In this paper, we study the risk-averse control problem for diffusion processes. We make use of a forward-backward system of stochastic differential equations to evaluate a fixed policy and to formulate the optimal control problem. Weak formulation is established to facilitate the derivation of the risk-averse dynamic programming equation. We prove that the value function of the risk-averse control problem is a viscosity solution of a risk-averse analog of the HamiltonJacobi-Bellman equation. On the other hand, a verification theorem is proved when the classical solution of the equation exists.
We propose a numerical recipe for risk evaluation defined by a backward stochastic differential equation. Using dual representation of the risk measure, we convert the risk evaluation to a simple stochastic control problem where the control is a certain Radon-Nikodym derivative process. By exploring the maximum principle, we show that a piecewise-constant dual control provides a good approximation on a short interval. A dynamic programming algorithm extends the approximation to a finite time horizon. Finally, we illustrate the application of the procedure to financial risk management in conjunction with nested simulation and on an multidimensional portfolio valuation problem.
We consider optimal control problems for diffusion processes, where the objective functional is defined by a time-consistent dynamic risk measure. We focus on coherent risk measures defined by g-evaluations. For such problems, we construct a family of time and space perturbed systems with piecewise-constant control functions. We obtain a regularized optimal value function by a special mollification procedure. This allows us to establish a bound on the difference between the optimal value functions of the original problem and of the problem with piecewise-constant controls.
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