  We introduce a new Langevin dynamics based algorithm, called
e-TH$\varepsilon$O POULA, to solve optimization problems with discontinuous
stochastic gradients which naturally appear in real-world applications such as
quantile estimation, vector quantization, CVaR minimization, and regularized
optimization problems involving ReLU neural networks. We demonstrate both
theoretically and numerically the applicability of the e-TH$\varepsilon$O POULA
algorithm. More precisely, under the conditions that the stochastic gradient is
locally Lipschitz in average and satisfies a certain convexity at infinity
condition, we establish non-asymptotic error bounds for e-TH$\varepsilon$O
POULA in Wasserstein distances and provide a non-asymptotic estimate for the
expected excess risk, which can be controlled to be arbitrarily small. Three
key applications in finance and insurance are provided, namely, multi-period
portfolio optimization, transfer learning in multi-period portfolio
optimization, and insurance claim prediction, which involve neural networks
with (Leaky)-ReLU activation functions. Numerical experiments conducted using
real-world datasets illustrate the superior empirical performance of
e-TH$\varepsilon$O POULA compared to SGLD, TUSLA, ADAM, and AMSGrad in terms of
model accuracy.
