Bernt Øksendal

We construct a white noise theory and white noise calculus for the (multi-parameter) L\' evy sheet and its compensated Poisson random measures. The theory applies to stochastic partial differential equations subject to L\' evy noise.

This paper studies a nonzero-sum stochastic differential game in the context of shared spatial-domain pollution control. The pollution dynamics are governed by a stochastic partial differential equation (SPDE) driven by a Brownian sheet, capturing the stochastic nature of environmental fluctuations. Two players, representing different regions, aim to minimize their respective cost functionals, which balance pollution penalties with the cost of implementing control strategies. The nonzero-sum framework reflects the interdependent yet conflicting objectives of the players, where both cooperation and competition influence the outcomes. We derive necessary and sufficient conditions for Nash equilibrium strategies, using a maximum principle approach. This approach involves the introduction of a new pair of adjoint variables, (L_1, L_2), which do not appear in a corresponding formulation with the classical (1-parameter) Brownian motion. Finally, we apply our results to two case studies in pollution control, demonstrating how spatial and stochastic dynamics shape the equilibrium strategies.

In this paper, we study the optimal control of systems where the state dynamics are governed by a stochastic partial differential equation (SPDE) driven by a two-parameter (time-space) Brownian motion, also referred to as a Brownian sheet. These equations can, for example, model the growth of an ecosystem under uncertainty. We first explore some fundamental properties of such linear SPDEs. Next, utilizing time-space white noise calculus, we derive both Pontryagin-type necessary and sufficient conditions for the optimality of the control. Finally, we illustrate our results by solving a linear-quadratic control problem and examining an optimal harvesting problem in the plane. Potential applications to machine learning and to managing random environmental influences are also discussed.

We investigate conditional McKean-Vlasov equations driven by time-space white noise, motivated by the propagation of chaos in an N-particle system with space-time Ornstein-Uhlenbeck dynamics. The framework builds on the stochastic calculus of time-space white noise, utilizing tools such as the two-parameter Ito formula, Malliavin calculus, and orthogonal decompositions to analyze convergence and stochastic properties. Existence and uniqueness of solutions for the associated stochastic partial differential equations (SPDEs) are rigorously established. Additionally, an integral stochastic Fokker-Planck equation is derived for the conditional law, employing Fourier transform methods and stochastic analysis in the plane. The framework is further applied to a partial observation control problem, showcasing its potential for analyzing stochastic systems with conditional dynamics.

We study a linear filtering problem where the signal and observation processes are described as solutions of linear stochastic differential equations driven by time-space Brownian sheets. We derive a stochastic integral equation for the conditional value of the signal given the observation, which can be considered a time-space analogue of the classical Kalman filter. The result is illustrated with examples of the filtering problem involving noisy observations.

We propose a deep learning approach to study the minimal variance pricing and hedging problem in an incomplete jump diffusion market. It is based on 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 feed-forward 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.