Deep ReLU network expression rates for option prices in high-dimensional, exponential Lévy models

Abstract

We study the expression rates of deep neural networks (DNNs for short) for option prices written on baskets of $d$ d risky assets whose log-returns are modelled by a multivariate Lévy process with general correlation structure of jumps. We establish sufficient conditions on the characteristic triplet of the Lévy process $X$ X that ensure $\varepsilon $ ε error of DNN expressed option prices with DNNs of size that grows polynomially with respect to ${\mathcal{O}}(\varepsilon ^{-1})$ O ( ε − 1 ) , and with constants implied in ${\mathcal{O}}(\, \cdot \, )$ O ( ⋅ ) which grow polynomially in $d$ d , thereby overcoming the curse of dimensionality (CoD) and justifying the use of DNNs in financial modelling of large baskets in markets with jumps.In addition, we exploit parabolic smoothing of Kolmogorov partial integro-differential equations for certain multivariate Lévy processes to present alternative architectures of ReLU (“rectified linear unit”) DNNs that provide $\varepsilon $ ε expression error in DNN size ${\mathcal{O}}(|\log (\varepsilon )|^{a})$ O ( | log ( ε ) | a ) with exponent $a$ a proportional to $d$ d , but with constants implied in ${\mathcal{O}}(\, \cdot \, )$ O ( ⋅ ) growing exponentially with respect to $d$ d . Under stronger, dimension-uniform non-degeneracy conditions on the Lévy symbol, we obtain algebraic expression rates of option prices in exponential Lévy models which are free from the curse of dimensionality. In this case, the ReLU DNN expression rates of prices depend on certain sparsity conditions on the characteristic Lévy triplet. We indicate several consequences and possible extensions of the presented results.

Deep ReLU neural network approximation in Bochner spaces and applications to parametric PDEs

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Lower bounds for artificial neural network approximations: A proof that shallow neural networks fail to overcome the curse of dimensionality

“…In particular, the results in such articles show that deep ANNs have the capacity to overcome the curse of dimensionality in the approximation of certain target function classes in the sense that the number of parameters of the approximating ANNs grows at most polynomially in the dimension d ∈ N of the target functions under considerations. For example, we refer to Elbrächter et al [15], Jentzen et al [33], Gonon et al [20,21], Grohs et al [22,23,25], Kutyniok et al [43], Reisinger & Zhang [49], Beneventano et al [6], Berner et al [7], Hornung et al [31], Hutzenthaler et al [32], and the overview articles Beck et al [4] and E et al [13] for such high-dimensional ANN approximation results in the numerical approximation of solutions of PDEs and we refer to Barron [1][2][3], Jones [34], Girosi & Anzellotti [19], Donahue et al [12], Gurvits & Koiran [28], Kůrková et al [39][40][41][42], Kainen et al [35,36], Klusowski & Barron [38], Li et al [45], and Cheridito et al [9] for such high-dimensional ANN approximation results in the numerical approximation of certain specific target function classes independent of solutions of PDEs (cf., e.g., also Maiorov & Pinkus [46], Pinkus [48], Guliyev & Ismailov [26], Petersen & Voigtlaender [47], and Bölcskei et al [8] for related results). In the proofs of several of the above named high-dimensional approximation results it is crucial that the involved ANNs ar...…”

High-dimensional approximation spaces of artificial neural networks and applications to partial differential equations

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