Contributions to risk management and mean-field type problems

Abstract

This thesis consists of four important topics about risk management and mean-field type problems.

For the first topic, we investigate the validity of diversification effect under extreme-value copulas, when the marginal risks of the portfolio are identically distributed, which can be any one having a finite endpoint or belonging to one of the three Maximum Domains of Attraction (MDA). We show that Value-at-Risk (V@R) under extreme-value copulas is asymptotically subadditive for marginal risks with finite mean, while asymptotically superadditive for risks with infinite mean.

For the second topic, we study the asymptotic sub/super-additivity of V@R under Archimedean copulas with regularly varying generator, where the marginal risks of the portfolio can be any one having a finite endpoint or belonging to one of the three MDAs. We show that V@R under Archimedean copulas is asymptotically subadditive for marginal risks with finite mean, while asymptotically superadditive if there is at least one risk with infinite mean. Our obtained results are universal and do not require the tail equivalence assumption as demanded in the existing literature.

For the third topic, we consider an energy system with $n$ consumers who are linked by a Demand Side Management (DSM) contract, i.e. they agreed to diminish, at random times, their aggregated power consumption by a predefined volume during a predefined duration. Their failure to deliver the service is penalized via the difference between the sum of the $n$ power consumptions and the contracted target. We are led to analyze a non-zero sum stochastic game with $n$ players, where the interaction takes place through a cost which involves a delay induced by the duration included in the DSM contract. When $n \to \infty$, we obtain a Mean-Field Game (MFG) with random jump time penalty and interaction on the control. We prove a stochastic maximum principle in this context, which allows to compare the MFG solution to the optimal strategy of a central planner. In a linear quadratic setting we obtain a semi-explicit solution through a system of decoupled forward-backward stochastic differential equations (BSDEs) with jumps, involving a Riccati BSDE with jumps. We show that it provides an approximate Nash equilibrium for the original $n$-player game for $n$ large. Finally, we propose a numerical algorithm to compute the MFG equilibrium and present several numerical experiments.

For the last topic, we study a class of mean-field reflected BSDEs with jumps, where the mean-field interaction in terms of the distribution of $Y$ (the first component of the solution) enters in both the driver and the lower obstacle. Under mild conditions on the coefficients, we prove the existence and uniqueness of such a class of equations. Furthermore, we also address the case of mean-field reflected BSDEs with the driver depending on the law of the whole solution and the obstacle independent of the solution.