Section: New Results

Backward stochastic (partial) differential equations with jumps, optimal stopping and stochastic control with nonlinear expectation, risk minimization

Participants : Roxana Dumitrescu, Marie-Claire Quenez [(Univ Paris 7)] , Arnaud Lionnet, Agnès Sulem.

R. Dumitrescu, M.C. Quenez and A. Sulem have provided a weak dynamic principle for Combined Optimal Stopping/Stochastic Control with ${ℰ}^f-$conditional Expectation . They have investigated the links between generalized Dynkin games and double barriers reflected BSDEs with jumps and also studied mixed generalized Dynkin games in a Markovian framework and associated nonlinear HJB equations with barriers .

In the recent paper [43] , they study game options in an imperfect market with default. They extend the results obtained by Kifer [68] in a perfect market model to the case of imperfections taken into account via the nonlinearity of the wealth dynamics. In this framework, the pricing system is expressed as a nonlinear $g$-expectation/evaluation induced by a nonlinear BSDE with jump. They prove that the superhedging price of a game option coincides with the value function of a corresponding generalized Dynkin game expressed in terms of the $g$-evaluation. They also address the case of ambiguity on the model, - for example an ambiguity on the default probability -, and characterize the superhedging price of a game option as the value function of a mixed generalized Dynkin game. They prove the existence of a cancellation time and a trading strategy which allows the seller to be super-hedged, whatever the model is. This study is introduced by the analysis of the simpler case of American options.

In collaboration with Jane Bielagt (Humbold Univ.) and Gona̧lo Dos Reis (Univ. of Edimburgh), Arnaud Lionnet investigates in the effects of the social interactions of a finite set of agents on an equilibrium pricing mechanism. They consider an incomplete market where agents invest so as to minimize their risk measure. Here, agents assess risk using convex dynamic risk measures expressed by Backward Stochastic Differential Equations (BSDE). Beside the risk associated with their own economic activity, the agents compare their trading gains to that of the others, and factor this relative performance in the evaluation of their risk/satisfaction. When a derivative product is introduced to complete the market and allow agents to trade a non-financial risk factor (such as temperature), the risk of each agent is lowered, as expected. However, agents then find it in their interest to be more concerned with their relative performance. This leads them to behave more like a herd and this destabilizes the previously stable, purely financial market.