Stochastics and Applications

TBA

We shall consider the similar and different properties of two closely related stochastic processes, namely, Cox-Ingersoll-Ross and Bessel processes, both of them being strictly positive solutions of the respective stochastic differential equations. Strictly positive values make them convenient to model real processes in physics, biology, economics. For example, in finances they are used to forecast interest rates and in bond pricing models. In our research we combine the methods of stochastic analysis and methods based on the explicit formulas for probability distributions of CIR and Bessel processes.

With the massive integration of renewable energies (photovoltaic (PV) and wind power) into the power grid, new uncertainties are impacting the power balance. At the same time, advances in « smart » technologies and batteries offer new flexibilities with the possibility of controlling the consumption of a large number of electrical appliances (electric vehicle recharging, heat pumps, etc.). In this framework, a major technical challenge is to optimize the management of this large number of heterogeneous flexibilities distributed across the network to help in balancing the system. This constitutes a large scale optimization problem under uncertainties, which can benefit from mean-field approximations.

We establish a stochastic maximum principle for controlled stochastic differential equations with delay and control-dependent noise, without convexity assumptions on the control space. The cost functional depends on both present and delayed states, modeled via general finite measures. For measures with square-integrable densities, we employ infinite-dimensional reformulation and BSDE techniques; for general measures, we apply anticipated BSDEs and weak convergence methods. We further analyze the case of delay measures with $L^p$-densities ($p \in (1,2)$), deriving a generalized mild backward equation beyond Hilbertian settings.

The aim of this work is to determine the optimal cyber-security investment strategy for an entity subject to cyber-attacks. Inspired by the Gordon-Loeb model, we assume that the success rate of cyber-attacks depends on the vulnerability of the security system under threat, which can be reduced investing in security measures. We introduce a dynamic version of the Gordon-Loeb setting, by exploiting Hawkes stochastic processes to model the arrival of attacks. This stochastic framework is crucial to rapidly react to the random changes which characterize cyber-risk. The problem is framed as a Markovian 2-dimensional stochastic control problem with jumps and it is addressed using dynamic programming techniques. The optimal value is characterized by a partial integro-differential equation, which is solved numerically. The corresponding optimal strategy is, hence, explicitly obtained by differentiating the optimal value function.

We frame dynamic persuasion in a partial observation stochastic control game with an ergodic criterion. The Receiver controls the dynamics of a multidimensional unobserved state process. Information is provided to the Receiver through a device designed by the Sender that generates the observation process. The commitment of the Sender is enforced. We develop this approach in the case where all dynamics are linear and the preferences of the Receiver are linear-quadratic. We prove a verification theorem for the existence and uniqueness of the solution of the HJB equation satisfied by the Receiver’s value function. An extension to the case of persuasion of a mean field of interacting Receivers is also provided. We illustrate this approach in two applications: the provision of information to electricity consumers with a smart meter designed by an electricity producer; the information provided by carbon footprint accounting rules to companies engaged in a best-in-class emissions reduction effort. In the first application, we link the benefits of information provision to the mispricing of electricity production. In the latter, we show that when firms declare a high level of best-in-class target, the information provided by stringent accounting rules offsets the Nash equilibrium effect that leads firms to increase pollution to make their target easier to achieve. This is a joint work with: R. Aïd (Paris Dauphine), O. Bonesini (LSE) and G. Callegaro (Padova).

We investigate the optimal self-protection problem, from the point of view of an insurance buyer, when the loss process is described by a Cox-shot-noise process and a Hawkes process with an exponential memory kernel. The insurance buyer chooses both the percentage of insured losses and the prevention effort. The latter term refers to actions aimed at reducing the frequency of the claim arrivals. The problem consists in maximizing the expected exponential utility of terminal wealth, in presence of a terminal reimbursement. We show that this problem can be formulated in terms of a suitable backward stochastic differential equation (BSDE), for which we prove existence and uniqueness of the solution. We extend in several directions the results obtained by Bensalem, Santibanez and Kazi-Tani [Finance Stoch. 2023] and compare our results with those presented therein. The talk is based on a joint paper with M. Brachetta, G. Callegaro and C. Sgarra.