arXiv subjects

Jan Rems

Publications and source records attributed to Jan Rems.

5 recordsLinked to original sources

arXiv2026

A cylindrical neural approximation theorem for conditional laws of McKean-Vlasov equations with common noise

We introduce conditional cylindrical neural networks for approximating functionals of conditional laws in McKean-Vlasov equations with common noise. Fourier moments of the initial law and truncated signatures of the time augmented common noise are mapped by a mixture density network to a Gaussian mixture approximation of the conditional law. A cylindrical neural network then evaluates the target functional through analytic integrals against this predicted measure. Rough path well posedness and stability provide a conditional law map that is continuous in the initial distribution and the rough driver and agrees almost surely with the classical conditional law at the It\^o Brownian lift. Combining this continuity with Fourier separation, signature uniqueness, Wasserstein density of Gaussian mixtures, and neural universal approximation, we prove an $L^2$ universal approximation theorem for continuous square integrable functionals. The numerical study implements the resulting two stage procedure on six examples, including non Gaussian initial laws, nonlinear drift, multiplicative common noise, and a two dimensional state. Independent particle references are used when no closed form law is available. The learned conditional law and functional approximations consistently improve on the empirical particle plug in, and additional experiments examine feature sensitivity, training from one terminal observation per common noise scenario, and It\^o--Stratonovich consistency.

math.PR↗

arXiv2025

A Deep Learning Approach to Renewable Capacity Installation under Jump Uncertainty

We study a stochastic model for the installation of renewable energy capacity under demand uncertainty and jump driven dynamics. The system is governed by a multidimensional Ornstein-Uhlenbeck (OU) process driven by a subordinator, capturing abrupt variations in renewable generation and electricity load. Installation decisions are modeled through control actions that increase capacity in response to environmental and economic conditions. We consider two distinct solution approaches. First, we implement a structured threshold based control rule, where capacity is increased proportionally when the stochastic capacity factor falls below a fixed level. This formulation leads to a nonlinear partial integro-differential equation (PIDE), which we solve by reformulating it as a backward stochastic differential equation with jumps. We extend the DBDP solver in \cite{hure2020deep} to the pure jump setting, employing a dual neural network architecture to approximate both the value function and the jump sensitivity. Second, we propose a fully data driven deep control algorithm that directly learns the optimal feedback policy by minimizing the expected cost functional using neural networks. This approach avoids assumptions on the form of the control rule and enables adaptive interventions based on the evolving system state. Numerical experiments highlight the strengths of both methods. While the threshold based BSDE approach offers interpretability and tractability, the deep control strategy achieves improved performance through flexibility in capacity allocation. Together, these tools provide a robust framework for decision support in long term renewable energy expansion under uncertainty.

eess.SY↗

arXiv2025

Deep Learning for Energy Market Contracts: Dynkin Game with Doubly RBSDEs

We formulate a Contract for Difference (CfD) with early exit options as a two-player zero-sum Dynkin game, reflecting the strategic interaction between an electricity producer and a regulatory entity. The game incorporates penalties for early termination and mean-reverting price dynamics, with the value characterized through a doubly reflected backward stochastic differential equation (DRBSDE). To compute the contract value and optimal stopping strategies, we develop a neural solver that approximates the DRBSDE solution using a sequence of neural networks trained on simulated trajectories. The method avoids discretizing the state space, supports time-dependent barriers, and scales to high-dimensional settings. We establish a convergence result and test the method on two scenarios: a benchmark symmetric game in 20 dimensions, and a CfD model with 24-dimensional electricity prices representing multiple European zones. The results demonstrate that the proposed solver accurately captures the contract's value and optimal stopping regions, with consistent performance across dimensional settings.

math.OC↗

arXiv2024

SIG-BSDE for Dynamic Risk Measures

In this paper, we consider dynamic risk measures induced by backward stochastic differential equations (BSDEs). We discuss different examples that come up in the literature, including the entropic risk measure and the risk measure arising from the ambiguous interest rate problem. We develop a numerical algorithm for solving a BSDE using the backward Euler-Maruyama scheme and the universal approximation theorem for the signature of a path. We prove the convergence theorem and use the algorithm to solve some examples of dynamic risk measures induced by BSDEs. At last a deep learning approach is included for solving the ambiguous interest rate problem as well.

math.PR↗

arXiv2024

Deep learning for quadratic hedging in incomplete jump market

We propose a deep learning approach to study the minimal variance pricing and hedging problem in an incomplete jump diffusion market. It is based upon a rigorous stochastic calculus derivation of the optimal hedging portfolio, optimal option price, and the corresponding equivalent martingale measure through the means of the Stackelberg game approach. A deep learning algorithm based on the combination of the feedforward and LSTM neural networks is tested on three different market models, two of which are incomplete. In contrast, the complete market Black-Scholes model serves as a benchmark for the algorithm's performance. The results that indicate the algorithm's good performance are presented and discussed. In particular, we apply our results to the special incomplete market model studied by Merton and give a detailed comparison between our results based on the minimal variance principle and the results obtained by Merton based on a different pricing principle. Using deep learning, we find that the minimal variance principle leads to typically higher option prices than those deduced from the Merton principle. On the other hand, the minimal variance principle leads to lower losses than the Merton principle.

q-fin.TR↗