MATH-SPA

This project concerns theoretical questions that arise in the probabilistic and PDE modelling of particle systems of "singular" interactions, motivated by applications in the fields of finance and biology/chemistry. Analogously to the spread of diseases, financial contagion attracts abundant attention and systemic risk in banking systems has emerged as a major problem for financial stability that yields significant difficulties both in terms of modelling and in terms of the effective exploitation of big data on balance sheets. In general mathematical language, we consider a large particle system ("particle" refers to "bank", "individual", "cell", etc.) with cross-interactions and the contagion that causes absorption ("absorption" refers to "default", "death", "disappearance", etc.) of a large number of particles always manifests itself through a multitude of different channels, with spillover or domino effects from one to another. In this project, we distinguish two mechanisms of interactions (denoted by (A) and (B)) between the particles : (A) Absorptions change surviving particles' status but have no impact on their evolution dynamics; (B) Absorptions do not change status but affect their dynamics. We focus on the following questions : 1. Computation of the survival probability (of representative particles) is of great importance to measure the system's state. Further, we are interested in anticipating the time of fragility, called systemic failure, at which the system transitions from a well-behaved regime (during which the system evolves softly and deaths are small or do not appear at all) to a systemic crisis regime (which occurs rarely and is characterised by lots of particles absorbed in a short period of time). It is crucial to detect the systemic failure. 2. This question is intimately linked to financial applications so we let "particle=bank" and "absorption=default" here for the sake of presentation. To damp out the systemic risk, two strategies are taken into account. 2.a. Regulators grant a budget that is divided and allocated to the banks. What is the optimal allocation strategy? Can the risk be significantly reduced? 2.b. The banks are allowed to purchase and hold the shares delivered by others. Does there exist a Nash equilibrium (where every bank adopts the same holding strategy)? If so, does the risk diminish? 3. This question arises in biology/chemistry (thanks to the discussions with biologists of MMB chair in which our team member Tomasevic is involved). We consider the phenomenon of chemotactic movement of cells. In this framework, cells emit a chemical product that diffuses and attracts cells, so that cells tend to concentrate around regions with high density of chemo-attractant. A typical model is the Keller-Segel one, where particles interact through a singular (attractive) drift that may lead to particle collisions. Hence, system can not only die of extinction of particles but also die of particle collisions, where the two dangers are of opposite traits. Setting an absorbing barrier makes the system more sparse and reduces the density of particles. Can this lengthens the system's lifespan? 4. Can we extend the analysis from perfect information to partial information (where cross-interactions depend not only on the "observable average" but also on some common noise)? Can we generalise the above results from homogeneous system to heterogeneous system? 5. Does the propagation of chaos (particle systems converge to a mean-field limit) hold? If so, how to estimate the convergence rate?