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.