Prof. Luciano Campi (London School of Economics)
Nonzero-sum stochastic differential games with impulse controls: a verification theorem with applications
We consider a general nonzero-sum impulse game with two players. The main mathematical contribution of the paper is a verification theorem which provides, under some regularity conditions, a suitable system of quasi-variational inequalities for the value functions and the optimal strategies of the two players. As an application, we study an impulse game with a one-dimensional state variable, following a real-valued scaled Brownian motion, and two players with linear and symmetric running payoffs. We fully characterize a Nash equilibrium and provide explicit expressions for the optimal strategies and the value functions. We also prove some asymptotic results with respect to the intervention costs. Finally, we consider two further non-symmetric examples where a Nash equilibrium is found numerically.
Identifying hierarchies and rankings of nodes in directed graphs is fundamental in many applications such as social network analysis, biology, economics, and finance. A recently proposed method identifies the hierarchy by finding the ordered partition of nodes which minimises a score function, termed agony. This function penalises the links violating the hierarchy in a way depending on the strength of the violation.
To investigate the resolution of ranking hierarchies we introduce an ensemble of random graphs, the Ranked Stochastic Block Model. We find that agony may fail to identify hierarchies when the structure is not strong enough and the size of the classes is small with respect to the whole network. We analytically characterise the resolution threshold and we show that an iterated version of agony can partly overcome this resolution limit.
This paper provides empirical evidences that corporate firms risk assessment could benefit from taking quantitatively into account the network of interactions among firms. Indeed, the structure of interactions between firms is critical to identify risk concentration and the possible pathways of propagation of financial distress. In this work, we consider the interactions by investigating a large proprietary dataset of payments between Italian firms. We first characterise the topological properties of the payment networks, and then we focus our attention on the relation between the networks and the risk of firms. Our main finding is to document the existence of an homophily of risk, i.e. the tendency of firms with similar risk profile to be statistically more connected among themselves. This effect is observed when considering both pairs of firms and communities or hierarchies identified in the network. We leverage this knowledge to demonstrate that network properties of a node can be used to predict the missing rating of a firm.