A method based on deep artificial neural networks and empirical risk minimization is developed to calculate the boundary separating the stopping and continuation regions in optimal stopping. The algorithm parameterizes the stopping boundary as the graph of a function and introduces relaxed stopping rules based on fuzzy boundaries to facilitate efficient optimization. Several financial instruments, some in high dimensions, are analyzed through this method, demonstrating its effectiveness. The existence of the stopping boundary is also proved under natural structural assumptions.

This paper outlines, and through stylized examples evaluates a novel and highly effective computational technique in quantitative finance. Empirical Risk Minimization (ERM) and neural networks are key to this approach. Powerful open source optimization libraries allow for efficient implementations of this algorithm making it viable in high-dimensional structures. The free-boundary problems related to American and Bermudan options showcase both the power and the potential difficulties that specific applications may face. The impact of the size of the training data is studied in a simplified Merton type problem. The classical option hedging problem exemplifies the need of market generators or large number of simulations.

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We develop an Itô calculus for functionals of the "time" spent by a path at arbitrary levels. A Markovian setting is recovered by lifting a process X with its flow of occupation measures O and call the pair (O, X) the occupied process. While the occupation measure erases the chronology of the path, we show that our framework still includes many relevant problems in stochastic analysis and financial mathematics. The study of occupied processes therefore strikes a middle ground between the path-independent case and Dupire's Functional Itô Calculus. We extend Itô's and Feynman-Kac's formula by introducing the occupation derivative, a projection of the functional linear derivative used extensively in mean field games and McKean-Vlasov optimal control. Importantly, we can recast through Feynman-Kac's theorem a large class of path-dependent PDEs as parabolic problems where the occupation measure plays the role of time. We apply the present tools to the optimal stopping of spot local time and discuss financial examples including exotic options, corridor variance swaps, and path-dependent volatility.