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Deep learning algorithms have been successfully applied to numerically solvelinear Kolmogorov partial differential equations (PDEs). A recent researchshows that the empirical risk minimization~(ERM) over deep artificial neuralnetworks overcomes the curse of dimensionality in the numerical approximationof linear Kolmogorov PDEs with bounded initial functions. However, the initialfunctions may be unbounded in many applications such as the Black Scholes PDEsin pricing call options. In this paper, we extend this result to the casesinvolving unbounded initial functions. We prove that for $d$-dimensional linearKolmogorov PDEs with unbounded initial functions, under suitable assumptions,the number of training data and the size of the artificial neural networkrequired to achieve an accuracy $\varepsilon$ for the ERM grow polynomially inboth $d$ and $\varepsilon^{-1}$. Moreover, we verify that the requiredassumptions hold for Black-Scholes PDEs and heat equations which are twoimportant cases of linear Kolmogorov PDEs.

Empirical Risk Minimization over Artificial Neural Networks Overcomes  the Curse of Dimensionality in the Numerical Approximation of Linear  Kolmogorov Partial Differential Equations with Unbounded Initial Functions