Context and Motivations

In a probabilistic model, a rare event is an event having a very small probability. Rare events are important in many areas. For instance, catastrophic failures in transporting systems or in nuclear power plants leading to human losses are represented by rare events. The failure of an information processing system in a bank, or in the communication network of a group of banks, leading to important money losses is also a rare event in the appropriate models. Some low level events in specific communication networks are also rare, or must be. A typical example is the loss of a small unit of information (a packet, or a cell in ATM networks). The loss of such an unit must be rare because the traffic intensities in these transportation systems are extremely high, and thus, if the loss probabilities are not small enough, important losses can occur in a small amount of time. Equivalently, the probability of ruin is an central issue for the overall wealth of a insurance company.

Being able to evaluate the probability of rare events is critical in many applications. Assessing that the probability of failure of control flight systems in aeronautics, for instance, is small enough, is necessary in the design of a new aircraft. The same typically happens in automatic transportation systems. Comparing the order of magnitude of the probability of rare events is needed when the designer must choose among different possible architectures for her future communication network. If the implemented fault tolerant mechanisms must provide very small probability of critical failure, it may happen that system A leads to a probability of failure of around 10^-8 while system B provides a failure probability of around 10^-9; evaluating accurately these numbers is crucial in these decision processes. Similar problems occur for managers of portfolios which have to maintain reserves against rare events such as large losses due to loan defaults; measuring those losses with enough precision and efficiently is thus of high importance.

Forecasting the behaviour of a system where the events of interest are rare is, in general, a formidable task, because different technical problems arise when trying to accurately evaluate very small probabilities. For instance, when using stochastic processes such a discrete state Markov chains to represent the system (either directly or through a high level model representation), numerical methods when applicable (the size of the state space must be moderate enough) can fail because of the stiffness of the model. From the representation power point of view, the most powerful techniques are simulation ones, and these suffer from a major drawback in rare event analysis: the standard approaches fail because of the rarity of the targets. If the event is rare and the system is simulated, the rarity makes the event difficult or impossible to observe, thus to analyze. This means that more sophisticate techniques must be used. The project concerns the development of such methods in the Monte Carlo area. The main idea is to design and evaluate various techniques coming from research work in different areas. This is possible because the participants of the project work in several important domains and follow specific approaches in their research work.

Scientific Objectives – Activities and Collaborations

The main goal is to put together people working on rare event simulation, but using different techniques, and with different application areas. A first step will be to confront/compare the methods. The project will then allow to come up with a set of new powerful techniques, that will be illustrated on applications in engineering and finance.