Worst case model risk management

We are interested in model risk control problems. We study a strategy for the trader which, in a sense, guarantees good performances whatever is the unknown model for the assets of his/her portfolio. The trader chooses trading strategies to decrease the risk and therefore acts as a minimizer; the market systematically acts against the interest of the trader, so that we consider it acts as a maximizer. Thus we consider the model risk control problem as a two players (Trader versus Market) zero-sum stochastic differential game problem. Therefore our construction corresponds to a ‘worst case’ worry and, in this sense, can be viewed as a continuous-time extension of discrete-time strategies based upon prescriptions issued from VaR analyses at the beginning of each period. In addition, the initial value of the optimal portfolio can be seen as the minimal amount of money which is needed to face the worst possible damage. We give a proper mathematical statement for such a game problem. We prove that the value function of this game problem is the unique viscosity solution to an Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation, and satisfies the Dynamic Programming Principle.