Stochastics for Time-Space Risk Models

STORM is now in its final phase and is constantly gathering results while we explore the new areas of sustainability and risk quantification of horizon risk. Time-space modeling is central to the analysis of many stochastic phenomena. As an illustration, let's think about renewable energy. The use of wind turbines, for example, introduces great uncertainty into the production side of electricity. This uncertainty comes from variation in weather conditions and the irregular behavior of the wind. This in turn has an impact on electricity prices, which also show peculiar behaviour. So we can say that coincidences in the wind conditions have a domino effect on the market's electricity prices, and effects beyond this as well. In fact, market price models are only a first step in the complex study that leads to decisions about strategies, financial portfolios and monetary risk assessments. On the electricity production side, we also have questions about optimal utilization of resources and optimal conversion of production types, for example from wind turbines to traditional gas production and back again. To tackle such problems, we need to study stochastic control, optimal stopping and designing risk measures for non-standard risk models. Our results include a new stochastic volatility model given by a stochastic differential equation with unbounded coefficients and noise carrying both memory and roughness. This process is squeezed within the limits, which is a decisive feature for pricing options and hedging derivatives in both finance and energy finance. With this, we have investigated numerical methods also with combinations of Malliavin methods and machine learning. STORM has also developed stochastic control for time-varying dynamics of the Volterra type within the search for Nash equilibria, at the same time we have proposed new applications of Markovian lifts to tackle control of Volterra dynamics with the dynamic principle. STORM can now celebrate the graduation of three new PhDs and three postdocs who are now permanent employees in academia in Norway and abroad.

Phenomena that evolve in time and spread in space are overall, sourced by nature, intrinsic in life, generated by human technology and construction. Whether generated by unknown past causes, present evaluations, or future unexpected events, randomness is a reality and it has to be properly dealt with. The overall goal in our research is to formulate and solve the key problems in stochastic modelling, analysis, and control of tempo-spatial random phenomena. To understand the significance of the STORM project we take illustrative examples from energy production and markets. In the present European energy forum the role of renewable resources in power generation is of the foremost importance. These resources introduce uncertainty from the production side as they depend on weather conditions: wind, cloud cover, precipitation, and temperatures. These environmental risk factors are clearly characterised by time-space dependences and show features that are well beyond the traditional structures: e.g. non-Gaussianity, non-Markovianity, non-semimartingality, fat-tails, jumps, and spikes. These influence the stream of production and electricity prices. Our research aims at providing a coherent framework to treat flexible classes of models able to accommodate the different features detected. Market price models are a first step towards the complex study of portfolio management and monetary risk evaluation. On the production side we have questions of optimal exploitation of resources and optimal switching of production lines. To solve these types of problems we shall study stochastic control, optimal stopping, and design of risk measures for space-time risk models that are non-standard e.g. delay and mean-field terms, irregular coefficients, asymmetric information. STORM takes up the challenge of studying these unconventional models believing that it is timely to introduce effective methods to provide solutions.