmogorov_train_and_test.The listing sets alpha=0.05 and K=110, and writes the training output to example3_5.csv.Source: §4.6 · pp. 50–51