Our experience with research in mathematics is that any new problem that we study, typically leads to several new and challenging problems. This was the case during our year at CAS as well. However, answers to these new problems shed light on the original ones in unexpected ways. For example, we developed some new mathematical models for cross-commodity markets using rather sophisticated random fields. The models came up as generalizations of the ones proposed in the research plan. These models open new questions concerning estimation and simulation, in order to be applicable in concrete market situations. We are following up these aspects in current research collaborations.

Another example comes from the need of pricing contracts in clearly incomplete markets where risks also come from exogenous environmental sources. We have adopted the so-called risk-indifference approach and we have developed techniques of stochastic analysis mixed with convex analysis to analyse the pricing systems. This led to new mathematical results that are now followed up into the application of financial pricing. Indeed this was in the original project description, but the research followed an unexpected line yielding to additional results.

A a new class of stochastic volatility models for commodity and energy futures markets was developed during our year at CAS. These models allow for a more flexible and realistic description of the stochasticity along the various futures contracts on the term structure curve. Our approach calls for new developments in the direction of pricing and hedging derivatives. Mathematical theory for pricing and hedging derivatives on futures contracts in commodity markets was also developed. We also introduced a new cross-commodity model, significantly going beyond the proposed models in the research plan. To fit such random field models to actual markets, we adopted the so-called kriging method to commodity markets. In a rather simplified framework, it shows how the econometric concept of co-integration works in commodity markets. Co-integration is a theoretical framework for linking the stochastic dynamics of two or more assets. We demonstrate how co-integrated commodity spot markets impacts the futures markets, in particular, leading to a specific correlation term structure which has huge consequences on spread option pricing compared with the conventional co-integration theory.

We also took up the investigation of the model and random fields presented in the research project. One of the results obtained at present is an extension of the Malliavin calculus techniques beyond the classical setting of square integrable random variables. This is suitable for the application to models that have so-called large-tails, that is, a comparatively large probability of happening of extreme events.

We have also worked on the management of risk developing a stochastic control for Volterra type equations, which are indeed models of the type presented in the research project. In doing this, we used techniques of Malliavin calculus. The studies will continue, to include application to large systems and network modelled by mean-field.

The financial means and administrative support of CAS made it possible to attract some of the world-leading researchers to Oslo to work with us on mathematical challenges in stochastic analysis, finance, energy and weather. To collaborate with excellent scientists sharpens the mind, as well as bringing up new and original ideas and solutions. A targeted group of international collaborators, working as one group in Oslo, enabled us to make significant breakthroughs on the problems suggested in the research plan. In addition, we have a bundle of new challenges that we will continue studying in our collaborations.