L4

In this talk, we consider the question of exact (boundary) controllability of wave equations: whether one can steer their solutions from any initial state to any final state using appropriate boundary data. In particular, we discuss new and fully general results for linear wave equations on time-dependent domains with moving boundaries. We also discuss the novel geometric Carleman estimates that are the main tools for proving these controllability results

Risk-indifference pricing is proposed as an alternative to utility indifference pricing, where a risk measure is used instead of a utility based preference. In this, we propose to include the possibility to change the attitude to risk evaluation as time progresses. This is particularly reasonable for long term investments and strategies. 

Then we introduce a fully-dynamic risk-indifference criteria, in which a whole family of risk measures is considered. The risk-indifference pricing system is studied from the point of view of its properties as a convex price system. We tackle questions of time-consistency in the risk evaluation and the corresponding prices. This analysis provides a new insight also to time-consistency for ordinary dynamic risk-measures.

Our techniques and results are set in the representation and extension theorems for convex operators. We shall argue and finally provide a setting in which fully-dynamic risk-indifference pricing is a well set convex price system.

The presentation is based on joint works with Jocelyne Bion-Nadal.

Welfare economics argues that competitive markets lead to efficient allocation of resources. The classical theorems are based on the Walrasian market model which assumes the existence of market clearing prices. The emergence of such prices remains debatable. We replace the Walrasian market model by double auctions and show that the conclusions of welfare economics remain largely the same. Double auctions are not only a more realistic description of real markets but they explain how equilibrium prices and efficient allocations emerge in practice. 

In this talk, I will present an incomplete equilibrium model to determine the price of an annuity.  A finite number of agents receive stochastic income streams and choose between consumption and investment in the traded annuity.  The novelty of this model is its ability to handle running consumption and general income streams.  In particular, the model incorporates mean reverting income, which is empirically relevant but historically too intractable in equilibrium.  The model is set in a Brownian framework, and equilibrium is characterized and proven to exist using a system of fully coupled quadratic BSDEs.  This work is joint with Gordan Zitkovic.

In my talk I will briefly introduce model-free approach to mathematical finance, which uses Vovk's outer measure. Then, using pathwise BDG inequality obtained by Beigbloeck and Siorpaes and modification of Vovk's measure, I will present and prove a model-free version of this inequality for continuous price paths. Finally, I will discuss possible applications, like the existence and uniqueness of solutions of SDEs driven by continuous, model-free price paths. The talk will be based on the joint work with Farai Mhlanga and Lesiba Galane (University of Limpopo, South Africa)

We study a family of optimal control problems under a set of controlled-loss constraints holding at different deterministic dates. The characterization of the associated value function by a Hamilton-Jacobi-Bellman equation usually calls for additional strong assumptions on the dynamics of the processes involved and the set of constraints. To treat this problem in absence of those assumptions, we first convert it into a state-constrained stochastic target problem and then apply a level-set approach to describe the reachable set. With this approach, the state constraints can be managed through an exact penalization technique. However, a new set of state and control variables enters the definition of this stochastic target problem. In particular, those controls are unbounded. A “compactification” of the problem is then performed. (joint work with Athena Picarelli)