We consider an infinite horizon, obliquely reflected backward stochastic differential equation (RBSDE). The main contribution of the present work is that we generalize previous results on infinite horizon RBSDEs to the setting where the driver has a stochastic Lipschitz coefficient. As an application, we consider robust optimal stopping problems for functional stochastic differential equations (FSDEs) where the driver has linear growth.

We propose a neural network using an unsupervised learning strategy for direct computation of closest saddle- node bifurcations, eliminating the need for labeled training data. Our method not only estimates the worst-case load increase scenarios but also significantly reduces the computational complexity traditionally associated with this task during inference time. Simulation results validate the effectiveness and real-time applicability of our approach, demonstrating its potential as a robust tool for modern power system analysis.

We consider impulse control of stochastic functional differential equations (SFDEs) driven by Lévy processes under an additional Lp-Lipschitz condition on the coefficients. Our results, which are first derived for a general stochastic optimization problem over infinite horizon impulse controls and then applied to the case of a controlled SFDE, apply to the infinite horizon as well as the random horizon settings. The methodology employed to show existence of optimal controls is a probabilistic one based on the concept of Snell envelopes.

In this paper, we introduce a non-linear Snell envelope which at each time represents the maximal value that can be achieved by stopping a BSDE with constrained jumps. We establish the existence of the Snell envelope by employing a penalization technique and the primary challenge we encounter is demonstrating the regularity of the limit for the scheme. Additionally, we relate the Snell envelope to a finite horizon, zero-sum stochastic differential game, where one player controls a path-dependent stochastic system by invoking impulses, while the opponent is given the opportunity to stop the game prematurely. Importantly, by developing new techniques within the realm of control randomization, we demonstrate that the value of the game exists and is precisely characterized by our non-linear Snell envelope.

Optimal Power Flow is a central tool for power system operation and planning. Given the substantial rise in intermittent power and shorter time windows in electricity markets, there’s a need for fast and efficient solutions to the Optimal Power Flow problem. With this in consideration, this paper propose an unsupervised deep learning approach to approximate the optimal solution of Optimal Power Flow problems. Once trained, deep learning models benefit from being several orders of magnitude faster during inference compared to conventional non-linear solvers.

In this work we show that one can solve a finite horizon non-Markovian impulse control problem with control dependent dynamics. This dynamic satisfies certain functional Lipschitz conditions and is path dependent in such a way that the resulting trajectory becomes a flow.

We consider a general type of non-Markovian impulse control problems under adverse non-linear expectation or, more specifically, the zero-sum game problem where the adversary player decides the probability measure. We show that the upper and lower value functions satisfy a dynamic programming principle (DPP). We first prove the dynamic programming principle (DPP) for a truncated version of the upper value function in a straightforward manner. Relying on a uniform convergence argument then enables us to show the DPP for the general setting. Following this, we use an approximation based on a combination of truncation and discretization to show that the upper and lower value functions coincide, thus establishing that the game has a value and that the DPP holds for the lower value function as well. Finally, we show that the DPP admits a unique solution and give conditions under which a saddle point for the game exists. As an example, we consider a stochastic differential game (SDG) of impulse versus classical control of path-dependent stochastic differential equations (SDEs).

We consider quasi-variational inequalities (QVIs) with general non-local drivers and related systems of reflected backward stochastic differential equations (BSDEs) in a Brownian filtration. We show existence and uniqueness of viscosity solutions to the QVIs by first considering the standard (local) setting and then applying a contraction argument. In addition, the contraction argument yields existence and uniqueness of solutions to the related systems of reflected BSDEs and extends the theory of probabilistic representations of PDEs in terms of BSDEs to our specific setting.

We formulate the day-ahead bidding problem for a hydropower producer having several hydropower plants residing in a river basin. We present a novel approach inspired by Dynamic programming with approximations in value and policy space by neural networks. This allows for more accurate modeling of the problem by avoiding linear approximations of the production function and bidding. Stochastic programming is a method frequently used in literature to solve the hydropower production planning problem. Stochastic programming is used on linearized systems and under assumptions of known distributions of the involved stochastic processes. We test the proposed algorithm on a simplified system, suitable for Stochastic Programming and compare the obtained policy with the results from Stochastic Programming. The results show that the algorithm obtains a policy similar to that of Stochastic Programming.

We consider a stochastic differential game in the context of forward-backward stochastic differential equations, where one player implements an impulse control while the opponent controls the system continuously. Utilizing the notion of "backward semigroups" we first prove the dynamic programming principle (DPP) for a truncated version of the problem in a straightforward manner. Relying on a uniform convergence argument then enables us to show the DPP for the general setting. Our approach avoids technical constraints imposed in previous works dealing with the same problem and, more importantly, allows us to consider impulse costs that depend on the present value of the state process in addition to unbounded coefficients. Using the dynamic programming principle we deduce that the upper and lower value functions are both solutions (in viscosity sense) to the same Hamilton-Jacobi-Bellman-Isaacs obstacle problem. By showing uniqueness of solutions to this partial differential inequality we conclude that the game has a value.(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).